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<big><b>API Documentation for Armadillo 7.600</b></big>
<br>
<br>
<br>


<b>Preamble</b>
<br>
<br>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="text-align: left; vertical-align: top; width: 45%;">
<ul>
<li>
For converting Matlab/Octave programs,
see the <a href="#syntax">syntax conversion table</a>
</li>
<br>
<li>
First time users: please see the short <a href="#example_prog">example program</a>
</li>
<br>
<li>
If you discover any bugs or regressions, please <a href="http://arma.sourceforge.net/support.html">report them</a>
</li>
<br>
<li>
History of <a href="#api_additions">API additions</a>
</li>
</ul>
</td>
<td>
&nbsp;
</td>
<td class="line" style="vertical-align: top;">
<br>
</td>
<td style="text-align: left; vertical-align: top; width: 50%;">
<ul>
<li>
Please cite the following article if you use Armadillo in your research or software.
Citations are useful for the continued development and maintenance of the library.
<br>
<br>
<font size=-1>
Conrad Sanderson and Ryan Curtin.
<br><i><a href="armadillo_joss_2016.pdf">Armadillo: a template-based C++ library for linear algebra</a></i>.
<br>Journal of Open Source Software, Vol.&nbsp;1, pp.&nbsp;26, 2016.
</font>
</li>
</ul>
</td>
</tr>
</tbody>
</table>

<br>
<br>

<b>Overview</b>
<ul>
<li><a href="#part_classes">matrix, vector, cube and field classes</a></li>
<li><a href="#part_membfns">member functions &amp; variables</a></li>
<br>
<li><a href="#part_gen">generated vectors&nbsp;/&nbsp;matrices&nbsp;/&nbsp;cubes</a></li>
<li><a href="#part_fns">functions of vectors&nbsp;/&nbsp;matrices&nbsp;/&nbsp;cubes</a></li>
<br>
<li><a href="#part_decompdense">decompositions, factorisations, inverses and equation solvers (dense matrices)</a></li>
<li><a href="#part_decompsparse">decompositions, factorisations, and equation solvers (sparse matrices)</a></li>
<br>
<li><a href="#part_sigproc">signal &amp; image processing</a></li>
<li><a href="#part_stats">statistics and clustering</a></li>
<li><a href="#part_misc">miscellaneous (constants, configuration)</a></li>
</ul>
<br>

<a name="part_classes"></a>
<b>Matrix, Vector, Cube and Field Classes</b>
<ul>
<table>
<tbody>
<tr><td><a href="#Mat">Mat&lt;<i>type</i>&gt;, mat, cx_mat</a></td><td>&nbsp;</td><td>dense matrix class</td></tr>
<tr><td><a href="#Col">Col&lt;<i>type</i>&gt;, colvec, vec</a></td><td>&nbsp;</td><td>dense column vector class</td></tr>
<tr><td><a href="#Row">Row&lt;<i>type</i>&gt;, rowvec</a></td><td>&nbsp;</td><td>dense row vector class</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
<tr><td><a href="#Cube">Cube&lt;<i>type</i>&gt;, cube, cx_cube</a></td><td>&nbsp;</td><td>dense cube class ("3D matrix")</td></tr>
<tr><td><a href="#field">field&lt;<i>object&nbsp;type</i>&gt;</a></td><td>&nbsp;</td><td>class for storing arbitrary objects in matrix-like or cube-like layouts</td></tr>
<tr><td><a href="#SpMat">SpMat&lt;<i>type</i>&gt;, sp_mat, sp_cx_mat</a></td><td>&nbsp;</td><td>sparse matrix class</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
<tr><td><a href="#operators">operators</a></td><td>&nbsp;</td><td><code><big>+</big>&nbsp; <big>-</big>&nbsp; <big>*</big>&nbsp; /&nbsp; %&nbsp; ==&nbsp; !=&nbsp; &lt;=&nbsp; &gt;=&nbsp; &lt;&nbsp; &gt;</code></td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_membfns"></a>
<b>Member Functions &amp; Variables</b>
<ul>
<table>
<tbody>
<tr><td><a href="#attributes">attributes</a></td><td>&nbsp;</td><td>.n_rows, .n_cols, .n_elem, .n_slices, ...</td></tr>
<tr><td><a href="#element_access">element&nbsp;access</a></td><td>&nbsp;</td><td>element/object access via (), [] and .at()</td></tr>
<tr><td><a href="#element_initialisation">element&nbsp;initialisation</a></td><td>&nbsp;</td><td>set elements via &lt;&lt; operator or initialiser list</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#zeros_member">.zeros</a></td><td>&nbsp;</td><td>set all elements to zero</td></tr>
<tr><td><a href="#ones_member">.ones</a></td><td>&nbsp;</td><td>set all elements to one</td></tr>
<tr><td><a href="#eye_member">.eye</a></td><td>&nbsp;</td><td>set elements along main diagonal to one and off-diagonal elements to zero</td></tr>
<tr><td><a href="#randu_randn_member">.randu&nbsp;/&nbsp;.randn</a></td><td>&nbsp;</td><td>set all elements to random values</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#fill">.fill</a></td><td>&nbsp;</td><td>set all elements to specified value</td></tr>
<tr><td><a href="#imbue">.imbue</a></td><td>&nbsp;</td><td>imbue (fill) with values provided by functor or lambda function</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#replace">.replace</a></td><td>&nbsp;</td><td>replace specific elements with a new value</td></tr>
<tr><td><a href="#transform">.transform</a></td><td>&nbsp;</td><td>transform each element via functor or lambda function</td></tr>
<tr><td><a href="#for_each">.for_each</a></td><td>&nbsp;</td><td>apply a functor or lambda function to each element</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#set_size">.set_size</a></td><td>&nbsp;</td><td>change size without keeping elements (fast)</td></tr>
<tr><td><a href="#reshape_member">.reshape</a></td><td>&nbsp;</td><td>change size while keeping elements</td></tr>
<tr><td><a href="#resize_member">.resize</a></td><td>&nbsp;</td><td>change size while keeping elements and preserving layout</td></tr>
<tr><td><a href="#copy_size">.copy_size</a></td><td>&nbsp;</td><td>change size to be same as given object</td></tr>
<tr><td><a href="#reset">.reset</a></td><td>&nbsp;</td><td>change size to empty</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#submat">submatrix&nbsp;views</a></td><td>&nbsp;</td><td>read/write access to contiguous and non-contiguous submatrices</td></tr>
<tr><td><a href="#subcube">subcube&nbsp;views</a></td><td>&nbsp;</td><td>read/write access to contiguous and non-contiguous subcubes</td></tr>
<tr><td><a href="#subfield">subfield&nbsp;views</a></td><td>&nbsp;</td><td>read/write access to contiguous subfields</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#diag">.diag</a></td><td>&nbsp;</td><td>read/write access to matrix diagonals</td></tr>
<tr><td><a href="#each_colrow">.each_col&nbsp;/&nbsp;.each_row</a></td><td>&nbsp;</td><td>repeated operations on each column or row of matrix</td></tr>
<tr><td><a href="#each_slice">.each_slice</a></td><td>&nbsp;</td><td>repeated operations on each slice of cube</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#set_imag">.set_imag&nbsp;/&nbsp;.set_real</a></td><td>&nbsp;</td><td>set imaginary/real part</td></tr>
<tr><td><a href="#insert">.insert_rows/cols/slices</a></td><td>&nbsp;</td><td>insert vector/matrix/cube at specified row/column/slice</td></tr>
<tr><td><a href="#shed">.shed_rows/cols/slices</a></td><td>&nbsp;</td><td>remove specified rows/columns/slices</td></tr>
<tr><td><a href="#swap_rows">.swap_rows/cols</a></td><td>&nbsp;</td><td>swap specified rows or columns</td></tr>
<tr><td><a href="#swap">.swap</a></td><td>&nbsp;</td><td>swap contents with given object</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#memptr">.memptr</a></td><td>&nbsp;</td><td>raw pointer to memory</td></tr>
<tr><td><a href="#colptr">.colptr</a></td><td>&nbsp;</td><td>raw pointer to memory used by specified column</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#iterators_mat">iterators&nbsp;(matrices)</a></td><td>&nbsp;</td><td>STL-style iterators and associated member functions for matrices and vectors</td></tr>
<tr><td><a href="#iterators_cube">iterators&nbsp;(cubes)</a></td><td>&nbsp;</td><td>STL-style iterators and associated member functions for cubes</td></tr>
<tr><td><a href="#stl_container_fns">STL&nbsp;container&nbsp;functions</a></td><td>&nbsp;</td><td>STL-style container functions</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#t_st_members">.t&nbsp;/&nbsp;.st&nbsp;</a></td><td>&nbsp;</td><td>return matrix transpose</td></tr>
<tr><td><a href="#i_member">.i&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</a></td><td>&nbsp;</td><td>return inverse of square matrix</td></tr>
<tr><td><a href="#min_and_max_member">.min&nbsp;/&nbsp;.max</a></td><td>&nbsp;</td><td>return extremum value</td></tr>
<tr><td><a href="#index_min_and_index_max_member">.index_min&nbsp;/&nbsp;.index_max</a></td><td>&nbsp;</td><td>return index of extremum value</td></tr>
<tr><td><a href="#eval_member">.eval</a></td><td>&nbsp;</td><td>force evaluation of delayed expression</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#in_range">.in_range</a></td><td>&nbsp;</td><td>check whether given location or span is valid</td></tr>
<tr><td><a href="#is_empty">.is_empty</a></td><td>&nbsp;</td><td>check whether object is empty</td></tr>
<tr><td><a href="#is_square">.is_square</a></td><td>&nbsp;</td><td>check whether matrix is square sized</td></tr>
<tr><td><a href="#is_vec">.is_vec</a></td><td>&nbsp;</td><td>check whether matrix is a vector</td></tr>
<tr><td><a href="#is_sorted">.is_sorted</a></td><td>&nbsp;</td><td>check whether vector or matrix is sorted</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#is_finite_member">.is_finite</a></td><td>&nbsp;</td><td>check whether all elements are finite</td></tr>
<tr><td><a href="#has_inf">.has_inf</a></td><td>&nbsp;</td><td>check whether any element is +-Inf</td></tr>
<tr><td><a href="#has_nan">.has_nan</a></td><td>&nbsp;</td><td>check whether any element is NaN</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#print">.print</a></td><td>&nbsp;</td><td>print object to <i>std::cout</i> or user specified stream</td></tr>
<tr><td><a href="#raw_print">.raw_print</a></td><td>&nbsp;</td><td>print object without formatting</td></tr>
<tr><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td><td><small><small>&nbsp;</small></small></td></tr>
<tr><td><a href="#save_load_mat">.save/.load&nbsp;(matrices&nbsp;&amp;&nbsp;cubes)</a></td><td>&nbsp;</td><td>save/load matrices and cubes in files or streams</td></tr>
<tr><td><a href="#save_load_field">.save/.load&nbsp;(fields)</a></td><td>&nbsp;</td><td>save/load fields in files or streams</td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_gen"></a>
<b>Generated Vectors/Matrices/Cubes</b>
<ul>
<table>
<tbody>
<tr style="background-color: #F5F5F5;"><td><a href="#eye_standalone">eye</a></td><td>&nbsp;</td><td>generate identity matrix</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#linspace">linspace</a></td><td>&nbsp;</td><td>generate vector with linearly spaced elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#logspace">logspace</a></td><td>&nbsp;</td><td>generate vector with logarithmically spaced elements</td></tr>
<tr><td><a href="#ones_standalone">ones</a></td><td>&nbsp;</td><td>generate object filled with ones</td></tr>
<tr><td><a href="#randi">randi</a></td><td>&nbsp;</td><td>generate object with random integer values in specified interval</td></tr>
<tr><td><a href="#randu_randn_standalone">randu&nbsp;/&nbsp;randn</a></td><td>&nbsp;</td><td>generate object with random values (uniform and normal distributions)</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#randg">randg</a></td><td>&nbsp;</td><td>generate object with random values (gamma distribution)</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#regspace">regspace</a></td><td>&nbsp;</td><td>generate vector with regularly spaced elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#speye">speye</a></td><td>&nbsp;</td><td>generate sparse identity matrix</td></tr>
<tr><td><a href="#spones">spones</a></td><td>&nbsp;</td><td>generate sparse matrix with non-zero elements set to one</td></tr>
<tr><td><a href="#sprandu_sprandn">sprandu&nbsp;/&nbsp;sprandn</a></td><td>&nbsp;</td><td>generate sparse matrix with non-zero elements set to random values</td></tr>
<tr><td><a href="#toeplitz">toeplitz</a></td><td>&nbsp;</td><td>generate Toeplitz matrix</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#zeros_standalone">zeros</a></td><td>&nbsp;</td><td>generate object filled with zeros</td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_fns"></a>
<b>Functions of Vectors/Matrices/Cubes</b>
<ul>
<table>
<tbody>
<tr style="background-color: #F5F5F5;"><td><a href="#abs">abs</a></td><td>&nbsp;</td><td>obtain magnitude of each element</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#accu">accu</a></td><td>&nbsp;</td><td>accumulate (sum) all elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#all">all</a></td><td>&nbsp;</td><td>check whether all elements are non-zero, or satisfy a relational condition</td></tr>
<tr><td><a href="#any">any</a></td><td>&nbsp;</td><td>check whether any element is non-zero, or satisfies a relational condition</td></tr>
<tr><td><a href="#approx_equal">approx_equal</a></td><td>&nbsp;</td><td>approximate equality</td></tr>
<tr><td><a href="#arg">arg</a></td><td>&nbsp;</td><td>phase angle of each element</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#as_scalar">as_scalar</a></td><td>&nbsp;</td><td>convert 1x1 matrix to pure scalar</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#clamp">clamp</a></td><td>&nbsp;</td><td>obtain clamped elements according to given limits</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#cond">cond</a></td><td>&nbsp;</td><td>condition number of matrix</td></tr>
<tr><td><a href="#conj">conj</a></td><td>&nbsp;</td><td>obtain complex conjugate of each element</td></tr>
<tr><td><a href="#conv_to">conv_to</a></td><td>&nbsp;</td><td>convert between matrix types</td></tr>
<tr><td><a href="#cross">cross</a></td><td>&nbsp;</td><td>cross product</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#cumsum">cumsum</a></td><td>&nbsp;</td><td>cumulative sum</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#cumprod">cumprod</a></td><td>&nbsp;</td><td>cumulative product</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#det">det / log_det</a></td><td>&nbsp;</td><td>determinant</td></tr>
<tr><td><a href="#diagmat">diagmat</a></td><td>&nbsp;</td><td>generate diagonal matrix from given matrix or vector</td></tr>
<tr><td><a href="#diagvec">diagvec</a></td><td>&nbsp;</td><td>extract specified diagonal</td></tr>
<tr><td><a href="#diff">diff</a></td><td>&nbsp;</td><td>differences between adjacent elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#dot">dot/cdot/norm_dot</a></td><td>&nbsp;</td><td>dot product</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#eps">eps</a></td><td>&nbsp;</td><td>obtain distance of each element to next largest floating point representation</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#expmat">expmat</a></td><td>&nbsp;</td><td>matrix exponential</td></tr>
<tr><td><a href="#expmat_sym">expmat_sym</a></td><td>&nbsp;</td><td>matrix exponential (symmetric)</td></tr>
<tr><td><a href="#find">find</a></td><td>&nbsp;</td><td>find indices of non-zero elements, or elements satisfying a relational condition</td></tr>
<tr><td><a href="#find_finite">find_finite</a></td><td>&nbsp;</td><td>find indices of finite elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#find_nonfinite">find_nonfinite</a></td><td>&nbsp;</td><td>find indices of non-finite elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#find_unique">find_unique</a></td><td>&nbsp;</td><td>find indices of unique elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#flip">fliplr&nbsp;/&nbsp;flipud</a></td><td>&nbsp;</td><td>reverse order of columns or rows</td></tr>
<tr><td><a href="#imag_real">imag / real</a></td><td>&nbsp;</td><td>extract imaginary/real part</td></tr>
<tr><td><a href="#ind2sub">ind2sub</a></td><td>&nbsp;</td><td>convert linear index to subscripts</td></tr>
<tr><td><a href="#index_min_and_index_max_standalone">index_min / index_max</a></td><td>&nbsp;</td><td>indices of extremum values</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#inplace_trans">inplace_trans</a></td><td>&nbsp;</td><td>in-place transpose</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#is_finite">is_finite</a></td><td>&nbsp;</td><td>check whether all elements are finite</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#join">join_rows / join_cols</a></td><td>&nbsp;</td><td>concatenation of matrices</td></tr>
<tr><td><a href="#join_slices">join_slices</a></td><td>&nbsp;</td><td>concatenation of cubes</td></tr>
<tr><td><a href="#kron">kron</a></td><td>&nbsp;</td><td>Kronecker tensor product</td></tr>
<tr><td><a href="#logmat">logmat</a></td><td>&nbsp;</td><td>matrix logarithm</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#logmat_sympd">logmat_sympd</a></td><td>&nbsp;</td><td>matrix logarithm (symmetric)</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#min_and_max">min&nbsp;/&nbsp;max</a></td><td>&nbsp;</td><td>return extremum values</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#nonzeros">nonzeros</a></td><td>&nbsp;</td><td>return non-zero values</td></tr>
<tr><td><a href="#norm">norm</a></td><td>&nbsp;</td><td>various norms of vectors and matrices</td></tr>
<tr><td><a href="#normalise">normalise</a></td><td>&nbsp;</td><td>normalise vectors to unit <i>p</i>-norm</td></tr>
<tr><td><a href="#prod">prod</a></td><td>&nbsp;</td><td>product of elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#rank">rank</a></td><td>&nbsp;</td><td>rank of matrix</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#rcond">rcond</a></td><td>&nbsp;</td><td>reciprocal of condition number</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#repmat">repmat</a></td><td>&nbsp;</td><td>replicate matrix in block-like fashion</td></tr>
<tr><td><a href="#reshape">reshape</a></td><td>&nbsp;</td><td>change size while keeping elements</td></tr>
<tr><td><a href="#resize">resize</a></td><td>&nbsp;</td><td>change size while keeping elements and preserving layout</td></tr>
<tr><td><a href="#shift">shift</a></td><td>&nbsp;</td><td>shift elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#shuffle">shuffle</a></td><td>&nbsp;</td><td>randomly shuffle elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#size">size</a></td><td>&nbsp;</td><td>obtain dimensions of given object</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#sort">sort</a></td><td>&nbsp;</td><td>sort elements</td></tr>
<tr><td><a href="#sort_index">sort_index</a></td><td>&nbsp;</td><td>vector describing sorted order of elements</td></tr>
<tr><td><a href="#sqrtmat">sqrtmat</a></td><td>&nbsp;</td><td>square root of matrix</td></tr>
<tr><td><a href="#sqrtmat_sympd">sqrtmat_sympd</a></td><td>&nbsp;</td><td>square root of matrix (symmetric)</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#sum">sum</a></td><td>&nbsp;</td><td>sum of elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#sub2ind">sub2ind</a></td><td>&nbsp;</td><td>convert subscripts to linear index</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#symmat">symmatu&nbsp;/&nbsp;symmatl</a></td><td>&nbsp;</td><td>generate symmetric matrix from given matrix</td></tr>
<tr><td><a href="#trace">trace</a></td><td>&nbsp;</td><td>sum of diagonal elements</td></tr>
<tr><td><a href="#trans">trans</a></td><td>&nbsp;</td><td>transpose of matrix</td></tr>
<tr><td><a href="#trapz">trapz</a></td><td>&nbsp;</td><td>trapezoidal numerical integration</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#trimat">trimatu&nbsp;/&nbsp;trimatl</a></td><td>&nbsp;</td><td>generate triangular matrix from given matrix</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#unique">unique</a></td><td>&nbsp;</td><td>return unique elements</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#vectorise">vectorise</a></td><td>&nbsp;</td><td>convert matrix to vector</td></tr>
<tr><td><a href="#misc_fns">misc&nbsp;functions</a></td><td>&nbsp;</td><td>miscellaneous element-wise functions:&nbsp;exp,&nbsp;log,&nbsp;pow,&nbsp;sqrt,&nbsp;round,&nbsp;sign,&nbsp;...</td></tr>
<tr><td><a href="#trig_fns">trig&nbsp;functions</a></td><td>&nbsp;</td><td>trigonometric element-wise functions:&nbsp;cos,&nbsp;sin,&nbsp;...</td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_decompdense"></a>
<b>Decompositions, Factorisations, Inverses and Equation Solvers (Dense Matrices)</b>
<ul>
<table>
<tbody>
<tr style="background-color: #F5F5F5;"><td><a href="#chol">chol</a></td><td>&nbsp;</td><td>Cholesky decomposition</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#eig_sym">eig_sym</a></td><td>&nbsp;</td><td>eigen decomposition of dense symmetric/hermitian matrix</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#eig_gen">eig_gen</a></td><td>&nbsp;</td><td>eigen decomposition of dense general square matrix</td></tr>
<tr><td><a href="#eig_pair">eig_pair</a></td><td>&nbsp;</td><td>eigen decomposition for pair of general dense square matrices</td></tr>
<tr><td><a href="#inv">inv</a></td><td>&nbsp;</td><td>inverse of general square matrix</td></tr>
<tr><td><a href="#inv_sympd">inv_sympd</a></td><td>&nbsp;</td><td>inverse of symmetric positive definite matrix</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#lu">lu&nbsp;&nbsp;</a></td><td>&nbsp;</td><td>lower-upper decomposition</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#null">null</a></td><td>&nbsp;</td><td>orthonormal basis of null space</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#orth">orth</a></td><td>&nbsp;</td><td>orthonormal basis of range space</td></tr>
<tr><td><a href="#pinv">pinv</a></td><td>&nbsp;</td><td>pseudo-inverse</td></tr>
<tr><td><a href="#qr">qr&nbsp;&nbsp;</a></td><td>&nbsp;</td><td>QR decomposition</td></tr>
<tr><td><a href="#qr_econ">qr_econ</a></td><td>&nbsp;</td><td>economical QR decomposition</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#qz">qz&nbsp;&nbsp;</a></td><td>&nbsp;</td><td>generalised Schur decomposition</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#schur">schur</a></td><td>&nbsp;</td><td>Schur decomposition</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#solve">solve</a></td><td>&nbsp;</td><td>solve systems of linear equations</td></tr>
<tr><td><a href="#svd">svd</a></td><td>&nbsp;</td><td>singular value decomposition</td></tr>
<tr><td><a href="#svd_econ">svd_econ</a></td><td>&nbsp;</td><td>economical singular value decomposition</td></tr>
<tr><td><a href="#syl">syl</a></td><td>&nbsp;</td><td>Sylvester equation solver</td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_decompsparse"></a>
<b>Decompositions, Factorisations and Equation Solvers (Sparse Matrices)</b>
<ul>
<table>
<tbody>
<tr style="background-color: #F5F5F5;"><td><a href="#eigs_sym">eigs_sym</a></td><td>&nbsp;</td><td>limited number of eigenvalues &amp; eigenvectors of sparse symmetric real matrix</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#eigs_gen">eigs_gen</a></td><td>&nbsp;</td><td>limited number of eigenvalues &amp; eigenvectors of sparse general square matrix</td></tr>
<tr><td><a href="#spsolve">spsolve</a></td><td>&nbsp;</td><td>solve sparse systems of linear equations</td></tr>
<tr><td><a href="#svds">svds</a></td><td>&nbsp;</td><td>limited number of singular values &amp; singular vectors of sparse matrix</td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_sigproc"></a>
<b>Signal &amp; Image Processing</b>
<ul>
<table>
<tbody>
<tr style="background-color: #F5F5F5;"><td><a href="#conv">conv</a></td><td>&nbsp;</td><td>1D convolution</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#conv2">conv2</a></td><td>&nbsp;</td><td>2D convolution</td></tr>
<tr><td><a href="#fft">fft&nbsp;/&nbsp;ifft</a></td><td>&nbsp;</td><td>1D fast Fourier transform and its inverse</td></tr>
<tr><td><a href="#fft2">fft2&nbsp;/&nbsp;ifft2</a></td><td>&nbsp;</td><td>2D fast Fourier transform and its inverse</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#interp1">interp1</a></td><td>&nbsp;</td><td>1D interpolation</td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_stats"></a>
<b>Statistics &amp; Clustering</b>
<ul>
<table>
<tbody>
<tr style="background-color: #F5F5F5;"><td><a href="#stats_fns">stats&nbsp;functions</a></td><td>&nbsp;</td><td>mean, median, standard deviation, variance</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#cov">cov</a></td><td>&nbsp;</td><td>covariance</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#cor">cor</a></td><td>&nbsp;</td><td>correlation</td></tr>
<tr><td><a href="#hist">hist</a></td><td>&nbsp;</td><td>histogram of counts</td></tr>
<tr><td><a href="#histc">histc</a></td><td>&nbsp;</td><td>histogram of counts with user specified edges</td></tr>
<tr><td><a href="#princomp">princomp</a></td><td>&nbsp;</td><td>principal component analysis</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#running_stat">running_stat</a></td><td>&nbsp;</td><td>running statistics of one dimensional process/signal</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#running_stat_vec">running_stat_vec</a></td><td>&nbsp;</td><td>running statistics of multi-dimensional process/signal</td></tr>
<tr><td><a href="#kmeans">kmeans</a></td><td>&nbsp;</td><td>cluster data into disjoint sets</td></tr>
<tr><td><a href="#gmm_diag">gmm_diag</a></td><td>&nbsp;</td><td>model data as a Gaussian Mixture Model (GMM)</td></tr>
</tbody>
</table>
</ul>
<br>

<a name="part_misc"></a>
<b>Miscellaneous</b>
<ul>
<table>
<tbody>
<tr style="background-color: #F5F5F5;"><td><a href="#constants">constants</a></td><td>&nbsp;</td><td>pi, inf, NaN, speed&nbsp;of&nbsp;light,&nbsp;...</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#wall_clock">wall_clock</a></td><td>&nbsp;</td><td>timer for measuring number of elapsed seconds</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#logging">logging&nbsp;of&nbsp;errors/warnings</a></td><td>&nbsp;</td><td>how to change the streams for displaying warnings and errors</td></tr>
<tr><td><a href="#uword">uword&nbsp;/&nbsp;sword</a></td><td>&nbsp;</td><td>shorthand for unsigned and signed integers</td></tr>
<tr><td><a href="#cx_double">cx_double&nbsp;/&nbsp;cx_float</a></td><td>&nbsp;</td><td>shorthand for std::complex&lt;double&gt; and std::complex&lt;float&gt;</td></tr>
<tr><td><a href="#syntax">Matlab/Armadillo&nbsp;syntax&nbsp;differences</a></td><td>&nbsp;</td><td>examples of Matlab syntax and conceptually corresponding Armadillo syntax</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#example_prog">example&nbsp;program</a></td><td>&nbsp;</td><td>short example program</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#config_hpp">config.hpp</a></td><td>&nbsp;</td><td>configuration options</td></tr>
<tr style="background-color: #F5F5F5;"><td><a href="#api_additions">API&nbsp;additions</a></td><td>&nbsp;</td><td>API version policy and list of API additions</td></tr>
<!--<tr><td><a href="#log_add">log_add</a></td><td>&nbsp;</td><td>TODO</td></tr>-->
<!--<tr><td><a href="#catching_exceptions">catching exceptions</a></td><td>&nbsp;</td><td>TODO</td></tr>-->
</tbody>
</table>
</ul>
<br>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Matrix, Vector, Cube and Field Classes</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="Mat"></a><b>Mat&lt;</b><i>type</i><b>&gt;</b>
<br><b>mat</b>
<br><b>cx_mat</b>
<ul>
<li>
The root matrix class is <b>Mat&lt;</b><i>type</i><b>&gt;</b>, where <i>type</i> is one of:
<ul>
<li>
<i>float</i>, <i>double</i>, <i>std::complex&lt;float&gt;</i>, <i>std::complex&lt;double&gt;</i>,
<i>short</i>, <i>int</i>, <i>long</i>, and unsigned versions of <i>short</i>, <i>int</i>, <i>long</i>
</li>
</ul>
</li>
<br>
<li>
For convenience the following typedefs have been defined:
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <code>mat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Mat&lt;double&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>fmat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Mat&lt;float&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_mat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Mat&lt;<a href="#cx_double">cx_double</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_fmat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Mat&lt;<a href="#cx_double">cx_float</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>umat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Mat&lt;<a href="#uword">uword</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>imat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Mat&lt;<a href="#uword">sword</a>&gt;</code>
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
In this documentation the <i>mat</i> type is used for convenience;
it is possible to use other types instead, eg. <i>fmat</i>
</li>
<br>
<li>
Functions which use LAPACK or ATLAS (generally matrix decompositions) are only valid for the following types:
<i>mat</i>, <i>fmat</i>, <i>cx_mat</i>, <i>cx_fmat</i>
</li>
<br>
<li>
Elements are stored with <a href="https://en.wikipedia.org/wiki/Column_major">column-major ordering</a> (ie. column by column)
</li>
<br>
<a name="constructors_mat"></a>
<li>
Constructors:
<ul>
<code>mat()</code>
<br><code>mat(<i>n_rows</i>, <i>n_cols</i>)</code>
<br><code>mat(<i>n_rows</i>, <i>n_cols</i>, <i>fill_type</i>)</code>
<br><code>mat(size(<i>X</i>))</code>
<br><code>mat(size(<i>X</i>), <i>fill_type</i>)</code>
<br><code>mat(mat)</code>
<br><code>mat(sp_mat)</code>
<br><code>mat(vec)</code>
<br><code>mat(rowvec)</code>
<br><code>mat(initializer_list)</code>
<br><code>mat(string)</code>
<br><code>mat(std::vector)</code> &nbsp; (treated as a column vector)
<br><code>cx_mat(mat,mat)</code> &nbsp; (for constructing a complex matrix out of two real matrices)
</ul>
</li>
<br>
<li>
When specifying the size with <i>n_rows</i> and <i>n_cols</i>, by default the memory is uninitialised;
memory can be initialised by specifying the <i>fill_type</i>,
which is one of:
<i>fill::zeros</i>,
<i>fill::ones</i>,
<i>fill::eye</i>,
<i>fill::randu</i>,
<i>fill::randn</i>,
<i>fill::none</i>,
with the following meanings:
<ul>
<table>
<tbody>
<tr><td><code>fill::zeros</code></td><td>&nbsp;=&nbsp;</td><td>set all elements to 0</td></tr>
<tr><td><code>fill::ones</code></td><td>&nbsp;=&nbsp;</td><td>set all elements to 1</td></tr>
<tr><td><code>fill::eye</code></td><td>&nbsp;=&nbsp;</td><td>set the elements along the main diagonal to 1 and off-diagonal elements to 0</td></tr>
<tr><td><code>fill::randu</code></td><td>&nbsp;=&nbsp;</td><td>set each element to a random value from a uniform distribution in the [0,1] interval</td></tr>
<tr><td><code>fill::randn</code></td><td>&nbsp;=&nbsp;</td><td>set each element to a random value from a normal/Gaussian distribution with zero mean and unit variance</td></tr>
<tr><td><code>fill::none</code></td><td>&nbsp;=&nbsp;</td><td>do not modify the elements</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
The string format for the constructor is elements separated by spaces, and rows denoted by semicolons.
For example, the 2x2 identity matrix can be created using <code>"1&nbsp;0;&nbsp;0&nbsp;1"</code>.
<br>
<b>Caveat:</b> string based initialisation is slower than directly <a href="#element_access">setting the elements</a> or using <a href="#element_initialisation">element initialisation</a>.
</li>
<br>
<a name="adv_constructors_mat"></a>
<li>
Advanced constructors:
<br>
<br>
<ul>
<code>mat(ptr_aux_mem, n_rows, n_cols, copy_aux_mem = true, strict = false)</code>
<br>
<br>
<ul>
Create a matrix using data from writable auxiliary (external) memory, where <i>ptr_aux_mem</i> is a pointer to the memory.
By default the matrix allocates its own memory and copies data from the auxiliary memory (for safety).
However, if <i>copy_aux_mem</i> is set to <i>false</i>,
the matrix will instead directly use the auxiliary memory (ie. no copying);
this is faster, but can be dangerous unless you know what you are doing!
<br>
<br>
The <i>strict</i> parameter comes into effect only when <i>copy_aux_mem</i> is set to <i>false</i>
(ie. the matrix is directly using auxiliary memory)
<ul>
<li>
when <i>strict</i> is set to <i>false</i>, the matrix will use the auxiliary memory until a size change
</li>
<li>
when <i>strict</i> is set to <i>true</i>, the matrix will be bound to the auxiliary memory for its lifetime;
the number of elements in the matrix can't be changed
</li>
<li>the default setting of <i>strict</i> in versions 6.000+ is <i>false</i></li>
<li>the default setting of <i>strict</i> in versions 5.600 and earlier is <i>true</i></li>
</ul>
</ul>
<br>
<code>mat(const ptr_aux_mem, n_rows, n_cols)</code>
<br>
<br>
<ul>
Create a matrix by copying data from read-only auxiliary memory,
where <i>ptr_aux_mem</i> is a pointer to the memory
</ul>
<a name="adv_constructors_mat_fixed"></a>
<br>
<code>mat::fixed&lt;n_rows, n_cols&gt;</code>
<br>
<br>
<ul>
Create a fixed size matrix, with the size specified via template arguments.
Memory for the matrix is allocated at compile time.
This is generally faster than dynamic memory allocation, but the size of the matrix can't be changed afterwards (directly or indirectly).
<br>
<br>
For convenience, there are several pre-defined typedefs for each matrix type
(where the types are: <i>umat</i>, <i>imat</i>, <i>fmat</i>, <i>mat</i>, <i>cx_fmat</i>, <i>cx_mat</i>).
The typedefs specify a square matrix size, ranging from 2x2 to 9x9.
The typedefs were defined by simply appending a two digit form of the size to the matrix type
-- for example, <i>mat33</i> is equivalent to <i>mat::fixed&lt;3,3&gt;</i>,
while <i>cx_mat44</i> is equivalent to <i>cx_mat::fixed&lt;4,4&gt;</i>.
</ul>
<br>
<code>mat::fixed&lt;n_rows, n_cols&gt;(const ptr_aux_mem)</code>
<br>
<br>
<ul>
Create a fixed size matrix, with the size specified via template arguments;
data is copied from auxiliary memory, where <i>ptr_aux_mem</i> is a pointer to the memory
</ul>
</ul>
</li>
<br>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5, 5, fill::randu);
double x = A(1,2);

mat B = A + A;
mat C = A * B;
mat D = A % B;

cx_mat X(A,B);

B.zeros();
B.set_size(10,10);
B.ones(5,6);

B.print("B:");

mat::fixed&lt;5,6&gt; F;

double aux_mem[24];
mat H(&amp;aux_mem[0], 4, 6, false);  // use auxiliary memory
</pre>
</ul>
</li>
<br>
<li><b>Caveat:</b>
For mathematical correctness, scalars are treated as 1x1 matrices during initialisation.
As such, the code below <b>will not</b> generate a 5x5 matrix with every element equal to 123.0:
<ul>
<pre>
mat A(5,5);  A = 123.0;
</pre>
</ul>
Use the following code instead:
<ul>
<pre>
mat A(5,5);  A.fill(123.0);
</pre>
</ul>
<br>
<li>
See also:
<ul>
<li><a href="#attributes">matrix attributes</a></li>
<li><a href="#element_access">accessing elements</a></li>
<li><a href="#element_initialisation">initialising elements</a></li>
<li><a href="#operators">math &amp; relational operators</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#save_load_mat">saving &amp; loading matrices</a></li>
<li><a href="#print">printing matrices</a></li>
<li><a href="#iterators_mat">STL-style element iterators</a></li>
<li><a href="#eval_member">.eval()</a></li>
<li><a href="#conv_to">conv_to()</a> (convert between matrix types)</li>
<li><a href="http://www.cplusplus.com/doc/tutorial/other_data_types/">explanation of <i>typedef</i></a> (cplusplus.com)
<li><a href="#Col">Col class</a></li>
<li><a href="#Row">Row class</a></li>
<li><a href="#Cube">Cube class</a></li>
<li><a href="#SpMat">SpMat class</a> (sparse matrix)</li>
<li><a href="#config_hpp">config.hpp</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="Col"></a><b>Col&lt;</b><i>type</i><b>&gt;</b>
<br><b>vec</b>
<br><b>cx_vec</b>
<ul>
<li>
Classes for column vectors (matrices with one column)
</li>
<br>
<li>The <b>Col&lt;</b><i>type</i><b>&gt;</b> class is derived from the <b>Mat&lt;</b><i>type</i><b>&gt;</b> class
and inherits most of the member functions
</li>
<br>
<li>
For convenience the following typedefs have been defined:
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <code>vec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>colvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Col&lt;double&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>fvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>fcolvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Col&lt;float&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_vec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>cx_colvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Col&lt;<a href="#cx_double">cx_double</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_fvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>cx_fcolvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Col&lt;<a href="#cx_double">cx_float</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>uvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>ucolvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Col&lt;<a href="#uword">uword</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>ivec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>icolvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Col&lt;<a href="#uword">sword</a>&gt;</code>
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
In this documentation, the <b><i>vec</i></b> and <b><i>colvec</i></b> types have the <b>same meaning</b> and are used <b>interchangeably</b>
</li>
<br>
<li>
In this documentation, the types <i>vec</i> or <i>colvec</i> are used for convenience; it is possible to use other types instead, eg.&nbsp;<i>fvec</i>, <i>fcolvec</i>
</li>
<br>
<li>
Functions which take <i>Mat</i> as input can generally also take <i>Col</i> as input.
Main exceptions are functions which require square matrices
</li>
<br>
<li>
Constructors
<ul>
<code>vec()</code>
<br><code>vec(<i>n_elem</i>)</code>
<br><code>vec(<i>n_elem</i>, <i>fill_type</i>)</code>
<br><code>vec(size(<i>X</i>))</code>
<br><code>vec(size(<i>X</i>), <i>fill_type</i>)</code>
<br><code>vec(vec)</code>
<br><code>vec(mat)</code> &nbsp; (a <i>std::logic_error</i> exception is thrown if the given matrix has more than one column)
<br><code>vec(initializer_list)</code>
<br><code>vec(string)</code> &nbsp; (elements separated by spaces)
<br><code>vec(std::vector)</code>
<br><code>cx_vec(vec,vec)</code> &nbsp; (for constructing a complex vector out of two real vectors)
</ul>
</li>
<br>
<li>
When specifying the size with <i>n_elem</i>, by default the memory is uninitialised;
memory can be initialised by specifying the <i>fill_type</i>,
as per the <a href="#Mat">Mat class</a>
</li>
<br>
<a name="adv_constructors_col"></a>
<li>
Advanced constructors:
<br>
<br>
<ul>
<code>vec(ptr_aux_mem, number_of_elements, copy_aux_mem = true, strict = false)</code>
<br>
<br>
<ul>
Create a column vector using data from writable auxiliary (external) memory, where <i>ptr_aux_mem</i> is a pointer to the memory.
By default the vector allocates its own memory and copies data from the auxiliary memory (for safety).
However, if <i>copy_aux_mem</i> is set to <i>false</i>,
the vector will instead directly use the auxiliary memory (ie. no copying);
this is faster, but can be dangerous unless you know what you are doing!
<br>
<br>
The <i>strict</i> parameter comes into effect only when <i>copy_aux_mem</i> is set to <i>false</i>
(ie. the vector is directly using auxiliary memory)
<ul>
<li>
when <i>strict</i> is set to <i>false</i>, the vector will use the auxiliary memory until a size change
</li>
<li>
when <i>strict</i> is set to <i>true</i>, the vector will be bound to the auxiliary memory for its lifetime;
the number of elements in the vector can't be changed
</li>
<li>the default setting of <i>strict</i> in versions 6.000+ is <i>false</i></li>
<li>the default setting of <i>strict</i> in versions 5.600 and earlier is <i>true</i></li>
</ul>
</ul>
<br>
<code>vec(const ptr_aux_mem, number_of_elements)</code>
<br>
<br>
<ul>
Create a column vector by copying data from read-only auxiliary memory,
where <i>ptr_aux_mem</i> is a pointer to the memory
</ul>
<a name="adv_constructors_col_fixed"></a>
<br>
<code>vec::fixed&lt;number_of_elements&gt;</code>
<br>
<br>
<ul>
Create a fixed size column vector, with the size specified via the template argument.
Memory for the vector is allocated at compile time.
This is generally faster than dynamic memory allocation, but the size of the vector can't be changed afterwards (directly or indirectly).
<br>
<br>
For convenience, there are several pre-defined typedefs for each vector type
(where the types are: <i>uvec</i>, <i>ivec</i>, <i>fvec</i>, <i>vec</i>, <i>cx_fvec</i>, <i>cx_vec</i> as well as the corresponding <i>colvec</i> versions).
The pre-defined typedefs specify vector sizes ranging from 2 to 9.
The typedefs were defined by simply appending a single digit form of the size to the vector type
-- for example, <i>vec3</i> is equivalent to <i>vec::fixed&lt;3&gt;</i>,
while <i>cx_vec4</i> is equivalent to <i>cx_vec::fixed&lt;4&gt;</i>.
</ul>
<br>
<code>vec::fixed&lt;number_of_elements&gt;(const ptr_aux_mem)</code>
<br>
<br>
<ul>
Create a fixed size column vector, with the size specified via the template argument;
data is copied from auxiliary memory, where <i>ptr_aux_mem</i> is a pointer to the memory
</ul>
</ul>
</li>
<br>
<br>
<li>
Examples:
<ul>
<pre>
vec x(10);
vec y = zeros&lt;vec&gt;(10);

mat A = randu&lt;mat&gt;(10,10);
vec z = A.col(5); // extract a column vector
</pre>
</ul>
</li>
<br>
<li><b>Caveat:</b>
For mathematical correctness, scalars are treated as 1x1 matrices during initialisation.
As such, the code below <b>will not</b> generate a column vector with every element equal to 123.0:
<ul>
<pre>
vec a(5);  a = 123.0;
</pre>
</ul>
Use the following code instead:
<ul>
<pre>
vec a(5);  a.fill(123.0);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#element_initialisation">element initialisation</a></li>
<li><a href="#Mat">Mat class</a></li>
<li><a href="#Row">Row class</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="Row"></a>
<b>Row&lt;</b><i>type</i><b>&gt;</b>
<br><b>rowvec</b>
<br><b>cx_rowvec</b>
<ul>
<li>
Classes for row vectors (matrices with one row)
</li>
<br>
<li>The template <b>Row&lt;</b><i>type</i><b>&gt;</b> class is derived from the <b>Mat&lt;</b><i>type</i><b>&gt;</b> class
and inherits most of the member functions
</li>
<br>
<li>
For convenience the following typedefs have been defined:
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <code>rowvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Row&lt;double&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>frowvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Row&lt;float&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_rowvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Row&lt;<a href="#cx_double">cx_double</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_frowvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Row&lt;<a href="#cx_double">cx_float</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>urowvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Row&lt;<a href="#uword">uword</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>irowvec</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Row&lt;<a href="#uword">sword</a>&gt;</code>
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
In this documentation, the <i>rowvec</i> type is used for convenience;
it is possible to use other types instead, eg. <i>frowvec</i>
</li>
<br>
<li>
Functions which take <i>Mat</i> as input can generally also take <i>Row</i> as input.
Main exceptions are functions which require square matrices
</li>
<br>
<li>
Constructors
<ul>
<code>rowvec()</code>
<br><code>rowvec(<i>n_elem</i>)</code>
<br><code>rowvec(<i>n_elem</i>, <i>fill_type</i>)</code>
<br><code>rowvec(size(<i>X</i>))</code>
<br><code>rowvec(size(<i>X</i>), <i>fill_type</i>)</code>
<br><code>rowvec(rowvec)</code>
<br><code>rowvec(mat)</code> &nbsp; (a <i>std::logic_error</i> exception is thrown if the given matrix has more than one row)
<br><code>rowvec(initializer_list)</code>
<br><code>rowvec(string)</code> &nbsp; (elements separated by spaces)
<br><code>rowvec(std::vector)</code>
<br><code>cx_rowvec(rowvec,rowvec)</code> &nbsp; (for constructing a complex row vector out of two real row vectors)
</ul>
</li>
<br>
<li>
When specifying the size with <i>n_elem</i>, by default the memory is uninitialised;
memory can be initialised by specifying the <i>fill_type</i>,
as per the <a href="#Mat">Mat class</a>
</li>
<br>
<a name="adv_constructors_row"></a>
<li>
Advanced constructors:
<br>
<br>
<ul>
<code>rowvec(ptr_aux_mem, number_of_elements, copy_aux_mem = true, strict = false)</code>
<br>
<br>
<ul>
Create a row vector using data from writable auxiliary (external) memory, where <i>ptr_aux_mem</i> is a pointer to the memory.
By default the vector allocates its own memory and copies data from the auxiliary memory (for safety).
However, if <i>copy_aux_mem</i> is set to <i>false</i>,
the vector will instead directly use the auxiliary memory (ie. no copying);
this is faster, but can be dangerous unless you know what you are doing!
<br>
<br>
The <i>strict</i> parameter comes into effect only when <i>copy_aux_mem</i> is set to <i>false</i>
(ie. the vector is directly using auxiliary memory)
<ul>
<li>
when <i>strict</i> is set to <i>false</i>, the vector will use the auxiliary memory until a size change
</li>
<li>
when <i>strict</i> is set to <i>true</i>, the vector will be bound to the auxiliary memory for its lifetime;
the number of elements in the vector can't be changed
</li>
<li>the default setting of <i>strict</i> in versions 6.000+ is <i>false</i></li>
<li>the default setting of <i>strict</i> in versions 5.600 and earlier is <i>true</i></li>
</ul>
</ul>
<br>
<code>rowvec(const ptr_aux_mem, number_of_elements)</code>
<br>
<br>
<ul>
Create a row vector by copying data from read-only auxiliary memory,
where <i>ptr_aux_mem</i> is a pointer to the memory
</ul>
<br>
<code>rowvec::fixed&lt;number_of_elements&gt;</code>
<br>
<br>
<ul>
Create a fixed size row vector, with the size specified via the template argument.
Memory for the vector is allocated at compile time.
This is generally faster than dynamic memory allocation, but the size of the vector can't be changed afterwards (directly or indirectly).
<br>
<br>
For convenience, there are several pre-defined typedefs for each vector type
(where the types are: <i>urowvec</i>, <i>irowvec</i>, <i>frowvec</i>, <i>rowvec</i>, <i>cx_frowvec</i>, <i>cx_rowvec</i>).
The pre-defined typedefs specify vector sizes ranging from 2 to 9.
The typedefs were defined by simply appending a single digit form of the size to the vector type
-- for example, <i>rowvec3</i> is equivalent to <i>rowvec::fixed&lt;3&gt;</i>,
while <i>cx_rowvec4</i> is equivalent to <i>cx_rowvec::fixed&lt;4&gt;</i>.
</ul>
<br>
<code>rowvec::fixed&lt;number_of_elements&gt;(const ptr_aux_mem)</code>
<br>
<br>
<ul>
Create a fixed size row vector, with the size specified via the template argument;
data is copied from auxiliary memory, where <i>ptr_aux_mem</i> is a pointer to the memory
</ul>
</ul>
</li>
<br>
<br>
<li>
Examples:
<ul>
<pre>
rowvec x(10);
rowvec y = zeros&lt;rowvec&gt;(10);

mat    A = randu&lt;mat&gt;(10,10);
rowvec z = A.row(5); // extract a row vector
</pre>
</ul>
</li>
<br>
<li>
<b>Caveat:</b>
For mathematical correctness, scalars are treated as 1x1 matrices during initialisation.
As such, the code below <b>will not</b> generate a row vector with every element equal to 123.0:
<ul>
<pre>
rowvec r(5);  r = 123.0;
</pre>
</ul>
Use the following code instead:
<ul>
<pre>
rowvec r(5);  r.fill(123.0);
</pre>
</ul>
<br>
<li>See also:
<ul>
<li><a href="#element_initialisation">element initialisation</a></li>
<li><a href="#Mat">Mat class</a></li>
<li><a href="#Col">Col class</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="Cube"></a>
<b>Cube&lt;</b><i>type</i><b>&gt;</b>
<br><b>cube</b>
<br><b>cx_cube</b>
<ul>
<li>
Classes for cubes, also known as "3D matrices" or 3rd order tensors
</li>
<br>
<li>
The cube class is <b>Cube&lt;</b><i>type</i><b>&gt;</b>, where <i>type</i> is one of:
<ul>
<li>
<i>float</i>, <i>double</i>, <i>std::complex&lt;float&gt;</i>, <i>std::complex&lt;double&gt;</i>,
<i>short</i>, <i>int</i>, <i>long</i> and unsigned versions of <i>short</i>, <i>int</i>, <i>long</i>
</li>
</ul>
</li>
<br>
<li>
For convenience the following typedefs have been defined:
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <code>cube</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Cube&lt;double&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>fcube</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Cube&lt;float&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_cube</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Cube&lt;<a href="#cx_double">cx_double</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>cx_fcube</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Cube&lt;<a href="#cx_double">cx_float</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>ucube</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Cube&lt;<a href="#uword">uword</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>icube</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>Cube&lt;<a href="#uword">sword</a>&gt;</code>
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
In this documentation the <i>cube</i> type is used for convenience;
it is possible to use other types instead, eg. <i>fcube</i>
</li>
<br>
<li>
Cube data is stored as a set of slices (matrices) stored contiguously within memory.
Within each slice, elements are stored with column-major ordering (ie. column by column)
</li>
<br>
<li>
Each slice can be interpreted as a matrix, hence functions which take <i>Mat</i> as input can generally also take cube slices as input
</li>
<br>
<a name="constructors_cube"></a>
<li>
Constructors:
<ul>
<code>cube()</code>
<br><code>cube(<i>n_rows</i>, <i>n_cols</i>, <i>n_slices</i>)</code>
<br><code>cube(<i>n_rows</i>, <i>n_cols</i>, <i>n_slices</i>, <i>fill_type</i>)</code>
<br><code>cube(size(<i>X</i>))</code>
<br><code>cube(size(<i>X</i>), <i>fill_type</i>)</code>
<br><code>cube(cube)</code>
<br><code>cx_cube(cube, cube)</code> &nbsp; (for constructing a complex cube out of two real cubes)
</ul>
</li>
<br>
<li>
When specifying the cube size with <i>n_rows</i>, <i>n_cols</i> and <i>n_slices</i>, by default the memory is uninitialised;
memory can be initialised by specifying the <i>fill_type</i>,
as per the <a href="#Mat">Mat class</a> (except for <i>fill::eye</i>)
</li>
<br>
<a name="adv_constructors_cube"></a>
<li>
Advanced constructors:
<br>
<br>
<ul>
<code>cube::fixed&lt;n_rows, n_cols, n_slices&gt;</code>
<br>
<br>
<ul>
Create a fixed size cube, with the size specified via template arguments.
Memory for the cube is allocated at compile time.
This is generally faster than dynamic memory allocation, but the size of the cube can't be changed afterwards (directly or indirectly).
</ul>
<br>
<code>cube(ptr_aux_mem, n_rows, n_cols, n_slices, copy_aux_mem = true, strict = false)</code>
<br>
<br>
<ul>
Create a cube using data from writable auxiliary (external) memory, where <i>ptr_aux_mem</i> is a pointer to the memory.
By default the cube allocates its own memory and copies data from the auxiliary memory (for safety).
However, if <i>copy_aux_mem</i> is set to <i>false</i>,
the cube will instead directly use the auxiliary memory (ie. no copying);
this is faster, but can be dangerous unless you know what you are doing!
<br>
<br>
The <i>strict</i> parameter comes into effect only when <i>copy_aux_mem</i> is set to <i>false</i>
(ie. the cube is directly using auxiliary memory)
<ul>
<li>
when <i>strict</i> is set to <i>false</i>, the cube will use the auxiliary memory until a size change
</li>
<li>
when <i>strict</i> is set to <i>true</i>, the cube will be bound to the auxiliary memory for its lifetime;
the number of elements in the cube can't be changed
</li>
<li>the default setting of <i>strict</i> in versions 6.000+ is <i>false</i></li>
<li>the default setting of <i>strict</i> in versions 5.600 and earlier is <i>true</i></li>
</ul>
</ul>
<br>
<code>cube(const ptr_aux_mem, n_rows, n_cols, n_slices)</code>
<br>
<br>
<ul>
Create a cube by copying data from read-only auxiliary memory,
where <i>ptr_aux_mem</i> is a pointer to the memory
</ul>
</ul>
</li>
<br>
<br>
<li>
Examples:
<ul>
<pre>
cube x(1,2,3);
cube y = randu&lt;cube&gt;(4,5,6);

mat A = y.slice(1);  // extract a slice from the cube
                     // (each slice is a matrix)

mat B = randu&lt;mat&gt;(4,5);
y.slice(2) = B;     // set a slice in the cube

cube q = y + y;     // cube addition
cube r = y % y;     // element-wise cube multiplication

cube::fixed&lt;4,5,6&gt; f;
f.ones();
</pre>
</ul>
</li>
<br>
<li>
<b>Caveats:</b>
<br>
<br>
<ul>
<li>
The size of individual slices can't be changed.
For example, the following <b>will not</b> work:
<ul>
<pre>
cube c(5,6,7);
c.slice(0) = randu&lt;mat&gt;(10,20); // wrong size
</pre>
</ul>
</li>
<li>
For mathematical correctness, scalars are treated as 1x1x1 cubes during initialisation.
As such, the code below <b>will not</b> generate a cube with every element equal to 123.0:
<ul>
<pre>
cube c(5,6,7);  c = 123.0;
</pre>
</ul>
Use the following code instead:
<ul>
<pre>
cube c(5,6,7);  c.fill(123.0);
</pre>
</ul>
<br>
</ul>
<li>
See also:
<ul>
<li><a href="#attributes">cube attributes</a></li>
<li><a href="#element_access">accessing elements</a></li>
<li><a href="#operators">math &amp; relational operators</a></li>
<li><a href="#subcube">subcube views and slices</a></li>
<li><a href="#save_load_mat">saving &amp; loading cubes</a></li>
<li><a href="#iterators_cube">STL-style element iterators</a></li>
<li><a href="#field">field class</a></li>
<li><a href="#Mat">Mat class</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="field"></a>
<b>field&lt;</b><i>object_type</i><b>&gt;</b>
<ul>
<li>
Class for storing arbitrary objects in matrix-like or cube-like layouts
</li>
<br>
<li>
Somewhat similar to a matrix or cube, but instead of each element being a scalar,
each element can be a vector, or matrix, or cube
</li>
<br>
<li>
Each element can have an arbitrary size (eg. in a field of matrices, each matrix can have a different size)
</li>
<br>
<li>
Constructors, where <i>object_type</i> is another class, eg. <i>vec</i>, <i>mat</i>, <i>std::string</i>, etc:
<ul>
<code>field&lt;<i>object_type</i>&gt;()</code>
<br><code>field&lt;<i>object_type</i>&gt;(<i>n_elem</i>)</code>
<br><code>field&lt;<i>object_type</i>&gt;(<i>n_rows</i>, <i>n_cols</i>)</code>
<br><code>field&lt;<i>object_type</i>&gt;(<i>n_rows</i>, <i>n_cols</i>, <i>n_slices</i>)</code>
<br><code>field&lt;<i>object_type</i>&gt;(size(<i>X</i>))</code>
<br><code>field&lt;<i>object_type</i>&gt;(field&lt;<i>object_type</i>&gt;)</code>
</ul>
</li>
<br>
<li>
<b>Caveat</b>: to store a set of matrices of the same size, the <a href="#Cube">Cube</a> class is more efficient
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randn(2,3);
mat B = randn(4,5);

field&lt;mat&gt; F(2,1);
F(0,0) = A;
F(1,0) = B; 

F.print("F:");

F.save("mat_field");
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#attributes">field attributes</a></li>
<li><a href="#subfield">subfield views</a></li>
<li><a href="#save_load_field">saving/loading fields</a></li>
<li><a href="#Cube">Cube class</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="SpMat"></a><b>SpMat&lt;</b><i>type</i><b>&gt;</b>
<br><b>sp_mat</b>
<br><b>sp_cx_mat</b>
<ul>
<li>
The root sparse matrix class is <b>SpMat&lt;</b><i>type</i><b>&gt;</b>, where <i>type</i> is one of:
<ul>
<li>
<i>float</i>, <i>double</i>, <i>std::complex&lt;float&gt;</i>, <i>std::complex&lt;double&gt;</i>,
<i>short</i>, <i>int</i>, <i>long</i> and unsigned versions of <i>short</i>, <i>int</i>, <i>long</i>
</li>
</ul>
</li>
<br>
<li>
For convenience the following typedefs have been defined:
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <code>sp_mat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>SpMat&lt;double&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>sp_fmat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>SpMat&lt;float&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>sp_cx_mat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>SpMat&lt;<a href="#cx_double">cx_double</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>sp_cx_fmat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>SpMat&lt;<a href="#cx_double">cx_float</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>sp_umat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>SpMat&lt;<a href="#uword">uword</a>&gt;</code>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <code>sp_imat</code>
      </td>
      <td style="vertical-align: top;">
      &nbsp;=&nbsp;
      </td>
      <td style="vertical-align: top;">
      <code>SpMat&lt;<a href="#uword">sword</a>&gt;</code>
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
In this documentation the <i>sp_mat</i> type is used for convenience;
it is possible to use other types instead, eg. <i>sp_fmat</i>
</li>
<br>
<a name="constructors_sp_mat"></a>
<li>
Constructors:
<ul>
<code>sp_mat()</code>
<br><code>sp_mat(<i>n_rows</i>, <i>n_cols</i>)</code>
<br><code>sp_mat(size(<i>X</i>))</code>
<br><code>sp_mat(sp_mat)</code>
<br><code>sp_mat(mat)</code>
<br><code>sp_mat(string)</code>
<br><code>sp_cx_mat(sp_mat,sp_mat)</code> &nbsp; (for constructing a complex matrix out of two real matrices)
</ul>
<br>
<li>
Elements are stored in the  <a href="http://en.wikipedia.org/wiki/Sparse_matrix">compressed sparse column (CSC) format</a>
</li>
<br>
<li>
All elements are treated as zero by default (ie. the matrix is initialised to contain zeros)
</li>
<br>
<li>
This class behaves in a similar manner to the <a href="#Mat">Mat</a> class,
however, member functions which set all elements to non-zero values (and hence do not make sense for sparse matrices) have been deliberately omitted;
examples of omitted functions: <a href="#fill">.fill()</a>, <a href="#ones_member">.ones()</a>, +=&nbsp;scalar, etc.
</li>
<br>
<li>
<a name="batch_constructors_sp_mat"></a>
Batch insertion constructors:
<ul>
<li>form&nbsp;1: <code>sp_mat(<i>locations</i>, <i>values</i>, <i>sort_locations = true</i>)</code></li>
<li>form&nbsp;2: <code>sp_mat(<i>locations</i>, <i>values</i>, <i>n_rows</i>, <i>n_cols</i>, <i>sort_locations&nbsp;=&nbsp;true</i>, <i>check_for_zeros&nbsp;=&nbsp;true</i>)</code></li>
<li>form&nbsp;3: <code>sp_mat(<i>add_values</i>, <i>locations</i>, <i>values</i>, <i>n_rows</i>, <i>n_cols</i>, <i>sort_locations&nbsp;=&nbsp;true</i>, <i>check_for_zeros&nbsp;=&nbsp;true</i>)</code></li>
<li>form&nbsp;4: <code>sp_mat(<i>rowind</i>, <i>colptr</i>, <i>values</i>, <i>n_rows</i>, <i>n_cols</i>)</code></li>
<br>
<ul>
<li>
Using batch insertion constructors is generally much faster than consecutively inserting values using <a href="#element_access">element access operators</a>
</li>
<br>
<li>
For forms&nbsp;1,&nbsp;2,&nbsp;3,
<i>locations</i> is a dense matrix of type <i>umat</i>, with a size of <i>2</i>&nbsp;x&nbsp;<i>N</i>, where <i>N</i> is the number of values to be inserted;
the location of the <i>i</i>-th element is specified by the contents of the <i>i</i>-th column of the <i>locations</i> matrix,
where the row is in <i>locations(0,i)</i>, and the column is in <i>locations(1,i)</i>
</li>
<br>
<li>
For form&nbsp;4,
<i>rowind</i> is a dense column vector of type <i>uvec</i> containing the row indices of the values to be inserted,
and
<i>colptr</i> is a dense column vector of type <i>uvec</i> (with length <i>n_cols&nbsp;+&nbsp;1</i>) containing indices of <i>values</i> corresponding to the start of new columns;
the vectors correspond to the arrays used by the <a href="http://en.wikipedia.org/wiki/Sparse_matrix">compressed sparse column format</a>;
this form is useful for copying data from other CSC sparse matrix containers
</li>
<br>
<li>
For all forms, <i>values</i> is a dense column vector containing the values to be inserted;
it must have the same element type as the sparse matrix.
For forms&nbsp;1 and&nbsp;2, the value in <i>values[i]</i> will be inserted at the location specified by the <i>i</i>-th column of the <i>locations</i> matrix.
</li>
<br>
<li>
For form&nbsp;3,
<i>add_values</i> is either <i>true</i> or <i>false</i>; when set to <i>true</i>, identical locations are allowed, and the values at identical locations are added 
</li>
<br>
<li>
The size of the constructed matrix is either 
automatically determined from the maximal locations in the <i>locations</i> matrix (form&nbsp;1),
or
manually specified via <i>n_rows</i> and <i>n_cols</i> (forms&nbsp;2,&nbsp;3,&nbsp;4)
</li>
<br>
<li>
If <i>sort_locations</i> is set to <i>false</i>, the <i>locations</i> matrix is assumed to contain locations that are already sorted according to column-major ordering
</li>
<br>
<li>
If <i>check_for_zeros</i> is set to <i>false</i>, the <i>values</i> vector is assumed to contain no zero values
</li>
</ul>
</ul>
<br>

<li>
<b>Caveats:</b>
<ul>
<li>support for sparse matrices in this version is <b>preliminary</b></li>
<li>the following subset of operations currently works with sparse matrices:
<ul>
<li><a href="#element_access">element access</a></li>
<li>fundamental arithmetic <a href="#operators">operations</a> (such as addition and multiplication)</li>
<li><a href="#submat">submatrix views</a> (contiguous forms only)</li>
<li><a href="#diag">diagonal views</a></li>
<li><a href="#save_load_mat">saving and loading</a> (using <i>arma_binary</i> format only)</li>
<li>element-wise functions: <a href="#abs">abs()</a>, <a href="#imag_real">imag()</a>, <a href="#imag_real">real()</a>, <a href="#conj">conj()</a>, <a href="#misc_fns">sqrt()</a>, <a href="#misc_fns">square()</a>, <a href="#misc_fns">floor()</a>, <a href="#misc_fns">ceil()</a>, <a href="#misc_fns">round()</a>, <a href="#misc_fns">trunc()</a>, <a href="#misc_fns">sign()</a></li>
<li>scalar functions of matrices: <a href="#accu">accu()</a>, <a href="#as_scalar">as_scalar()</a>, <a href="#dot">dot()</a>, <a href="#norm">norm()</a>, <a href="#trace">trace()</a></li>
<li>vector valued functions of matrices: <a href="#min_and_max">min()</a>, <a href="#min_and_max">max()</a>, <a href="#nonzeros">nonzeros()</a>, <a href="#sum">sum()</a>, <a href="#stats_fns">mean()</a>, <a href="#stats_fns">var()</a></li>
<li>matrix valued functions of matrices: <a href="#diagmat">diagmat()</a>, <a href="#join">join_rows()</a>, <a href="#join">join_cols()</a>, <a href="#repmat">repmat()</a>, <a href="#reshape">reshape()</a>, <a href="#resize">resize()</a>, <a href="#t_st_members">.t()</a>, <a href="#trans">trans()</a></li>
<li>generated matrices: <a href="#speye">speye()</a>, <a href="#spones">spones()</a>, <a href="#sprandu_sprandn">sprandu()/sprandn()</a></li>
<li>eigen and svd decomposition: <a href="#eigs_sym">eigs_sym()</a>, <a href="#eigs_gen">eigs_gen()</a>, <a href="#svds">svds()</a></li>
<li>solution of sparse linear systems: <a href="#spsolve">spsolve()</a>
<li>miscellaneous: <a href="#print">print()</a></li>
</ul>
</li>
</ul>
</li>
<br>

<li>
Examples:
<ul>
<pre>
sp_mat A(5,6);
sp_mat B(6,5);

A(0,0) = 1;
A(1,0) = 2;

B(0,0) = 3;
B(0,1) = 4;

sp_mat C = 2*B;

sp_mat D = A*C;


// batch insertion of two values at (5, 6) and (9, 9)
umat locations;
locations &lt;&lt; 5 &lt;&lt; 9 &lt;&lt; endr
          &lt;&lt; 6 &lt;&lt; 9 &lt;&lt; endr;

vec values;
values &lt;&lt; 1.5 &lt;&lt; 3.2 &lt;&lt; endr;

sp_mat X(locations, values);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#element_access">accessing elements</a></li>
<li><a href="#print">printing matrices</a></li>
<!--
<li><a href="#SpCol">SpCol class</a> (TODO: add to documentation)</li>
<li><a href="#SpRow">SpRow class</a> (TODO: add to documentation)</li>
-->
<li><a href="http://en.wikipedia.org/wiki/Sparse_matrix">Sparse Matrix in Wikipedia</a></li>
<li><a href="#Mat">Mat class</a> (dense matrix)</li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="operators"></a>
<b>operators:&nbsp; <code><big>+</big>&nbsp; <big>&minus;</big>&nbsp; <big>*</big>&nbsp; /&nbsp; %&nbsp; ==&nbsp; !=&nbsp; &lt;=&nbsp; &gt;=&nbsp; &lt;&nbsp; &gt;</code></b>
<ul>
<li>
Overloaded operators for <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i> classes
</li>
<br>
<li>
Meanings:
<br>
<br>
<ul>
<table style="text-align: left; width: 100%;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><code><b><big>+</big></b></code></td>
      <td style="vertical-align: top;">&nbsp;&nbsp;&nbsp;<br>
      </td>
      <td style="vertical-align: top;">Addition of two objects</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b><big>&minus;</big></b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">Subtraction of one object from another or negation of an object</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>/</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">Element-wise division of an object by another object or a scalar</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b><big>*</big></b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">Matrix multiplication of two objects; not applicable to the <i>Cube</i> class unless multiplying a cube by a scalar</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>%</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;"><a name="schur_product"></a>Schur product: element-wise multiplication of two objects</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>==</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">Element-wise equality evaluation of two objects; generates a matrix of type <i>umat</i> with entries that indicate whether at a given position the two elements from the two objects are equal (1) or not equal (0)</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>!=</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">Element-wise non-equality evaluation of two objects</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>&gt;=</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">As for <code>==</code>, but the check is for "greater than or equal to"</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>&lt;=</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">As for <code>==</code>, but the check is for "less than or equal to"</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>&gt;</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">As for <code>==</code>, but the check is for "greater than"</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><code><b>&lt;</b></code></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td style="vertical-align: top;">As for <code>==</code>, but the check is for "less than"</td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
<b>Caveat:</b> operators involving an equality comparison (ie. <code><b>==</b></code>, <code><b>!=</b></code>, <code><b>&gt;=</b></code>, <code><b>&lt;=</b></code>)
are not recommended for matrices of type <i>mat</i> or <i>fmat</i>, due to the necessarily <a href="https://en.wikipedia.org/wiki/Floating_point">limited precision</a> of the underlying element types;
you may wish to use <a href="#approx_equal">approx_equal()</a> instead
</li>
<br>
<li>
A <i>std::logic_error</i> exception is thrown if incompatible object sizes are used
</li>
<br>
<li>
If the <code><b>+</b></code>, <code><b>&minus;</b></code> and <code><b>%</b></code> operators are chained, Armadillo will try to avoid the generation of temporaries;
no temporaries are generated if all given objects are of the same type and size
</li>
<br>
<li>
If the <code><b>*</b></code> operator is chained, Armadillo will try to find an efficient ordering of the matrix multiplications
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,10);
mat B = randu&lt;mat&gt;(5,10);
mat C = randu&lt;mat&gt;(10,5);

mat P = A + B;
mat Q = A - B;
mat R = -B;
mat S = A / 123.0;
mat T = A % B;
mat U = A * C;

// V is constructed without temporaries
mat V = A + B + A + B;

imat AA = "1 2 3; 4 5 6; 7 8 9;";
imat BB = "3 2 1; 6 5 4; 9 8 7;";

// compare elements
umat ZZ = (AA >= BB);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#any">any()</a></li>
<li><a href="#all">all()</a></li>
<li><a href="#accu">accu()</a></li>
<li><a href="#approx_equal">approx_equal()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
<li><a href="#find">find()</a></li>
<li><a href="#replace">.replace()</a></li>
<li><a href="#transform">.transform()</a></li>
<li><a href="#each_colrow">.each_col() &amp; .each_row()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a> &nbsp; (exp, log, pow, sqrt, square, round, ...)</li>
<li><a href="https://en.wikipedia.org/wiki/Floating_point">floating point representation in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/Floating-PointRepresentation.html">floating point representation in MathWorld</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Member Functions &amp; Variables</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="attributes"></a>
<b>attributes</b>
<ul>
<table style="text-align: left;" border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td>
<code><b>.n_rows</b></code>
</td>
<td>&nbsp;&nbsp;&nbsp;</td>
<td>
number of rows; present in <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>field</i> and <i>SpMat</i>
</td>
</tr>
<tr>
<td>
<code><b>.n_cols</b></code>
</td>
<td>&nbsp;&nbsp;&nbsp;</td>
<td>
number of columns; present in <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>field</i> and <i>SpMat</i>
</td>
</tr>
<tr>
<td>
<code><b>.n_elem</b></code>
</td>
<td>&nbsp;&nbsp;&nbsp;</td>
<td>
total number of elements; present in <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>field</i> and <i>SpMat</i>
</td>
</tr>
<tr>
<td>
<code><b>.n_slices</b></code>
</td>
<td>&nbsp;&nbsp;&nbsp;</td>
<td>
number of slices; present in <i>Cube</i>
</td>
</tr>
<tr>
<td>
<code><b>.n_nonzero</b></code>
</td>
<td>&nbsp;&nbsp;&nbsp;</td>
<td>
number of non-zero elements; present in <i>SpMat</i>
</td>
</tr>
</tbody>
</table>
</ul>
<ul>
<li>The variables are of type <a href="#uword">uword</a></li>
<br>
<li>
The variables are read-only;
to change the size, use
<a href="#set_size">.set_size()</a>,
<a href="#copy_size">.copy_size()</a>,
<a href="#zeros_member">.zeros()</a>,
<a href="#ones_member">.ones()</a>,
or
<a href="#reset">.reset()</a>
</li>
<br>
<li>
For the <i>Col</i> and <i>Row</i> classes, <i>n_elem</i> also indicates vector length</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X(4,5);
cout &lt;&lt; "X has " &lt;&lt; X.n_cols &lt;&lt; " columns" &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#set_size">.set_size()</a></li>
<li><a href="#copy_size">.copy_size()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#ones_member">.ones()</a></li>
<li><a href="#reset">.reset()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="element_access"></a>
<b>element/object access via (), [] and .at()</b>
<ul>
<li>
Provide access to individual elements or objects stored in a container object
(ie. <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>field</i>)<br>
                    <br>
<ul>
                    <table style="text-align: left; width: 100%;"
 border="0" cellpadding="2" cellspacing="2">
                      <tbody>
                        <tr>
                          <td style="vertical-align: top;">
                          <code>(n)</code>
                          </td>
                          <td style="vertical-align: top;">&nbsp;<br>
                          </td>
                          <td style="vertical-align: top;">
For <i>vec</i> and <i>rowvec</i>, access the <i>n</i>-th element.
For <i>mat</i>, <i>cube</i> and <i>field</i>, access the <i>n</i>-th element/object under the assumption of a flat layout,
with column-major ordering of data (ie. column by column).
A&nbsp;<i>std::logic_error</i> exception is thrown if the requested element is out of bounds.
The bounds check can be <a href="#element_access_bounds_check_note">optionally disabled</a> at compile-time to get more speed.
                          </td>
                        </tr>
<tr>
<td>&nbsp;</td>
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
                        <tr>
                          <td style="vertical-align: top;">
                          <code>.at(n)</code>&nbsp;&nbsp;or&nbsp;&nbsp;<code>[n]</code>&nbsp;
                          </td>
                          <td style="vertical-align: top;"><br>
                          </td>
                          <td style="vertical-align: top;">
As for <i>(n)</i>, but without a bounds check.
Not recommended for use unless your code has been thoroughly debugged.
                          </td>
                        </tr>
<tr>
<td>&nbsp;</td>
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
                        <tr>
                          <td style="vertical-align: top;">
                          <code>(i,j)</code>
                          </td>
                          <td style="vertical-align: top;"><br>
                          </td>
                          <td style="vertical-align: top;">
For <i>mat</i> and <i>2D field</i> classes, access the element/object stored at the <i>i</i>-th row and <i>j</i>-th column.
A&nbsp;<i>std::logic_error</i> exception is thrown if the requested element is out of bounds.
The bounds check can be <a href="#element_access_bounds_check_note">optionally disabled</a> at compile-time to get more speed.
                          </td>
                        </tr>
<tr>
<td>&nbsp;</td>
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
                        <tr>
                          <td style="vertical-align: top;">
                          <code>.at(i,j)</code>
                          </td>
                          <td style="vertical-align: top;"><br>
                          </td>
                          <td style="vertical-align: top;">
As for <i>(i,j)</i>, but without a bounds check.
Not recommended for use unless your code has been thoroughly debugged.
</td>
                          <td style="vertical-align: top;"><br>
                          </td>
                        </tr>
<tr>
<td>&nbsp;</td>
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
                        <tr>
                          <td style="vertical-align: top;">
                          <code>(i,j,k)</code>
                          </td>
                          <td style="vertical-align: top;"><br>
                          </td>
                          <td style="vertical-align: top;">
For <i>cube</i> and <i>3D field</i> classes, access the element/object stored at the <i>i</i>-th row, <i>j</i>-th column and <i>k</i>-th slice.
A&nbsp;<i>std::logic_error</i> exception is thrown if the requested element is out of bounds.
The bounds check can be <a href="#element_access_bounds_check_note">optionally disabled</a> at compile-time to get more speed.
                          </td>
                        </tr>
<tr>
<td>&nbsp;</td>
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
                        <tr>
                          <td style="vertical-align: top;">
                          <code>.at(i,j,k)</code>
                          </td>
                          <td style="vertical-align: top;"><br>
                          </td>
                          <td style="vertical-align: top;">
As for <i>(i,j,k)</i>, but without a bounds check.
Not recommended for use unless your code has been thoroughly debugged.</td>
                        </tr>
                      </tbody>
                    </table>
</ul>
</li>
<br>
<a name="element_access_bounds_check_note"></a>
<li>
The bounds checks used by the <i>(n)</i>, <i>(i,j)</i> and <i>(i,j,k)</i> access forms
can be disabled by defining the <a href="#config_hpp_arma_no_debug">ARMA_NO_DEBUG</a> macro
before including the <i>armadillo</i> header file (eg. <i>#define ARMA_NO_DEBUG</i>).
Disabling the bounds checks is not recommended until your code has been thoroughly debugged
-- it's better to write correct code first, and then maximise its speed.
</li>
<br>
<li>
The indices of elements are specified via the <a href="#uword">uword</a> type, which is a typedef for an <a href="#uword">unsigned integer type</a>.
When using loops to access elements, it's best to use <i>uword</i> instead of <i>int</i>.
For example: <code>for(uword&nbsp;i=0;&nbsp;i&lt;X.n_elem;&nbsp;++i)&nbsp;{&nbsp;X(i)&nbsp;=&nbsp;...&nbsp;}</code>
</li>
<br>
<li>
<b>Caveat</b>: for <a href="#SpMat">sparse matrices</a>, using element access operators to insert values via loops can be inefficient;
you may wish to use <a href="#batch_constructors_sp_mat">batch insertion constructors</a> instead
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(10,10);
A(9,9) = 123.0;
double x = A.at(9,9);
double y = A[99];

vec p = randu&lt;vec&gt;(10,1);
p(9) = 123.0;
double z = p[9];
</pre>
</ul>
</li>                  
<br>
<li>See also:
<ul>
<li><a href="#in_range">.in_range()</a></li>
<li><a href="#element_initialisation">element initialisation</a></li>
<li><a href="#ind2sub">ind2sub()</a></li>
<li><a href="#sub2ind">sub2ind()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#memptr">.memptr()</a></li>
<li><a href="#iterators_mat">iterators (matrices)</a></li>
<li><a href="#iterators_cube">iterators (cubes)</a></li>
<li><a href="#config_hpp">config.hpp</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="element_initialisation"></a>
<b>element initialisation</b>
<ul>
<li>
When using the C++11 standard, elements in <i>Mat</i>, <i>Col</i>, <i>Row</i> can be set via initialiser lists
</li>
<br>
<li>
When using the old C++98 standard, elements can be set via the <code>&lt;&lt;</code> operator; special element <i>endr</i> indicates "end of row" (conceptually similar to <i>std::endl</i>)
</li>
<br>
<li>
<b>Caveat:</b> using the <code>&lt;&lt;</code> operator is slower than using initialiser lists
</li>
<br>
<li>
Examples:
<ul>
<pre>
// C++11

vec v = { 1, 2, 3 };

mat A = { {1, 3, 5},
          {2, 4, 6} };


// C++98

mat B;

B &lt;&lt; 1 &lt;&lt; 3 &lt;&lt; 5 &lt;&lt; endr
  &lt;&lt; 2 &lt;&lt; 4 &lt;&lt; 6 &lt;&lt; endr;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#element_access">element access</a></li>
<li><a href="#reshape_member">.reshape()</a></li>
<li><a href="#print">.print()</a></li>
<li><a href="#save_load_mat">saving &amp; loading matrices</a></li>
<li><a href="#adv_constructors_mat">advanced constructors (matrices)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="zeros_member"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.zeros()</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i>, <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.zeros( n_elem )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Col</i> and <i>Row</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.zeros( n_rows, n_cols )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i> and <i>SpMat</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.zeros( n_rows, n_cols, n_slices )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.zeros( size(X) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>
Set the elements of an object to zero,
optionally first changing the size to specified dimensions
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5,10);  A.zeros();   // or:  mat A(5,10,fill::zeros);

mat B;  B.zeros(10,20);

mat C;  C.zeros( size(B) );
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#zeros_standalone">zeros()</a> (standalone function)</li>
<li><a href="#ones_member">.ones()</a></li>
<li><a href="#randu_randn_member">.randu()</a></li>
<li><a href="#fill">.fill()</a></li>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#reset">.reset()</a></li>
<li><a href="#set_size">.set_size()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="ones_member"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.ones()</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.ones( n_elem )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Col</i> and <i>Row</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.ones( n_rows, n_cols )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.ones( n_rows, n_cols, n_slices )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.ones( size(X) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>
Set all the elements of an object to one,
optionally first changing the size to specified dimensions
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5,10);  A.ones();   // or:  mat A(5,10,fill::ones);

mat B;  B.ones(10,20);

mat C;  C.ones( size(B) );
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#ones_standalone">ones()</a> (standalone function)</li>
<li><a href="#eye_member">.eye()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#fill">.fill()</a></li>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#randu_randn_member">.randu()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eye_member"></a>
<b>.eye()</b>
<br>
<b>.eye( n_rows, n_cols )</b>
<br>
<b>.eye( size(X) )</b>
<ul>
<li>
Member functions of <i>Mat</i> and <i>SpMat</i>
</li>
<br>
<li>
Set the elements along the main diagonal to one and off-diagonal elements to zero,
optionally first changing the size to specified dimensions
</li>
<br>
<li>
An identity matrix is generated when <i>n_rows</i> = <i>n_cols</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5,5);  A.eye();  // or:  mat A(5,5,fill::eye);

mat B;  B.eye(5,5);

mat C;  C.eye( size(B) );
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#ones_member">.ones()</a></li>
<li><a href="#diag">.diag()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="#diagvec">diagvec()</a></li>
<li><a href="#eye_standalone">eye()</a> (standalone function)</li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="randu_randn_member"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.randu()</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randu( n_elem )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Col</i> and <i>Row</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randu( n_rows, n_cols )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randu( n_rows, n_cols, n_slices )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randu( size(X) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>)
      </td>
    </tr>
  </tbody>
</table>
<br>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.randn()</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randn( n_elem )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Col</i> and <i>Row</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randn( n_rows, n_cols )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randn( n_rows, n_cols, n_slices )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.randn( size(X) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>
Set all the elements to random values,
optionally first changing the size to specified dimensions
</li>
<br>
<li><i>.randu()</i> uses a uniform distribution in the [0,1] interval
</li>
<br>
<li><i>.randn()</i> uses a normal/Gaussian distribution with zero mean and unit variance
</li>
<br>
<li>
To change the RNG seed, use <i>arma_rng::set_seed(value)</i> or <i>arma_rng::set_seed_random()</i> functions
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5,10);  A.randu();   // or:  mat A(5,10,fill::randu);

mat B;  B.randu(10,20);

mat C;  C.randu( size(B) );

arma_rng::set_seed_random();  // set the seed to a random value
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#randu_randn_standalone">randu() &amp; randn()</a> (standalone functions)</li>
<li><a href="#fill">.fill()</a></li>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#ones_member">.ones()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#size">size()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)">uniform distribution in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Normal_distribution">normal distribution in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="fill"></a>
<b>.fill( value )</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>
</li>
<br>
<li>Sets the elements to a specified value</li>
<br>
<li>The type of value must match the type of elements used by the container object (eg. for <i>mat</i> the type is <i>double</i>)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5,5);
A.fill(123.0);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#ones_member">.ones()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#randu_randn_member">.randu() &amp; .randn()</a></li>
<li><a href="#replace">.replace()</a></li>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="imbue"></a>
<b>.imbue(&nbsp;</b>functor<b>&nbsp;)</b>
<br>
<b>.imbue(&nbsp;</b>lambda_function<b>&nbsp;)</b>&nbsp;&nbsp;&nbsp;<i>(C++11 only)</i>
<br>
<ul>
<li>
Member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i>
</li>
<br>
<li>
Imbue (fill) with values provided by a functor or lambda function
</li>
<br>
<li>
For matrices, filling is done column-by-column (ie. column 0 is filled, then column 1, ...)
</li>
<br>
<li>
For cubes, filling is done slice-by-slice, with each slice treated as a matrix
</li>
<br>
<li>
Examples:
<ul>
<pre>
// C++11 only example
// need to include &lt;random&gt;

std::mt19937 engine;  // Mersenne twister random number engine

std::uniform_real_distribution&lt;double&gt; distr(0.0, 1.0);
  
mat A(4,5);
  
A.imbue( [&amp;]() { return distr(engine); } );
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#fill">.fill()</a></li>
<li><a href="#transform">.transform()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Function_object">function object</a> at Wikipedia</li>
<li><a href="http://en.wikipedia.org/wiki/C%2B%2B11#Lambda_functions_and_expressions">C++11 lambda functions</a> at Wikipedia</li>
<li><a href="http://www.cprogramming.com/c++11/c++11-lambda-closures.html">lambda function</a> at cprogramming.com</li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="replace"></a>
<b>.replace( old_value, new_value )</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i> and <i>SpMat</i>
</li>
<br>
<li>For all elements equal to <i>old_value</i>, set them to <i>new_value</i></li>
<br>
<li>The type of <i>old_value</i> and <i>new_value</i> must match the type of elements used by the container object (eg. for <i>mat</i> the type is <i>double</i>)
</li>
<br>
<li>
<b>Caveats:</b>
<ul>
<li>floating point numbers (<i>float</i> and <i>double</i>) are approximations due to their <a href="https://en.wikipedia.org/wiki/Floating_point">necessarily limited precision</a></li>
<li>for sparse matrices (<i>SpMat</i>), replacement is not done when <i>old_value = 0</i></li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5,6,fill::randu);

A.diag().fill(datum::nan);

A.replace(datum::nan, 0);  // replace each NaN with 0
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#transform">.transform()</a></li>
<li><a href="#fill">.fill()</a></li>
<li><a href="#find">find()</a></li>
<li><a href="#operators">relational operators</a></li>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="transform"></a>
<b>.transform(&nbsp;</b>functor<b>&nbsp;)</b>
<br>
<b>.transform(&nbsp;</b>lambda_function<b>&nbsp;)</b>&nbsp;&nbsp;&nbsp;<i>(C++11 only)</i>
<br>
<ul>
<li>
Member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i>
</li>
<br>
<li>
Transform each element using a functor or lambda function
</li>
<br>
<li>
For matrices, transformation is done column-by-column (ie. column 0 is transformed, then column 1, ...)
</li>
<br>
<li>
For cubes, transformation is done slice-by-slice, with each slice treated as a matrix
</li>
<br>
<li>
Examples:
<ul>
<pre>
// C++11 only example

mat A = ones&lt;mat&gt;(4,5);

// add 123 to every element
A.transform( [](double val) { return (val + 123.0); } );
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#for_each">.for_each()</a></li>
<li><a href="#replace">.replace()</a></li>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#fill">.fill()</a></li>
<li><a href="#operators">overloaded operators</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a> &nbsp; (exp, log, pow, sqrt, square, round, ...)</li>
<li><a href="http://en.wikipedia.org/wiki/Function_object">function object</a> at Wikipedia</li>
<li><a href="http://en.wikipedia.org/wiki/C%2B%2B11#Lambda_functions_and_expressions">C++11 lambda functions</a> at Wikipedia</li>
<li><a href="http://www.cprogramming.com/c++11/c++11-lambda-closures.html">lambda function</a> at cprogramming.com</li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="for_each"></a>
<b>.for_each(&nbsp;</b>functor<b>&nbsp;)</b>
<br>
<b>.for_each(&nbsp;</b>lambda_function<b>&nbsp;)</b>&nbsp;&nbsp;&nbsp;<i>(C++11 only)</i>
<br>
<ul>
<li>
Member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i> and <i>field</i>
</li>
<br>
<li>
For each element, pass its reference to a functor or lambda function
</li>
<br>
<li>
For matrices, the processing is done column-by-column
</li>
<br>
<li>
For cubes, processing is done slice-by-slice, with each slice treated as a matrix
</li>
<br>
<li>
Examples:
<ul>
<pre>
// C++11 only examples

mat A = ones&lt;mat&gt;(4,5);

// add 123 to each element
A.for_each( [](mat::elem_type&amp; val) { val += 123.0; } );  // NOTE: the '&amp;' is crucial!


field&lt;mat&gt; F(2,3);

// set the size of all matrices in field F
F.for_each( [](mat&amp; X) { X.zeros(4,5); } );  // NOTE: the '&amp;' is crucial!
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#transform">.transform()</a></li>
<li><a href="#each_colrow">.each_col() &amp; .each_row()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a> &nbsp; (exp, log, pow, sqrt, square, round, ...)</li>
<li><a href="http://en.wikipedia.org/wiki/Function_object">function object</a> at Wikipedia</li>
<li><a href="http://en.wikipedia.org/wiki/C%2B%2B11#Lambda_functions_and_expressions">C++11 lambda functions</a> at Wikipedia</li>
<li><a href="http://www.cprogramming.com/c++11/c++11-lambda-closures.html">lambda function</a> at cprogramming.com</li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="set_size"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.set_size( n_elem )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Col</i>, <i>Row</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.set_size( n_rows, n_cols )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.set_size( n_rows, n_cols, n_slices )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Cube</i> and <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.set_size( size(X) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>Change the size of an object, without explicitly preserving data and without initialising the elements</li>
<br>
<li>
If you need to initialise the elements to zero while changing the size, use <a href="#zeros_member">.zeros()</a> instead
</li>
<br>
<li>
If you need to explicitly preserve data while changing the size, use <a href="#reshape_member">.reshape()</a> or <a href="#resize_member">.resize()</a> instead;
<br><b>caveat</b>: <i>.reshape()</i> and <i>.resize()</i> are considerably slower than <i>.set_size()</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A;  A.set_size(5,10);       // or:  mat A(5,10);

mat B;  B.set_size( size(A) );  // or:  mat B( size(A) );

vec v;  v.set_size(100);        // or:  vec v(100);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#reset">.reset()</a></li>
<li><a href="#copy_size">.copy_size()</a></li>
<li><a href="#reshape_member">.reshape()</a></li>
<li><a href="#resize_member">.resize()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="reshape_member"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.reshape( n_rows, n_cols )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i> and <i>SpMat</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.reshape( n_rows, n_cols, n_slices )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.reshape( size(X) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Cube</i>, <i>SpMat</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>
Recreate the object according to given size specifications,
with the elements taken from the previous version of the object in a column-wise manner;
the elements in the generated object are placed column-wise (ie. the first column is filled up before filling the second column)
</li>
<br>
<li>
The layout of the elements in the recreated object will be different to the layout in the previous version of the object
</li>
<br>
<li>
If the total number of elements in the previous version of the object is less than the specified size,
the extra elements in the recreated object are set to zero
</li>
<br>
<li>
If the total number of elements in the previous version of the object is greater than the specified size,
only a subset of the elements is taken
</li>
<br>
<li>
<b>Caveats:</b>
<ul>
<li>
do not use <i>.reshape()</i> if you simply want to change the size without preserving data;
use <a href="#set_size">.set_size()</a> instead, which is much faster
</li>
<li>
to grow/shrink the object while preserving the elements <b>as well as</b> the layout of the elements,
use <a href="#resize_member">.resize()</a> instead
</li>
<li>
to create a vector representation of a matrix (ie. concatenate all the columns or rows),
use <a href="#vectorise">vectorise()</a> instead
</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);

A.reshape(5,4);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#resize_member">.resize()</a></li>
<li><a href="#set_size">.set_size()</a></li>
<li><a href="#copy_size">.copy_size()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#reset">.reset()</a></li>
<li><a href="#reshape">reshape()</a> (standalone function)</li>
<li><a href="#vectorise">vectorise()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="resize_member"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.resize( n_elem )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Col</i>, <i>Row</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.resize( n_rows, n_cols )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i> and <i>SpMat</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.resize( n_rows, n_cols, n_slices )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Cube</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.resize( size(X) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>
Recreate the object according to given size specifications, while preserving the elements as well as the layout of the elements
</li>
<br>
<li>
Can be used for growing or shrinking an object (ie. adding/removing rows, and/or columns, and/or slices)
</li>
<br>
<li>
<b>Caveat:</b>
do not use <i>.resize()</i> if you simply want to change the size without preserving data;
use <a href="#set_size">.set_size()</a> instead, which is much faster
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);
A.resize(7,6);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#reshape_member">.reshape()</a></li>
<li><a href="#set_size">.set_size()</a></li>
<li><a href="#copy_size">.copy_size()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#reset">.reset()</a></li>
<li><a href="#insert">.insert_rows/cols/slices</a></li>
<li><a href="#shed">.shed_rows/cols/slices</a></li>
<li><a href="#resize">resize()</a> (standalone function)</li>
<li><a href="#vectorise">vectorise()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="copy_size"></a>
<b>.copy_size( A )</b>
<ul>
<li>
Set the size to be the same as object <i>A</i>
</li>
<br>
<li>
Object <i>A</i> must be of the same root type as the object being modified
(eg. you can't set the size of a matrix by providing a cube)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,6);
mat B;
B.copy_size(A);

cout &lt;&lt; B.n_rows &lt;&lt; endl;
cout &lt;&lt; B.n_cols &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#reset">.reset()</a></li>
<li><a href="#set_size">.set_size()</a></li>
<li><a href="#reshape_member">.reshape()</a></li>
<li><a href="#resize_member">.resize()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
<li><a href="#repmat">repmat()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="reset"></a>
<b>
.reset()
</b>
<ul>
<li>
Reset the size to zero (the object will have no elements)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5, 5);
A.reset();
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#set_size">.set_size()</a></li>
<li><a href="#is_empty">.is_empty()</a></li>
<li><a href="#zeros_member">.zeros()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="submat"></a>
<b>submatrix views</b>
<ul>
<li>A collection of member functions of <i>Mat</i>, <i>Col</i> and <i>Row</i> classes that provide read/write access to submatrix views<br>
<br>
<table>
<tbody>
<tr>
<td>

<table>
<tbody>
<tr>
<td>

<li>contiguous views for matrix X:
<br>
<br>
<ul>
X.<b>col(&nbsp;</b>col_number<b>&nbsp;)</b><br>
X.<b>row(&nbsp;</b>row_number<b>&nbsp;)</b><br>
<br>
X.<b>cols(&nbsp;</b>first_col<b>,</b>&nbsp;last_col<b>&nbsp;)</b><br>
X.<b>rows(&nbsp;</b>first_row<b>,</b>&nbsp;last_row<b>&nbsp;)</b><br>
<br>
X.<b>submat(&nbsp;</b>first_row<b>,</b>&nbsp;first_col<b>,</b>&nbsp;last_row<b>,</b>&nbsp;last_col<b>&nbsp;)</b><br>
<br>
X<b>(&nbsp;span(</b>first_row<b>,</b>&nbsp;last_row<b>),&nbsp;span(</b>first_col<b>,</b>&nbsp;last_col<b>)&nbsp;)</b><br>
<br>
X<b>(&nbsp;</b>first_row<b>,</b>&nbsp;first_col<b>,&nbsp;size(</b>n_rows<b>,&nbsp;</b>n_cols<b>)&nbsp;)</b><br>
X<b>(&nbsp;</b>first_row<b>,</b>&nbsp;first_col<b>,&nbsp;size(</b>Y<b>)&nbsp;)</b>&nbsp;&nbsp;&nbsp;&nbsp;<i>(Y&nbsp;is&nbsp;a&nbsp;mat)</i><br>
<br>
X<b>(</b>&nbsp;<b>span(</b>first_row<b>,</b>&nbsp;last_row<b>),</b>&nbsp;col_number&nbsp;<b>)</b><br>
X<b>(</b>&nbsp;row_number<b>,</b>&nbsp;<b>span(</b>first_col<b>,</b>&nbsp;last_col<b>)&nbsp;)</b><br>
<br>
X.<b>head_cols(&nbsp;</b>number_of_cols<b>&nbsp;)</b><br>
X.<b>head_rows(&nbsp;</b>number_of_rows<b>&nbsp;)</b><br>
<br>
X.<b>tail_cols(&nbsp;</b>number_of_cols<b>&nbsp;)</b><br>
X.<b>tail_rows(&nbsp;</b>number_of_rows<b>&nbsp;)</b><br>
<br>
X.<b>unsafe_col(&nbsp;</b>col_number<b>&nbsp;)</b><br>
</ul>
</li>

<br>

<li>
contiguous views for vector V:
<br>
<br>
<ul>
V<b>(&nbsp;span(</b>first_index<b>,</b> last_index<b>)&nbsp;)</b><br>
V.<b>subvec(&nbsp;</b>first_index<b>,</b> last_index<b>&nbsp;)</b><br>
<br>
V.<b>subvec(&nbsp;</b>first_index<b>, size(</b>W<b>)&nbsp;)</b>&nbsp;&nbsp;&nbsp;&nbsp;<i>(W&nbsp;is&nbsp;a&nbsp;vector)</i><br>
<br>
V.<b>head(&nbsp;</b>number_of_elements<b>&nbsp;)</b><br>
V.<b>tail(&nbsp;</b>number_of_elements<b>&nbsp;)</b>
</ul>
</li>


</td>
</tr>
</tbody>
</table>

</td>

<td>&nbsp;&nbsp;&nbsp;</td>
<td class="line" style="vertical-align: top;">&nbsp;</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td style="vertical-align: top;">

<table>
<tbody>
<tr>
<td>

<li>non-contiguous views for matrix or vector X:
<ul>
<br>
X.<b>elem(</b>&nbsp;vector_of_indices&nbsp;<b>)</b><br>
X<b>(</b>&nbsp;vector_of_indices&nbsp;<b>)</b><br>
<br>
X.<b>cols(&nbsp;</b>vector_of_column_indices<b>&nbsp;)</b><br>
X.<b>rows(&nbsp;</b>vector_of_row_indices<b>&nbsp;)</b><br>
<br>
X.<b>submat(</b>&nbsp;vector_of_row_indices<b>,</b>&nbsp;vector_of_column_indices&nbsp;<b>)</b><br>
X<b>(</b>&nbsp;vector_of_row_indices<b>,</b>&nbsp;vector_of_column_indices&nbsp;<b>)</b><br>
<!--X.<b>elem(</b>&nbsp;vector_of_row_indices<b>,</b>&nbsp;vector_of_column_indices&nbsp;<b>)</b><br>-->
</ul>
</li>
<br>
<br>

<li>related views (documented separately)
<ul>
<br>
X.<a href="#diag">diag()</a><br>
X.<a href="#each_colrow">each_row()</a><br>
X.<a href="#each_colrow">each_col()</a><br>
</ul>
</li>

</td>
</tr>
</tbody>
</table>

</td>
</tr>
</tbody>
</table>


<br>
<li>
Instances of <i>span(start,end)</i> can be replaced by <i>span::all</i> to indicate the entire range
</li>
<br>
<li>
For functions requiring one or more vector of indices,
eg. <i>X.submat(vector_of_row_indices,&nbsp;vector_of_column_indices)</i>,
each vector of indices must be of type <i><a href="#Col">uvec</a></i>
</li>
<br>
<li>
In the function <i>X.elem(vector_of_indices)</i>,
elements specified in <i>vector_of_indices</i> are accessed.
<i>X</i> is interpreted as one long vector,
with column-by-column ordering of the elements of <i>X</i>.
The <i>vector_of_indices</i> must evaluate to a vector of type <a href="#Col">uvec</a>
(eg. generated by the <a href="#find">find()</a> function).
The aggregate set of the specified elements is treated as a column vector
(ie. the output of <i>X.elem()</i> is always a column vector).
</li>
<br>
<li>
The function <i>.unsafe_col()</i> is provided for speed reasons and should be used only if you know what you are doing.
It creates a seemingly independent <i>Col</i> vector object (eg.&nbsp;<i>vec</i>),
but uses memory from the existing matrix object.
As such, the created vector is not alias safe,
and does not take into account that the underlying matrix memory could be freed
(eg. due to any operation involving a size change of the matrix).
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = zeros&lt;mat&gt;(5,10);

A.submat( 0,1, 2,3 )      = randu&lt;mat&gt;(3,3);
A( span(0,2), span(1,3) ) = randu&lt;mat&gt;(3,3);
A( 0,1, size(3,3) )       = randu&lt;mat&gt;(3,3);

mat B = A.submat( 0,1, 2,3 );
mat C = A( span(0,2), span(1,3) );
mat D = A( 0,1, size(3,3) );

A.col(1)        = randu&lt;mat&gt;(5,1);
A(span::all, 1) = randu&lt;mat&gt;(5,1);

mat X = randu&lt;mat&gt;(5,5);

// get all elements of X that are greater than 0.5
vec q = X.elem( find(X &gt; 0.5) );

// add 123 to all elements of X greater than 0.5
X.elem( find(X &gt; 0.5) ) += 123.0;

// set four specific elements of X to 1
uvec indices;
indices &lt;&lt; 2 &lt;&lt; 3 &lt;&lt; 6 &lt;&lt; 8;

X.elem(indices) = ones&lt;vec&gt;(4);

// add 123 to the last 5 elements of vector a
vec a(10, fill::randu);
a.tail(5) += 123.0;

// add 123 to the first 3 elements of column 2 of X
X.col(2).head(3) += 123;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#diag">diagonal views</a></li>
<li><a href="#each_colrow">.each_col() &amp; .each_row()</a> &nbsp;&nbsp; (vector operations repeated on each column or row)</li>
<li><a href="#colptr">.colptr()</a></li>
<li><a href="#in_range">.in_range()</a></li>
<li><a href="#find">find()</a></li>
<li><a href="#join">join_rows/columns/slices</a></li>
<li><a href="#shed">shed_rows/columns/slices</a></li>
<li><a href="#insert">.insert_rows/cols/slices</a></li>
<li><a href="#size">size()</a></li>
<li><a href="#subcube">subcube views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="subcube"></a>
<b>subcube views and slices</b>
<ul>
<li>A collection of member functions of the <i>Cube</i> class that provide subcube views</li>
<br>
<table>
<tr>

<td style="vertical-align: top;">
<li>contiguous views for cube Q:
<br>
<br>
<ul>
Q.<b>slice(&nbsp;</b>slice_number&nbsp;<b>)</b><br>
Q.<b>slices(&nbsp;</b>first_slice<b>,</b> last_slice&nbsp;<b>)</b><br>
<br>
Q.<b>subcube(&nbsp;</b>first_row<b>,</b> first_col<b>,</b> first_slice<b>, </b>last_row<b>,</b> last_col<b>, </b>last_slice&nbsp;<b>)</b><br>
<br>
Q<b>(&nbsp;span(</b>first_row<b>,</b> last_row<b>), span(</b>first_col<b>,</b> last_col<b>), span(</b>first_slice<b>,</b> last_slice<b>)&nbsp;)</b><br>
<br>
Q<b>(&nbsp;</b>first_row<b>,</b> first_col<b>,</b> first_slice<b>, size(</b>n_rows<b>,</b> n_cols<b>, </b>n_slices<b>)&nbsp;)</b><br>
Q<b>(&nbsp;</b>first_row<b>,</b> first_col<b>,</b> first_slice<b>, size(</b>R<b>)&nbsp;)</b>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>(R is a cube)</i><br>
<br>
Q.<b>head_slices(&nbsp;</b>number_of_slices<b>&nbsp;)</b><br>
Q.<b>tail_slices(&nbsp;</b>number_of_slices<b>&nbsp;)</b><br>
<br>
Q.<b>tube(&nbsp;</b>row<b>,</b> col&nbsp;<b>)</b><br>
Q.<b>tube(&nbsp;</b>first_row<b>,</b> first_col<b>,</b> last_row<b>,</b> last_col&nbsp;<b>)</b><br>
Q.<b>tube(&nbsp;span(</b>first_row<b>,</b> last_row<b>), span(</b>first_col<b>,</b> last_col<b>)&nbsp;)</b><br>
Q.<b>tube(&nbsp;</b>first_row<b>,</b> first_col<b>, size(</b>n_rows<b>,</b> n_cols<b>)&nbsp;)</b><br>
</ul>
</li>
</td>

<td>&nbsp;&nbsp;&nbsp;</td>
<td class="line" style="vertical-align: top;">&nbsp;</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;</td>

<td style="vertical-align: top;">
<li>non-contiguous views for cube Q:
<ul>
<br>
Q.<b>elem(</b>&nbsp;vector_of_indices&nbsp;<b>)</b><br>
Q<b>(</b>&nbsp;vector_of_indices&nbsp;<b>)</b>
</ul>
</li>
<br>
<li>related views (documented separately)
<ul>
<br>
Q.<a href="#each_slice">each_slice()</a><br>
</ul>
</li>
</td>

</tr>
</table>
<br>
<li>
Instances of <i>span(a,b)</i> can be replaced by:
<ul>
<li><i>span()</i> or <i>span::all</i>, to indicate the entire range</li>
<li><i>span(a)</i>, to indicate a particular row, column or slice</li>
</ul>
</li>
<br>
<li>
An individual slice, accessed via <i>.slice()</i>, is an instance of the <i>Mat</i> class
(a reference to a matrix is provided)
</li>
<br>
<li>
All <i>.tube()</i> forms are variants of <i>.subcube()</i>, using <i>first_slice&nbsp;=&nbsp;0</i> and <i>last_slice&nbsp;=&nbsp;Q.n_slices-1</i>
</li>
<br>
<li>
The <i>.tube(row,col)</i> form uses <i>row&nbsp;=&nbsp;first_row&nbsp;=&nbsp;last_row</i>, and <i>col&nbsp;=&nbsp;first_col&nbsp;=&nbsp;last_col</i>  
</li>
<br>
<li>
In the function <i>Q.elem(vector_of_indices)</i>,
elements specified in <i>vector_of_indices</i> are accessed.
<i>Q</i> is interpreted as one long vector,
with slice-by-slice and column-by-column ordering of the elements of <i>Q</i>.
The <i>vector_of_indices</i> must evaluate to a vector of type <a href="#Col">uvec</a>
(eg. generated by the <a href="#find">find()</a> function).
The aggregate set of the specified elements is treated as a column vector
(ie. the output of <i>Q.elem()</i> is always a column vector).
</li>
<br>
<li>
Examples:
<ul>
<pre>
cube A = randu&lt;cube&gt;(2,3,4);

mat  B = A.slice(1); // each slice is a matrix

A.slice(0) = randu&lt;mat&gt;(2,3);
A.slice(0)(1,2) = 99.0;

A.subcube(0,0,1,  1,1,2)             = randu&lt;cube&gt;(2,2,2);
A( span(0,1), span(0,1), span(1,2) ) = randu&lt;cube&gt;(2,2,2);
A( 0,0,1, size(2,2,2) )              = randu&lt;cube&gt;(2,2,2);

// add 123 to all elements of A greater than 0.5
A.elem( find(A &gt; 0.5) ) += 123.0;

cube C = A.head_slices(2);  // get first two slices

A.head_slices(2) += 123.0;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#in_range">.in_range()</a></li>
<li><a href="#each_slice">.each_slice()</a></li>
<li><a href="#join_slices">join_slices()</a></li>
<li><a href="#shed">shed_slices()</a></li>
<li><a href="#insert">insert_slices()</a></li>
<li><a href="#size">size()</a></li>
<li><a href="#submat">submatrix views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="subfield"></a>
<b>subfield views</b>
<ul>
<li>A collection of member functions of the <i>field</i> class that provide subfield views<br>
<br>
<li>For a 2D field <i>F</i>, the subfields are accessed as:</li>
<br>
<ul>
F.<b>row(&nbsp;</b>row_number&nbsp;<b>)</b><br>
F.<b>col(&nbsp;</b>col_number&nbsp;<b>)</b><br>
<br>
F.<b>rows(&nbsp;</b>first_row<b>,</b> last_row&nbsp;<b>)</b><br>
F.<b>cols(&nbsp;</b>first_col<b>,</b> last_col&nbsp;<b>)</b><br>
<br>
F.<b>subfield(&nbsp;</b>first_row<b>,</b>&nbsp;first_col<b>,</b> last_row<b>,</b>&nbsp;last_col <b>)</b><br>
<br>
F<b>(&nbsp;span(</b>first_row<b>,</b> last_row<b>), span(</b>first_col<b>,</b>&nbsp;last_col<b>) )</b><br>
<br>
F<b>(&nbsp;</b>first_row<b>,</b> first_col<b>, size(</b>G<b>)&nbsp;)</b> &nbsp;&nbsp;&nbsp;<i>(G is a 2D field)</i><br>
F<b>(&nbsp;</b>first_row<b>,</b> first_col<b>, size(</b>n_rows<b>, </b>n_cols<b>)&nbsp;)</b><br>
</ul>
<br>
<li>For a 3D field <i>F</i>, the subfields are accessed as:</li>
<br>
<ul>
F.<b>slice(&nbsp;</b>slice_number&nbsp;<b>)</b><br>
<br>
F.<b>slices(&nbsp;</b>first_slice<b>,</b> last_slice&nbsp;<b>)</b><br>
<br>
F.<b>subfield(&nbsp;</b>first_row<b>,</b>&nbsp;first_col<b>,</b>&nbsp;first_slice<b>,</b> last_row<b>,</b>&nbsp;last_col<b>,</b>&nbsp;last_slice <b>)</b><br>
<br>
F<b>(&nbsp;span(</b>first_row<b>,</b> last_row<b>), span(</b>first_col<b>,</b>&nbsp;last_col<b>), span(</b>first_slice<b>,</b>&nbsp;last_slice<b>) )</b><br>
<br>
F<b>(&nbsp;</b>first_row<b>,</b> first_col<b>,</b> first_slice<b>, size(</b>G<b>)&nbsp;)</b> &nbsp;&nbsp;&nbsp;<i>(G is a 3D field)</i><br>
F<b>(&nbsp;</b>first_row<b>,</b> first_col<b>,</b> first_slice<b>, size(</b>n_rows<b>, </b>n_cols<b>, </b>n_slices<b>)&nbsp;)</b><br>
</ul>
</li>
<br>
<li>
Instances of <i>span(a,b)</i> can be replaced by:
<ul>
<li><i>span()</i> or <i>span::all</i>, to indicate the entire range</li>
<li><i>span(a)</i>, to indicate a particular row or column</li>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#in_range">.in_range()</a></li>
<li><a href="#size">size()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#subcube">subcube views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="diag"></a>
<b>.diag()</b>
<br><b>.diag( k )</b>
<ul>
<li>
Member function of <i>Mat</i> and <i>SpMat</i>
</li>
<br>
<li>
Read/write access to a diagonal in a matrix
</li>
<br>
<li>The argument <i>k</i> is optional; by default the main diagonal is accessed (<i>k=0</i>)</li>
<br>
<li>For <i>k &gt; 0</i>, the <i>k</i>-th super-diagonal is accessed (top-right corner)</li>
<br>
<li>For <i>k &lt; 0</i>, the <i>k</i>-th sub-diagonal is accessed (bottom-left corner)</li>
<br>
<li>
The diagonal is interpreted as a <b>dense</b> column vector within expressions
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(5,5);

vec a = X.diag();
vec b = X.diag(1);
vec c = X.diag(-2);

X.diag()  = randu&lt;vec&gt;(5);
X.diag() += 6;
X.diag().ones();
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#eye_member">.eye()</a></li>
<li><a href="#diagvec">diagvec()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#each_colrow">.each_col() &amp; .each_row()</a></li>
<li><a href="#trace">trace()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="each_colrow"></a>
<b>.each_col()</b>
<br>
<b>.each_row()</b>
<br>
<br>
<b>.each_col(</b>&nbsp;vector_of_indices&nbsp;<b>)</b>
<br>
<b>.each_row(</b>&nbsp;vector_of_indices&nbsp;<b>)</b>
<br>
<br>
<b>.each_col(</b> lambda_function <b>)</b>&nbsp;&nbsp;&nbsp;<i>(C++11 only)</i>
<br>
<b>.each_row(</b> lambda_function <b>)</b>&nbsp;&nbsp;&nbsp;<i>(C++11 only)</i>
<ul>
<li>
Member functions of <i>Mat</i>
</li>
<br>
<li>
Repeat a vector operation on each column or row of a matrix
</li>
<br>
<li>
Supported operations for <i>.each_col()&nbsp;/&nbsp;.each_row()</i> and <i>.each_col(vector_of_indices)&nbsp;/&nbsp;.each_row(vector_of_indices)</i> forms:
<br>
<br>
<ul>
<table>
<tr><td><code><b>+</b></code></td><td>&nbsp;</td><td>addition</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>+=</b></code></td><td>&nbsp;</td><td>in-place addition</td></tr>
<tr><td><code><b>-</b></code></td><td>&nbsp;</td><td>subtraction</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>-=</b></code></td><td>&nbsp;</td><td>in-place subtraction</td></tr>
<tr><td><code><b>%</b></code></td><td>&nbsp;</td><td>element-wise multiplication</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>%=</b></code></td><td>&nbsp;</td><td>in-place element-wise multiplication</td></tr>
<tr><td><code><b>/</b></code></td><td>&nbsp;</td><td>element-wise division</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>/=</b></code></td><td>&nbsp;</td><td>in-place element-wise division</td></tr>
<tr><td><code><b>=</b></code></td><td>&nbsp;</td><td>assignment (copy)</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
</table>
</ul>
</li>
<br>
<li>The argument <i>vector_of_indices</i> is optional; by default all columns or rows are used</li>
<br>
<li>
If the argument <i>vector_of_indices</i> is specified, it must evaluate to a vector of type <i><a href="#Col">uvec</a></i>;
the vector contains a list of indices of the columns or rows to be used
</li>
<br>
<li>
If the <i>lambda_function</i> is specified,
the function must accept a reference to a <a href="#Col">Col</a> or <a href="#Row">Row</a> object with the same element type as the underlying matrix
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = ones&lt;mat&gt;(6,5);
vec v = linspace&lt;vec&gt;(10,15,6);

X.each_col() += v;         // in-place addition of v to each column vector of X

mat Y = X.each_col() + v;  // generate Y by adding v to each column vector of X

// subtract v from columns 0 through to 3 in X
X.cols(0,3).each_col() -= v;


uvec indices(2);
indices(0) = 2;
indices(1) = 4;

X.each_col(indices) = v;   // copy v to columns 2 and 4 in X


X.each_col( [](vec&amp; a){ a.print(); } );     // lambda function with non-const vector

const mat&amp; XX = X;
XX.each_col( [](const vec&amp; b){ b.print(); } );  // lambda function with const vector
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#diag">diagonal views</a></li>
<li><a href="#repmat">repmat()</a></li>
<li><a href="#for_each">.for_each()</a></li>
<li><a href="#each_slice">.each_slice()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="each_slice"></a>
<b>.each_slice()</b>
<br>
<b>.each_slice(</b>&nbsp;vector_of_indices&nbsp;<b>)</b>
<br>
<b>.each_slice(</b>&nbsp;lambda_function&nbsp;<b>)</b>&nbsp;&nbsp;&nbsp;<i>(C++11 only)</i>
<ul>
<li>
Member function of <i>Cube</i>
</li>
<br>
<li>
Repeat a matrix operation on each slice of a cube, with each slice treated as a matrix
</li>
<br>
<li>
Supported operations for the <i>.each_slice()</i> form:
<br>
<br>
<ul>
<table>
<tr><td><code><b>+</b></code></td><td>&nbsp;</td><td>addition</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>+=</b></code></td><td>&nbsp;</td><td>in-place addition</td></tr>
<tr><td><code><b>-</b></code></td><td>&nbsp;</td><td>subtraction</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>-=</b></code></td><td>&nbsp;</td><td>in-place subtraction</td></tr>
<tr><td><code><b>%</b></code></td><td>&nbsp;</td><td>element-wise multiplication</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>%=</b></code></td><td>&nbsp;</td><td>in-place element-wise multiplication</td></tr>
<tr><td><code><b>/</b></code></td><td>&nbsp;</td><td>element-wise division</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>/=</b></code></td><td>&nbsp;</td><td>in-place element-wise division</td></tr>
<tr><td><code><b>*</b></code></td><td>&nbsp;</td><td>matrix multiplication</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td><code><b>*=</b></code></td><td>&nbsp;</td><td>in-place matrix multiplication</td></tr>
<tr><td><code><b>=</b></code></td><td>&nbsp;</td><td>assignment (copy)</td><td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
</table>
</ul>
</li>
<br>
<li>
The <i>.each_slice(vector_of_indices)</i> form also supports the above operations, except for <code><b>*</b></code> and <code><b>*=</b></code> (ie. matrix multiplication)
</li>
<br>
<li>The argument <i>vector_of_indices</i> is optional; by default all slices are used</li>
<br>
<li>
If the argument <i>vector_of_indices</i> is specified, it must evaluate to a vector of type <i><a href="#Col">uvec</a></i>;
the vector contains a list of indices of the slices to be used
</li>
<br>
<li>
If the <i>lambda_function</i> is specified,
the function must accept a reference to a <a href="#Mat">Mat</a> object with the same element type as the underlying cube
</li>
<br>
<li>
Examples:
<ul>
<pre>
cube C(4,5,6, fill::randu);

mat M = repmat(linspace&lt;vec&gt;(1,4,4), 1, 5);

C.each_slice() += M;          // in-place addition of M to each slice of C

cube D = C.each_slice() + M;  // generate D by adding M to each slice of C


uvec indices(2);
indices(0) = 2;
indices(1) = 4;

C.each_slice(indices) = M;    // copy M to slices 2 and 4 in C


C.each_slice( [](mat&amp; X){ X.print(); } );     // lambda function with non-const matrix

const cube&amp; CC = C;
CC.each_slice( [](const mat&amp; X){ X.print(); } );  // lambda function with const matrix
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#subcube">subcube views</a></li>
<li><a href="#for_each">.for_each()</a></li>
<li><a href="#each_colrow">.each_col() &amp; .each_row()</a> &nbsp;</li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="set_imag"></a>
<b>.set_imag( X )</b>
<br>
<b>.set_real( X )</b>
<br>  
<ul>
<li>
Set the imaginary/real part of an object
</li>
<br>
<li>
<i>X</i> must have the same size as the recipient object
</li>
<br>
<li>
Examples:
<ul>
<pre>
   mat A = randu&lt;mat&gt;(4,5);
   mat B = randu&lt;mat&gt;(4,5);

cx_mat C = zeros&lt;cx_mat&gt;(4,5);

C.set_real(A);
C.set_imag(B);
</pre>
</ul>
</li>
<br>
<li>
<b>Caveat:</b>
to directly construct a complex matrix out of two real matrices,
the following code is faster:
<ul>
<pre>
   mat A = randu&lt;mat&gt;(4,5);
   mat B = randu&lt;mat&gt;(4,5);

cx_mat C = cx_mat(A,B);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#constructors_mat">matrix constructors</a></li>
<li><a href="#constructors_cube">cube constructors</a></li>
<li><a href="#imag_real">imag()&nbsp;/&nbsp;real()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="insert"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>.insert_rows(&nbsp;</b>row_number, X<b>&nbsp;)</b>
      <br>
      <b>.insert_rows(&nbsp;</b>row_number, number_of_rows&nbsp;<b>)</b>
      <br>
      <b>.insert_rows(&nbsp;</b>row_number, number_of_rows, set_to_zero&nbsp;<b>)</b>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member functions of <i>Mat</i> and <i>Col</i>)
      </td>
    </tr>
    <tr>
    <td>&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.insert_cols(&nbsp;</b>col_number<b>,</b> X<b>&nbsp;)</b>
      <br>
      <b>.insert_cols(&nbsp;</b>col_number<b>,</b> number_of_cols&nbsp;<b>)</b>
      <br>
      <b>.insert_cols(&nbsp;</b>col_number<b>,</b> number_of_cols<b>,</b> set_to_zero&nbsp;<b>)</b>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member functions of <i>Mat</i> and <i>Row</i>)
      </td>
    </tr>
    <tr>
    <td>&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.insert_slices(&nbsp;</b>slice_number<b>,</b> X<b>&nbsp;)</b>
      <br>
      <b>.insert_slices(&nbsp;</b>slice_number<b>,</b> number_of_slices&nbsp;<b>)</b>
      <br>
      <b>.insert_slices(&nbsp;</b>slice_number<b>,</b> number_of_slices<b>,</b> set_to_zero&nbsp;<b>)</b>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member functions of <i>Cube</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>
Functions with the <i>X</i> argument: insert a copy of <i>X</i> at the specified row/column/slice
<ul>
<li>if inserting rows, <i>X</i> must have the same number of columns as the recipient object</li>
<li>if inserting columns, <i>X</i> must have the same number of rows as the recipient object</li>
<li>if inserting slices, <i>X</i> must have the same number of rows and columns as the recipient object (ie. all slices must have the same size)</li>
</ul>
</li>
<br>
<li>
Functions with the <i>number_of_...</i> argument: expand the object by creating new rows/columns/slices.
By default, the new rows/columns/slices are set to zero.
If <i>set_to_zero</i> is <i>false</i>, the memory used by the new rows/columns/slices will not be initialised.
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,10);
mat B = ones&lt;mat&gt;(5,2);

// at column 2, insert a copy of B;
// A will now have 12 columns
A.insert_cols(2, B);

// at column 1, insert 5 zeroed columns;
// B will now have 7 columns
B.insert_cols(1, 5);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#shed">shed_rows/columns/slices</a></li>
<li><a href="#join">join_rows/columns/slices</a></li>
<li><a href="#resize_member">.resize()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#subcube">subcube views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="shed"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>.shed_row(&nbsp;</b>row_number<b>&nbsp;)</b>
      <br>
      <b>.shed_rows(&nbsp;</b>first_row, last_row<b>&nbsp;)</b>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member functions of <i>Mat</i>, <i>Col</i> and <i>SpMat</i>)
      </td>
    </tr>
    <tr>
    <td>&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.shed_col(&nbsp;</b>column_number<b>&nbsp;)</b>
      <br>
      <b>.shed_cols(&nbsp;</b>first_column, last_column<b>&nbsp;)</b>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member functions of <i>Mat</i>, <i>Row</i> and <i>SpMat</i>)
      </td>
    </tr>
    <tr>
    <td>&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.shed_slice(&nbsp;</b>slice_number<b>&nbsp;)</b>
      <br>
      <b>.shed_slices(&nbsp;</b>first_slice, last_slice<b>&nbsp;)</b>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member functions of <i>Cube</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>
Single argument functions:
remove the specified row/column/slice
</li>
<br>
<li>
Two argument functions:
remove the specified range of rows/columns/slices
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,10);

A.shed_row(2);
A.shed_cols(2,4);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#insert">insert_rows/columns/slices</a></li>
<li><a href="#join">join_rows/columns/slices</a></li>
<li><a href="#resize_member">.resize()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#subcube">subcube views</a></li>
<li><a href="http://thesaurus.com/browse/shed"><i>shed</i> in thesaurus.com</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="swap_rows"></a>
<b>
.swap_rows( row1, row2 )
<br>.swap_cols( col1, col2 )
</b>
<ul>
<li>
Member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>SpMat</i>
</li>
<br>
<li>
Swap the contents of specified rows or columns
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(5,5);
X.swap_rows(0,4);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#flip">fliplr() &amp; flipud()</a></li>
<li><a href="#swap">.swap()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="swap"></a>
<b>.swap( X )</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i>
</li>
<br>
<li>
Swap contents with object <i>X</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = zeros&lt;mat&gt;(4,5);
mat B =  ones&lt;mat&gt;(6,7);

A.swap(B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#swap_rows">.swap_rows() &amp; .swap_cols()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="memptr"></a>
<b>.memptr()</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i> 
</li>
<br>
<li>
Obtain a raw pointer to the memory used for storing elements
</li>
<br>
<li>
The function can be used for interfacing with libraries such as <a href="http://www.fftw.org/">FFTW</a>
</li>
<br>
<li>
Data for matrices is stored in a column-by-column order
</li>
<br>
<li>
Data for cubes is stored in a slice-by-slice (matrix-by-matrix) order
</li>
<br>
<li>
<b>Caveat:</b> the pointer becomes invalid after any operation involving a size change or aliasing
</li>
<br>
<li>
<b>Caveat:</b> this function is not recommended for use unless you know what you are doing!
</li>
<br>
<li>
Examples:
<ul>
<pre>
      mat A = randu&lt;mat&gt;(5,5);
const mat B = randu&lt;mat&gt;(5,5);

      double* A_mem = A.memptr();
const double* B_mem = B.memptr();
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#colptr">.colptr()</a></li>
<li><a href="#element_access">element access</a></li>
<li><a href="#iterators_mat">iterators (matrices)</a></li>
<li><a href="#iterators_cube">iterators (cubes)</a></li>
<li><a href="#adv_constructors_mat">advanced constructors (matrices)</a></li>
<li><a href="#adv_constructors_cube">advanced constructors (cubes)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="colptr"></a>
<b>.colptr( col_number )</b>
<ul>
<li>
Member function of <i>Mat</i>
</li>
<br>
<li>
Obtain a raw pointer to the memory used by the specified column
</li>
<br>
<li>
<b>Caveat:</b> the pointer becomes invalid after any operation involving a size change or aliasing
</li>
<br>
<li>
<b>Caveat:</b> this function is not recommended for use unless you know what you are doing
-- it is safer to use <a href="#submat">submatrix views</a> instead
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

double* mem = A.colptr(2);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#memptr">.memptr()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#element_access">element access</a></li>
<li><a href="#iterators_mat">iterators (matrices)</a></li>
<li><a href="#adv_constructors_mat">advanced constructors (matrices)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="iterators_mat"></a>
<b>iterators (matrices &amp; vectors)</b>
<ul>
<li>
STL-style iterators and associated member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>SpMat</i>
</li>
<br>
<li>
Member functions:
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>.begin()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the first element
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.end()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the <i>past-the-end</i> element
      </td>
    </tr>
    <tr>
      <td>
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.begin_row(&nbsp;row_number&nbsp;)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the first element of the specified row
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.end_row(&nbsp;row_number&nbsp;)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the <i>past-the-end</i> element of the specified row
      </td>
    </tr>
    <tr>
      <td>
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.begin_col(&nbsp;col_number&nbsp;)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the first element of the specified column
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.end_col(&nbsp;col_number&nbsp;)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the <i>past-the-end</i> element of the specified column
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
Iterator types:
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>mat::iterator</b>
      <br>
      <b>vec::iterator</b>
      <br>
      <b>rowvec::iterator</b>
      <br>
      <b>sp_mat::iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read/write access to elements
        (which are stored column by column)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>mat::const_iterator</b>
      <br>
      <b>vec::const_iterator</b>
      <br>
      <b>rowvec::const_iterator</b>
      <br>
      <b>sp_mat::const_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read-only access to elements
        (which are stored column by column)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>mat::col_iterator</b>
      <br>
      <b>vec::col_iterator</b>
      <br>
      <b>rowvec::col_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read/write access to the elements of a specific column
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>mat::const_col_iterator</b>
      <br>
      <b>vec::const_col_iterator</b>
      <br>
      <b>rowvec::const_col_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read-only access to the elements of a specific column
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>mat::row_iterator</b>
      <br>
      <b>sp_mat::row_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        rudimentary forward iterator, for read/write access to the elements of a specific row
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>mat::const_row_iterator</b>
      <br>
      <b>sp_mat::const_row_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        rudimentary forward iterator, for read-only access to the elements of a specific row
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>vec::row_iterator</b>
      <br>
      <b>rowvec::row_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read/write access to the elements of a specific row
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>vec::const_row_iterator</b>
      <br>
      <b>rowvec::const_row_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read-only access to the elements of a specific row
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(5,5);


mat::iterator a = X.begin();
mat::iterator b = X.end();

for(mat::iterator i=a; i!=b; ++i)
  {
  cout &lt;&lt; *i &lt;&lt; endl;
  }


mat::col_iterator c = X.begin_col(1);  // start of column 1
mat::col_iterator d = X.end_col(3);    // end of column 3

for(mat::col_iterator i=c; i!=d; ++i)
  {
  cout &lt;&lt; *i &lt;&lt; endl;
  (*i) = 123.0;
  }
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#stl_container_fns">STL-style container functions</a></li>
<li><a href="http://cplusplus.com/reference/std/iterator/">iterator at cplusplus.com</a></li>
<li><a href="#element_access">element access</a></li>
<li><a href="#memptr">.memptr()</a></li>
<li><a href="#colptr">.colptr()</a></li>
<li><a href="#submat">submatrix views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="iterators_cube"></a>
<b>iterators (cubes)</b>
<ul>
<li>
STL-style iterators and associated member functions of <i>Cube</i>
</li>
<br>
<li>
Member functions:
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>.begin()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the first element
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.end()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the <i>past-the-end</i> element
      </td>
    </tr>
    <tr>
      <td>
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.begin_slice( slice_number )</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the first element of the specified slice
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.end_slice( slice_number )</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        iterator referring to the <i>past-the-end</i> element of the specified slice
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
Iterator types:
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>cube::iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterator, for read/write access to elements;
        the elements are ordered slice by slice;
        the elements within each slice are ordered column by column
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>cube::const_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read-only access to elements
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>cube::slice_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterator, for read/write access to the elements of a particular slice;
        the elements are ordered column by column
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>cube::const_slice_iterator</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        random access iterators, for read-only access to the elements of a particular slice
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
cube X = randu&lt;cube&gt;(2,3,4);


cube::iterator a = X.begin();
cube::iterator b = X.end();

for(cube::iterator i=a; i!=b; ++i)
  {
  cout &lt;&lt; *i &lt;&lt; endl;
  }


cube::slice_iterator c = X.begin_slice(1);  // start of slice 1
cube::slice_iterator d = X.end_slice(2);    // end of slice 2

for(cube::slice_iterator i=c; i!=d; ++i)
  {
  cout &lt;&lt; *i &lt;&lt; endl;
  (*i) = 123.0;
  }
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="http://cplusplus.com/reference/std/iterator/">iterator at cplusplus.com</a></li>
<li><a href="#element_access">element access</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="stl_container_fns"></a>
<b>STL-style container functions</b>
<ul>
<li>Member functions that mimic the containers in the C++ Standard Template Library:<br>
<br>

<table style="text-align: left; width: 100%;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>.clear()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        causes an object to have no elements
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.empty()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        returns <i>true</i> if the object has no elements; returns <i>false</i> if the object has one or more elements
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>.size()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
        returns the total number of elements
      </td>
    </tr>
  </tbody>
</table>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
cout &lt;&lt; A.size() &lt;&lt; endl;

A.clear();
cout &lt;&lt; A.empty() &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#iterators_mat">iterators (matrices)</a></li>
<li><a href="#attributes">matrix and vector attributes</a></li>
<li><a href="#is_empty">.is_empty()</a></li>
<li><a href="#reset">.reset()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="t_st_members"></a>
<b>
.t()
<br>.st()
</b>
<ul>
<li>
Member functions of any matrix or vector expression
</li>
<br>
<li>
For real (non-complex) matrix:
<ul>
<li><i>.t()</i> provides a transposed copy of the matrix</li>
<li><i>.st()</i> not applicable</li>
</ul>
</li>
<br>
<li>
For complex matrix:
<ul>
<li><i>.t()</i> provides a Hermitian transpose (ie. the conjugate of the elements is taken during the transpose)</li>
<li><i>.st()</i> provides a transposed copy without taking the conjugate of the elements</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);
mat B = A.t();
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#trans">trans()</a></li>
<li><a href="#flip">fliplr() &amp; flipud()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="i_member"></a>
<b>.i()</b>
<ul>
<li>
Member function of any matrix expression
</li>
<br>
<li>
Provides an inverse of the matrix expression
</li>
<br>
<li>
If the matrix expression is not square sized, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>
If the matrix expression appears to be singular, the output matrix is reset and a <i>std::runtime_error</i> exception is thrown
</li>
<br>
<li>
<b>Caveats:</b>
<ul>
<li>
if matrix <i>A</i> is know to be symmetric positive definite, it's faster to use <a href="#inv_sympd">inv_sympd()</a> instead 
</li>
<li>
to solve a system of linear equations, such as <i>Z&nbsp;=&nbsp;inv(X)*Y</i>,
it is faster and more accurate to use <a href="#solve">solve()</a> instead
</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,4);
mat B = randu&lt;mat&gt;(4,1);

mat X = A.i();

mat Y = (A+A).i();
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#inv">inv()</a></li>
<li><a href="#rcond">rcond()</a></li>
<li><a href="#pinv">pinv()</a></li>
<li><a href="#solve">solve()</a></li>
<li><a href="#spsolve">spsolve()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="min_and_max_member"></a>
<b>.min()</b>
<br><b>.max()</b>
<ul>
<li>
Return the extremum value of any matrix or cube expression
</li>
<br>
<li>
For objects with complex numbers, absolute values are used for comparison
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

double max_val = A.max();
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#index_min_and_index_max_member">.index_min() &amp; .index_max()</a></li>
<li><a href="#min_and_max">min() &amp; max()</a> (standalone functions)</li>
<li><a href="#clamp">clamp()</a>
<li><a href="#running_stat">running_stat</a></li>
<li><a href="#running_stat_vec">running_stat_vec</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="index_min_and_index_max_member"></a>
<b>.index_min()</b>
<br><b>.index_max()</b>
<ul>
<li>
Return the linear index of the extremum value of any matrix or cube expression
</li>
<br>
<li>
For objects with complex numbers, absolute values are used for comparison
</li>
<br>
<li>
The returned index is of type <a href="#uword">uword</a>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

uword i = A.index_max();

double max_val = A(i);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#min_and_max_member">.min() &amp; .max()</a></li>
<li><a href="#index_min_and_index_max_standalone">index_min() &amp; index_max()</a> (standalone functions)</li>
<li><a href="#ind2sub">ind2sub()</a></li>
<li><a href="#sort_index">sort_index()</a></li>
<li><a href="#element_access">element access</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eval_member"></a>
<b>.eval()</b>
<br>
<ul>
<li>
Member function of any matrix or vector expression
</li>
<br>
<li>
Explicitly forces the evaluation of a delayed expression and outputs a matrix
</li>
<br>
<li>
This function should be used sparingly and only in cases where it is absolutely necessary; indiscriminate use can cause slow downs
</li>
<br>
<li>
Examples:
<ul>
<pre>
cx_mat A( randu&lt;mat&gt;(4,4), randu&lt;mat&gt;(4,4) );

real(A).eval().save("A_real.dat", raw_ascii);
imag(A).eval().save("A_imag.dat", raw_ascii);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#Mat">Mat class</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="in_range"></a>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;"><b>.in_range(</b> i <b>)</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range( span(</b>start<b>,</b> end<b>) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td>
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range(</b> row<b>,</b> col <b>)</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range( <font size=-1>span(</b>start_row<b>,</b> end_row<b>), span(</b>start_col<b>,</b> end_col<b>)</font> )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td>
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range(</b> row<b>,</b> col<b>,</b> slice <b>)</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Cube</i> and <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range( <font size=-1>span(</b>start_row<b>,</b> end_row<b>), span(</b>start_col<b>,</b> end_col<b>), span(</b>start_slice<b>,</b> end_slice<b>)</font> )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Cube</i> and <i>field</i>)
      </td>
    </tr>
    <tr>
      <td>
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range(</b> first_row<b>,</b> first_col<b>, size(</b>X<b>) )</b> &nbsp;&nbsp;<i>(X is a matrix or field)</i></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range(</b> first_row<b>,</b> first_col<b>, size(</b>n_rows<b>,</b> n_cols<b>) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i>, <i>field</i>)
      </td>
    </tr>
    <tr>
      <td>
      &nbsp;
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range(</b> first_row<b>,</b> first_col<b>, </b> first_slice<b>, size(</b>Q<b>) )</b> &nbsp;&nbsp;<i>(Q is a cube or field)</i></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Cube</i> and <i>field</i>)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><b>.in_range(</b> first_row<b>,</b> first_col<b>, </b> first_slice<b>, size(</b>n_rows<b>,</b> n_cols<b></b> n_slices<b>) )</b></td>
      <td style="vertical-align: top;"><br>
      </td>
      <td>&nbsp;</td>
      <td style="vertical-align: top;">(member of <i>Cube</i> and <i>field</i>)
      </td>
    </tr>
  </tbody>
</table>
<ul>
<li>Returns <i>true</i> if the given location or span is currently valid
</li>
<br>
<li>Returns <i>false</i> if the object is empty, the location is out of bounds, or the span is out of bounds
</li>
<br>
<li>
Instances of <i>span(a,b)</i> can be replaced by:
<ul>
<li><i>span()</i> or <i>span::all</i>, to indicate the entire range</li>
<li><i>span(a)</i>, to indicate a particular row, column or slice</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);

cout &lt;&lt; A.in_range(0,0) &lt;&lt; endl;  // true
cout &lt;&lt; A.in_range(3,4) &lt;&lt; endl;  // true
cout &lt;&lt; A.in_range(4,5) &lt;&lt; endl;  // false
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#element_access">element access</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#subcube">subcube views</a></li>
<li><a href="#subfield">subfield views</a></li>
<li><a href="#set_size">.set_size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="is_empty"></a>
<b>.is_empty()</b>
<ul>
<li>Returns <i>true</i> if the object has no elements
</li>
<br>
<li>Returns <i>false</i> if the object has one or more elements
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
cout &lt;&lt; A.is_empty() &lt;&lt; endl;

A.reset();
cout &lt;&lt; A.is_empty() &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#is_square">.is_square()</a></li>
<li><a href="#is_vec">.is_vec()</a></li>
<li><a href="#is_finite_member">.is_finite()</a></li>
<li><a href="#reset">.reset()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="is_square"></a>
<b>.is_square()</b>
<ul>
<li>
Member function of <i>Mat</i> and <i>SpMat</i>
</li>
<br>
<li>Returns <i>true</i> if the matrix is square, ie. number of rows is equal to the number of columns
</li>
<br>
<li>Returns <i>false</i> if the matrix is not square
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = randu&lt;mat&gt;(6,7);

cout &lt;&lt; A.is_square() &lt;&lt; endl;
cout &lt;&lt; B.is_square() &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#is_empty">.is_empty()</a></li>
<li><a href="#is_vec">.is_vec()</a></li>
<li><a href="#is_finite_member">.is_finite()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="is_vec"></a>
<b>.is_vec()</b>
<br><b>.is_colvec()</b>
<br><b>.is_rowvec()</b>
<ul>
<li>
Member functions of <i>Mat</i> and <i>SpMat</i>
</li>
<br>

<li>.is_vec():
<ul>
<li>returns <i>true</i> if the matrix can be interpreted as a vector (either column or row vector)
</li>
<li>returns <i>false</i> if the matrix does not have exactly one column or one row
</li>
</ul>
</li>
<br>

<li>.is_colvec():
<ul>
<li>returns <i>true</i> if the matrix can be interpreted as a column vector
</li>
<li>returns <i>false</i> if the matrix does not have exactly one column
</li>
</ul>
</li>
<br>

<li>.is_rowvec():
<ul>
<li>returns <i>true</i> if the matrix can be interpreted as a row vector
</li>
<li>returns <i>false</i> if the matrix does not have exactly one row
</li>
</ul>
</li>
<br>

<li><b>Caveat:</b> do not assume that the vector has elements if these functions return <i>true</i>; it is possible to have an empty vector (eg. 0x1)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(1,5);
mat B = randu&lt;mat&gt;(5,1);
mat C = randu&lt;mat&gt;(5,5);

cout &lt;&lt; A.is_vec() &lt;&lt; endl;
cout &lt;&lt; B.is_vec() &lt;&lt; endl;
cout &lt;&lt; C.is_vec() &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#is_empty">.is_empty()</a></li>
<li><a href="#is_square">.is_square()</a></li>
<li><a href="#is_finite_member">.is_finite()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="is_sorted"></a>
<b>.is_sorted()</b>
<br><b>.is_sorted( sort_direction )</b>
<br><b>.is_sorted( sort_direction, dim )</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Row</i> and <i>Col</i>
</li>
<br>
<li>
If the object is a vector, return a <i>bool</i> indicating whether the elements are sorted
</li>
<br>
<li>
If the object is a matrix, return a <i>bool</i> indicating whether the elements are sorted in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
The <i>sort_direction</i> argument is optional; <i>sort_direction</i> is either <code>"ascend"</code> or <code>"descend"</code>; by default <code>"ascend"</code> is used
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
For matrices and vectors with complex numbers, order is checked via absolute values
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec a = randu&lt;vec&gt;(10);
vec b = sort(a);

bool check1 = a.is_sorted();
bool check2 = b.is_sorted();


mat A = randu&lt;mat&gt;(10,10);

// check whether each column is sorted in descending manner
cout &lt;&lt; A.is_sorted("descend") &lt;&lt; endl;

// check whether each row is sorted in ascending manner
cout &lt;&lt; A.is_sorted("ascend", 1) &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#sort">sort()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="is_finite_member"></a>
<b>.is_finite()</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>
</li>
<br>
<li>Returns <i>true</i> if all elements of the object are finite
</li>
<br>
<li>Returns <i>false</i> if at least one of the elements of the object is non-finite (&plusmn;infinity or NaN)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = randu&lt;mat&gt;(5,5);

B(1,1) = datum::inf;

cout &lt;&lt; A.is_finite() &lt;&lt; endl;
cout &lt;&lt; B.is_finite() &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#is_finite">is_finite()</a> (standalone function)</li>
<li><a href="#find_finite">find_finite()</a></li>
<li><a href="#find_nonfinite">find_nonfinite()</a></li>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="has_inf"></a>
<b>.has_inf()</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>
</li>
<br>
<li>Returns <i>true</i> if at least one of the elements of the object is &plusmn;infinity
</li>
<br>
<li>Returns <i>false</i> otherwise
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = randu&lt;mat&gt;(5,5);

B(1,1) = datum::inf;

cout &lt;&lt; A.has_inf() &lt;&lt; endl;
cout &lt;&lt; B.has_inf() &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#has_nan">.has_nan()</a>
<li><a href="#is_finite_member">.is_finite()</a>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="has_nan"></a>
<b>.has_nan()</b>
<ul>
<li>
Member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>Cube</i>, <i>SpMat</i>
</li>
<br>
<li>Returns <i>true</i> if at least one of the elements of the object is NaN (not-a-number)
</li>
<br>
<li>Returns <i>false</i> otherwise
</li>
<br>
<li>
<b>Caveat:</b>
<code>NaN</code> is not equal to anything, even itself
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = randu&lt;mat&gt;(5,5);

B(1,1) = datum::nan;

cout &lt;&lt; A.has_nan() &lt;&lt; endl;
cout &lt;&lt; B.has_nan() &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#has_inf">.has_inf()</a>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="print"></a>
<b>.print()</b>
<br><b>.print( header )</b>
<br>
<br><b>.print( stream )</b>
<br><b>.print( stream, header )</b>
<ul>
<li>
Member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i>, <i>Cube</i> and <i>field</i>
</li>
<br>
<li>
Print the contents of an object to the <i>std::cout</i> stream (default),
or a user specified stream, with an optional header string
</li>
<br>
<li>
Objects can also be printed using the &lt;&lt; stream operator
</li>
<br>
<li>
Elements of a field can only be printed if there is an associated <i>operator&lt;&lt;</i> function defined
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = randu&lt;mat&gt;(6,6);

A.print();

// print a transposed version of A
A.t().print();

// "B:" is the optional header line
B.print("B:");

cout &lt;&lt; A &lt;&lt; endl;

cout &lt;&lt; "B:" &lt;&lt; endl;
cout &lt;&lt; B &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#raw_print">.raw_print()</a></li>
<li><a href="#save_load_mat">saving &amp; loading matrices</a></li>
<li><a href="#element_initialisation">initialising elements</a></li>
<li><a href="#logging">logging of errors and warnings</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="raw_print"></a>
<b>.raw_print()</b>
<br><b>.raw_print( header )</b>
<br>
<br><b>.raw_print( stream )</b>
<br><b>.raw_print( stream, header )</b>
<ul>
<li>
Member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i> and <i>Cube</i>
</li>
<br>
<li>
Similar to the <a href="#print">.print()</a> member function,
with the difference that no formatting of the output is done; the stream's parameters such as precision, cell width, etc. can be set manually
</li>
<br>
<li>
If the cell width is set to zero, a space is printed between the elements
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

cout.precision(11);
cout.setf(ios::fixed);

A.raw_print(cout, "A:");
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="http://en.cppreference.com/w/cpp/io/ios_base/fmtflags">std::ios_base::fmtflags</a> (cppreference.com)</li>
<li><a href="http://www.cplusplus.com/reference/ios/ios_base/fmtflags/">std::ios_base::fmtflags</a> (cplusplus.com)</li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="save_load_mat"></a>
<b>saving/loading matrices &amp; cubes</b>
<br>
<br>
<table>
<tbody>
<tr>
<td>
<b>.save( name )</b>
<br><b>.save( name, file_type )</b>
<br>
<br><b>.save( stream )</b>
<br><b>.save( stream, file_type )</b>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<b>.load( name )</b>
<br><b>.load( name, file_type )</b>
<br>
<br><b>.load( stream )</b>
<br><b>.load( stream, file_type )</b>
</td>
</tr>
</tbody>
</table>
<br>
<ul>
<li>Member functions of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i></li>
<br>
<li>
Store/retrieve data in a file or stream
<ul>
<li><b>caveat:</b> the stream must be opened in binary mode</li>
</ul>
</li>
<br>
<li>
The default <i>file_type</i> for <i>.save()</i> is <i>arma_binary</i> (see below) 
</li>
<br>
<li>
The default <i>file_type</i> for <i>.load()</i> is <i>auto_detect</i> (see below) 
</li>
<br>
<li>On success, <i>.save()</i> and <i>.load()</i> will return a <i>bool</i> set to <i>true</i></li>
<br>
<li><i>.save()</i> will return a <i>bool</i> set to <i>false</i> if the saving process fails</li>
<br>
<li>
<i>.load()</i> will return a <i>bool</i> set to <i>false</i> if the loading process fails;
additionally, the object will be reset so it has no elements
</li>
<br>
<li>
<i>file_type</i> can be one of the following:
<br>
<br>
<ul>
                  <table style="text-align: left; width: 100%;" border="0" cellpadding="2" cellspacing="2">
                    <tbody>
                      <tr>
                        <td style="vertical-align: top;"><b>auto_detect</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Used by <i>.load()</i> only:
try to automatically detect the file type as one of the formats described below.
This is the default operation.
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>raw_ascii</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Numerical data stored in raw ASCII format, without a header.
The numbers are separated by whitespace.
The number of columns must be the same in each row.
Cubes are loaded as one slice.
Data which was saved in Matlab/Octave using the <i>-ascii</i> option can be read in Armadillo, except for complex numbers.
Complex numbers are stored in standard C++ notation, which is a tuple surrounded by brackets: eg. (1.23,4.56) indicates 1.24&nbsp;+&nbsp;4.56i.
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>raw_binary</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Numerical data stored in machine dependent raw binary format, without a header.
Matrices are loaded to have one column,
while cubes are loaded to have one slice with one column.
The <a href="#reshape_member">.reshape()</a> function can be used to alter the size of the loaded matrix/cube without losing data.
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>arma_ascii</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Numerical data stored in human readable text format, with a simple header to speed up loading.
The header indicates the type of matrix as well as the number of rows and columns.
For cubes, the header additionally specifies the number of slices.
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>arma_binary</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Numerical data stored in machine dependent binary format, with a simple header to speed up loading.
The header indicates the type of matrix as well as the number of rows and columns.
For cubes, the header additionally specifies the number of slices.
<i>arma_binary</i> is the default <i>file_type</i> for <i>.save()</i>
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>csv_ascii</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Numerical data stored in comma separated value (CSV) text format, without a header.
Applicable to <i>Mat</i> only.
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>hdf5_binary</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Numerical data stored in portable HDF5 binary format.
<br>
<b>Caveat</b>:
support for HDF5 must be enabled within Armadillo's <a href="#config_hpp">configuration</a>;
the <i>hdf5.h</i> header file must be available on your system and you will need to link with the hdf5 library (eg. -lhdf5)
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>hdf5_binary_trans</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
As per <i>hdf5_binary</i>, but save/load the data with columns transposed to rows (and vice versa)
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>pgm_binary</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Image data stored in Portable Gray Map (PGM) format.
Applicable to <i>Mat</i> only.
Saving <i>int</i>, <i>float</i> or <i>double</i> matrices is a lossy operation, as each element is copied and converted to an 8 bit representation.
As such the matrix should have values in the [0,255] interval, otherwise the resulting image may not display correctly.
<br>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>ppm_binary</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
Image data stored in Portable Pixel Map (PPM) format.
Applicable to <i>Cube</i> only.
Saving <i>int</i>, <i>float</i> or <i>double</i> matrices is a lossy operation, as each element is copied and converted to an 8 bit representation.
As such the cube/field should have values in the [0,255] interval, otherwise the resulting image may not display correctly.
                        </td>
                      </tr>
                    </tbody>
                  </table>
</ul>
</li>
<br>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

A.save("A1.mat");  // default save format is arma_binary

A.save("A2.mat", arma_ascii);  // force saving in arma_ascii format

mat B;
B.load("A1.mat");  // automatically detect format type

mat C;
C.load("A2.mat", arma_ascii);  // force loading in arma_ascii format


// example of testing for success
mat D;
bool status = D.load("A2.mat");

if(status == true)
  {
  cout &lt;&lt; "loaded okay" &lt;&lt; endl;
  }
else
  {
  cout &lt;&lt; "problem with loading" &lt;&lt; endl;
  }
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#save_load_field">saving/loading fields</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="save_load_field"></a>
<b>saving/loading fields</b>
<br>
<br>
<table>
<tbody>
<tr>
<td>
<b>.save( name )</b>
<br><b>.save( name, file_type )</b>
<br>
<br><b>.save( stream )</b>
<br><b>.save( stream, file_type )</b>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<b>.load( name )</b>
<br><b>.load( name, file_type )</b>
<br>
<br><b>.load( stream )</b>
<br><b>.load( stream, file_type )</b>
</td>
</tr>
</tbody>
</table>
<br>
<ul>
<li>
Store/retrieve data in a file or stream
<ul>
<li><b>caveat:</b> the stream must be opened in binary mode</li>
</ul>
</li>
<br>
<li>On success, <i>.save()</i> and <i>.load()</i> will return a <i>bool</i> set to <i>true</i></li>
<br>
<li><i>.save()</i> will return a <i>bool</i> set to <i>false</i> if the saving process fails</li>
<br>
<li>
<i>.load()</i> will return a <i>bool</i> set to <i>false</i> if the loading process fails;
additionally, the field will be reset so it has no elements
</li>
<br>
<li>
Fields with objects of type <i>std::string</i> are saved and loaded as raw text files.
The text files do not have a header.
Each string is separated by a whitespace.
<i>load()</i> will only accept text files that have the same number of strings on each line.
The strings can have variable lengths.
</li>
<br>
<li>
Other than storing string fields as text files, the following file formats are supported:
<br>
<br>
<ul>
                  <table style="text-align: left; width: 100%;" border="0" cellpadding="2" cellspacing="2">
                    <tbody>
                      <tr>
                        <td style="vertical-align: top;"><b>auto_detect</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
<br>
<li>
<i>.load()</i>: try to automatically detect the field format type as one of the formats described below;
this is the default operation.
</li>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>arma_binary</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
<br>
<li>
Objects are stored in machine dependent binary format
<li>
Default type for fields of type <i>Mat</i>, <i>Col</i>, <i>Row</i> or <i>Cube</i>
</li>
<li>
Only applicable to fields of type <i>Mat</i>, <i>Col</i>, <i>Row</i> or <i>Cube</i>
</li>
<br>
                        </td>
                      </tr>
                      <tr>
                        <td style="vertical-align: top;"><b>ppm_binary</b></td>
                        <td style="vertical-align: top;"><br>
                        </td>
                        <td style="vertical-align: top;">
<br>
<li>
Image data stored in Portable Pixmap Map (PPM) format.
</li>
<li>
Only applicable to fields of type <i>Mat</i>, <i>Col</i> or <i>Row</i>
</li>
<li>
<i>.load()</i>: loads the specified image and stores the red, green and blue components as three separate matrices;
the resulting field is comprised of the three matrices,
with the red, green and blue components in the first, second and third matrix, respectively
</li>
<li>
<i>.save()</i>: saves a field with exactly three matrices of equal size as an image;
it is assumed that the red, green and blue components are stored in the first, second and third matrix, respectively;
saving <i>int</i>, <i>float</i> or <i>double</i> matrices is a lossy operation,
as each matrix element is copied and converted to an 8 bit representation
</li>

                        </td>
                      </tr>
                    </tbody>
                  </table>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#save_load_mat">saving/loading matrices and cubes</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Generated Vectors/Matrices/Cubes</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eye_standalone"></a>
<b>eye( n_rows, n_cols )</b>
<br>
<b>eye( size(X) )</b>
<ul>
<li>
Generate a matrix with the elements along the main diagonal set to one
and off-diagonal elements set to zero
</li>
<br>
<li>
An identity matrix is generated when <i>n_rows</i> = <i>n_cols</i>
</li>
<br>
<li>
Usage:
<ul>
<li><i>matrix_type</i> X = eye&lt;<i>matrix_type</i>&gt;( n_rows, n_cols )</li>
<li><i>matrix_type</i> Y = eye&lt;<i>matrix_type</i>&gt;( size(X) )</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = eye&lt;mat&gt;(5,5);  // or:  mat A(5,5,fill::eye);

mat B = 123.0 * eye&lt;mat&gt;(5,5);

mat C = eye&lt;mat&gt;( size(B) );
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#eye_member">.eye()</a> (member function of Mat)</li>
<li><a href="#diag">.diag()</a></li>
<li><a href="#ones_standalone">ones()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="#diagvec">diagvec()</a></li>
<li><a href="#speye">speye()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="linspace"></a>
<b>linspace( start, end )</b>
<br><b>linspace( start, end, N )</b>
<ul>
<li>
Generate a vector with <i>N</i> elements;
the values of the elements are linearly spaced from <i>start</i> to (and including) <i>end</i>
</li>
<br>
<li>
The argument <i>N</i> is optional; by default <i>N = 100</i>
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = linspace&lt;<i>vector_type</i>&gt;(start, end, N)</li>
</ul>
</li>
<br>
<li>
<b>Caveat:</b> if N &le; 1, the generated vector will have a size of one, with the single element equal to <i>end</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec a = linspace&lt;vec&gt;(0, 5, 6);

vec b = linspace&lt;vec&gt;(5, 0, 6);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#regspace">regspace()</a></li>
<li><a href="#logspace">logspace()</a></li>
<li><a href="#ones_standalone">ones()</a></li>
<li><a href="#interp1">interp1()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="logspace"></a>
<b>logspace( A, B )</b>
<br><b>logspace( A, B, N )</b>
<ul>
<li>
Generate a vector with <i>N</i> elements;
the values of the elements are logarithmically spaced from <i>10<sup><font size=-1>A</font></sup></i> to (and including) <i>10<sup><font size=-1>B</font></sup></i>
</li>
<br>
<li>
The argument <i>N</i> is optional; by default <i>N = 50</i>
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = logspace&lt;<i>vector_type</i>&gt;(A, B, N)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec a = logspace&lt;vec&gt;(0, 5, 6);

vec b = logspace&lt;vec&gt;(5, 0, 6);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#linspace">linspace()</a></li>
<li><a href="#regspace">regspace()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="ones_standalone"></a>
<b>
ones( n_elem )
<br>ones( n_rows, n_cols )
<br>ones( n_rows, n_cols, n_slices )
<br>ones( size(X) )
</b>
<ul>
<li>
Generate a vector, matrix or cube with all elements set to one
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = ones&lt;<i>vector_type</i>&gt;( n_elem )</li>
<li><i>matrix_type</i> X = ones&lt;<i>matrix_type</i>&gt;( n_rows, n_cols )</li>
<li><i>matrix_type</i> Y = ones&lt;<i>matrix_type</i>&gt;( size(X) )</li>
<li><i>cube_type</i> Q = ones&lt;<i>cube_type</i>&gt;( n_rows, n_cols, n_slices )</li>
<li><i>cube_type</i> R = ones&lt;<i>cube_type</i>&gt;( size(Q) )</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec  v = ones&lt;vec&gt;(10);
uvec u = ones&lt;uvec&gt;(11);
mat  A = ones&lt;mat&gt;(5,6);
cube Q = ones&lt;cube&gt;(5,6,7);

mat  B = 123.0 * ones&lt;mat&gt;(5,6);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#ones_member">.ones()</a> (member function of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i>)</li>
<li><a href="#eye_standalone">eye()</a></li>
<li><a href="#linspace">linspace()</a></li>
<li><a href="#regspace">regspace()</a></li>
<li><a href="#zeros_standalone">zeros()</a></li>
<li><a href="#randu_randn_standalone">randu() &amp; randn()</a></li>
<li><a href="#spones">spones()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="randi"></a>
<b>randi( n_elem )</b> 
<br><b>randi( n_elem, distr_param(a,b) )</b> 
<br>
<br><b>randi( n_rows, n_cols )</b>
<br><b>randi( n_rows, n_cols, distr_param(a,b) )</b>
<br>
<br><b>randi( n_rows, n_cols, n_slices )</b>
<br><b>randi( n_rows, n_cols, n_slices, distr_param(a,b) )</b>
<br>
<br><b>randi( size(X) )</b>
<br><b>randi( size(X), distr_param(a,b) )</b>
<ul>
<li>
Generate a vector, matrix or cube with the elements set to random <b>integer</b> values in the [a,b] interval
</li>
<br>
<li>The default distribution parameters are <i>a=0</i> and <i>b=maximum_int</i>
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = randi&lt;<i>vector_type</i>&gt;( n_elem )</li>
<li><i>vector_type</i> v = randi&lt;<i>vector_type</i>&gt;( n_elem, distr_param(a,b) )</li>
<br>
<li><i>matrix_type</i> X = randi&lt;<i>matrix_type</i>&gt;( n_rows, n_cols )</li>
<li><i>matrix_type</i> X = randi&lt;<i>matrix_type</i>&gt;( n_rows, n_cols, distr_param(a,b) )</li>
<li><i>matrix_type</i> Y = randi&lt;<i>matrix_type</i>&gt;( size(X) )</li>
<li><i>matrix_type</i> Y = randi&lt;<i>matrix_type</i>&gt;( size(X), distr_param(a,b) )</li>
<br>
<li><i>cube_type</i> Q = randi&lt;<i>cube_type</i>&gt;( n_rows, n_cols, n_slices )</li>
<li><i>cube_type</i> Q = randi&lt;<i>cube_type</i>&gt;( n_rows, n_cols, n_slices, distr_param(a,b) )</li>
<li><i>cube_type</i> R = randi&lt;<i>cube_type</i>&gt;( size(Q) )</li>
<li><i>cube_type</i> R = randi&lt;<i>cube_type</i>&gt;( size(Q), distr_param(a,b) )</li>
</ul>
</li>
<br>
<li>
To change the RNG seed, use <i>arma_rng::set_seed(value)</i> or <i>arma_rng::set_seed_random()</i> functions
</li>
<br>
<li>
<b>Caveat:</b> to generate a continuous distribution with floating point values (ie. <i>float</i> or <i>double</i>),
use <a href="#randu_randn_standalone">randu()</a> or <a href="#randu_randn_standalone">randn()</a> instead 
</li>
<br>
<li>
Examples:
<ul>
<pre>
imat A = randi&lt;imat&gt;(5, 6);

imat A = randi&lt;imat&gt;(6, 7, distr_param(-10, +20));

arma_rng::set_seed_random();  // set the seed to a random value
</pre>
</ul>
</li>
<li>See also:
<ul>
<li><a href="#randu_randn_standalone">randu() &amp; randn()</a></li>
<li><a href="#randg">randg()</a></li>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#ones_standalone">ones()</a></li>
<li><a href="#zeros_standalone">zeros()</a></li>
<li><a href="#shuffle">shuffle()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="randu_randn_standalone"></a>
<b>randu( n_elem )</b> 
<br><b>randu( n_rows, n_cols )</b>
<br><b>randu( n_rows, n_cols, n_slices )</b>
<br><b>randu( size(X) )</b>
<br>
<br><b>randn( n_elem )</b>
<br><b>randn( n_rows, n_cols )</b>
<br><b>randn( n_rows, n_cols, n_slices )</b>
<br><b>randn( size(X) )</b>
<ul>
<li>
Generate a vector, matrix or cube with the elements set to random <b>floating point</b> values
</li>
<br>
<li><i>randu()</i> uses a uniform distribution in the [0,1] interval
</li>
<br>
<li><i>randn()</i> uses a normal/Gaussian distribution with zero mean and unit variance
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = randu&lt;<i>vector_type</i>&gt;( n_elem )</li>
<li><i>matrix_type</i> X = randu&lt;<i>matrix_type</i>&gt;( n_rows, n_cols )</li>
<li><i>matrix_type</i> Y = randu&lt;<i>matrix_type</i>&gt;( size(X) )</li>
<li><i>cube_type</i> Q = randu&lt;<i>cube_type</i>&gt;( n_rows, n_cols, n_slices )</li>
<li><i>cube_type</i> R = randu&lt;<i>cube_type</i>&gt;( size(Q) )</li>
</ul>
</li>
<br>
<li>
To change the RNG seed, use <i>arma_rng::set_seed(value)</i> or <i>arma_rng::set_seed_random()</i> functions
</li>
<br>
<li>
<b>Caveat:</b> to generate a matrix with random integer values instead of floating point values,
use <a href="#randi">randi()</a> instead 
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec  v = randu&lt;vec&gt;(5);
mat  A = randu&lt;mat&gt;(5,6);
cube Q = randu&lt;cube&gt;(5,6,7);

arma_rng::set_seed_random();  // set the seed to a random value
</pre>
</ul>
</li>
<li>See also:
<ul>
<li><a href="#randu_randn_member">.randu() &amp; .randn()</a> (member functions)</li>
<li><a href="#randi">randi()</a></li>
<li><a href="#randg">randg()</a></li>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#ones_standalone">ones()</a></li>
<li><a href="#zeros_standalone">zeros()</a></li>
<li><a href="#shuffle">shuffle()</a></li>
<li><a href="#sprandu_sprandn">sprandu() &amp; sprandn()</a></li>
<li><a href="#size">size()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)">uniform distribution in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Normal_distribution">normal distribution in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="randg"></a>
<b>randg( n_elem )</b> 
<br><b>randg( n_elem, distr_param(a,b) )</b> 
<br>
<br><b>randg( n_rows, n_cols )</b>
<br><b>randg( n_rows, n_cols, distr_param(a,b) )</b>
<br>
<br><b>randg( n_rows, n_cols, n_slices )</b>
<br><b>randg( n_rows, n_cols, n_slices, distr_param(a,b) )</b>
<br>
<br><b>randg( size(X) )</b>
<br><b>randg( size(X), distr_param(a,b) )</b>
<ul>
<li>
Generate a vector, matrix or cube with the elements set to random values from a gamma distribution:
<ul>
<table style="text-align: left;" border="0" cellpadding="0" cellspacing="0">
  <tbody>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top;"><i>x<sup>&thinsp;a-1</sup> exp(&thinsp;-x&thinsp;/&thinsp;b&thinsp;)</i></td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>p<font size=+1>(</font>x|a,b<font size=+1>)</font></i></td><td style="vertical-align: top;">&nbsp;<font size=+1>=</font>&nbsp;</td><td style="vertical-align: top;"><font size=+1><b><hr></b></font></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top; text-align: center;"><i>b<sup>&thinsp;a</sup>&nbsp;&Gamma;(a)</i></td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>The default distribution parameters are <i>a=1</i> and <i>b=1</i>
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = randg&lt;<i>vector_type</i>&gt;( n_elem )</li>
<li><i>vector_type</i> v = randg&lt;<i>vector_type</i>&gt;( n_elem, distr_param(a,b) )</li>
<br>
<li><i>matrix_type</i> X = randg&lt;<i>matrix_type</i>&gt;( n_rows, n_cols )</li>
<li><i>matrix_type</i> X = randg&lt;<i>matrix_type</i>&gt;( n_rows, n_cols, distr_param(a,b) )</li>
<li><i>matrix_type</i> Y = randg&lt;<i>matrix_type</i>&gt;( size(X) )</li>
<li><i>matrix_type</i> Y = randg&lt;<i>matrix_type</i>&gt;( size(X), distr_param(a,b) )</li>
<br>
<li><i>cube_type</i> Q = randg&lt;<i>cube_type</i>&gt;( n_rows, n_cols, n_slices )</li>
<li><i>cube_type</i> Q = randg&lt;<i>cube_type</i>&gt;( n_rows, n_cols, n_slices, distr_param(a,b) )</li>
<li><i>cube_type</i> R = randg&lt;<i>cube_type</i>&gt;( size(Q) )</li>
<li><i>cube_type</i> R = randg&lt;<i>cube_type</i>&gt;( size(Q), distr_param(a,b) )</li>
</ul>
</li>
<br>
<li>
<b>Caveat:</b> <i>randg()</i> is available only when using a C++11 compiler
</li>
<br>
<li>
To change the RNG seed, use <i>arma_rng::set_seed(value)</i> or <i>arma_rng::set_seed_random()</i> functions
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec v = randg&lt;vec&gt;(100, distr_param(2,1));

mat X = randg&lt;mat&gt;(10, 10, distr_param(2,1));
</pre>
</ul>
</li>
<li>See also:
<ul>
<li><a href="#randu_randn_standalone">randu() &amp; randn()</a></li>
<li><a href="#randi">randi()</a></li>
<li><a href="#imbue">.imbue()</a></li>
<li><a href="#size">size()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Gamma_distribution">gamma distribution in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="regspace"></a>
<b>regspace( start, end )</b>
<br><b>regspace( start, delta, end )</b>
<ul>
<li>
Generate a vector with regularly spaced elements:
<br>[&nbsp;&nbsp;<i>(start&nbsp;+&nbsp;0*delta)</i>,&nbsp;&nbsp;<i>(start&nbsp;+&nbsp;1*delta)</i>,&nbsp;&nbsp;<i>(start&nbsp;+&nbsp;2*delta)</i>,&nbsp;&nbsp;<i>&#8943;</i>,&nbsp;&nbsp;<i>(start&nbsp;+&nbsp;M*delta)</i>&nbsp;&nbsp;]
<br>where <i>M&nbsp;=&nbsp;floor((end-start)/delta)</i>, so that <i>(start&nbsp;+&nbsp;M*delta)&nbsp;&le;&nbsp;end</i>
</li>
<br>
<li>
Similar in operation to the Matlab/Octave colon operator, ie. <i>start:end</i>&nbsp;&nbsp;and&nbsp;&nbsp;<i>start:delta:end</i>
</li>
<br>
<li>
If <i>delta</i> is not specified:
<ul>
<li><i>delta</i>&nbsp;=&nbsp;+1,       if <i>start&nbsp;&le;&nbsp;end</i></li>
<li><i>delta</i>&nbsp;=&nbsp;&minus;1, if <i>start&nbsp;&gt;&nbsp;end</i> &nbsp; (<b>caveat:</b> this is different to Matlab/Octave)</li>
</ul>
</li>
<br>
<li>
An empty vector is generated when one of the following conditions is true:
<ul>
<li><i>start&nbsp;&lt;&nbsp;end</i>, and <i>delta&nbsp;&lt;&nbsp;0</i></li>
<li><i>start&nbsp;&gt;&nbsp;end</i>, and <i>delta&nbsp;&gt;&nbsp;0</i></li>
<li><i>delta&nbsp;=&nbsp;0</i></li>
</ul>
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = regspace&lt;<i>vector_type</i>&gt;(start, end)</li>
<li><i>vector_type</i> v = regspace&lt;<i>vector_type</i>&gt;(start, delta, end)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
 vec a = regspace&lt; vec&gt;(0,  9);       // 0,  1, ...,   9
uvec b = regspace&lt;uvec&gt;(2,  2,  10);  // 2,  4, ...,  10
ivec c = regspace&lt;ivec&gt;(0, -1, -10);  // 0, -1, ..., -10
</pre>
</ul>
</li>
<br>
<li>
<b>Caveat:</b> do not use <i>regspace()</i> to specify ranges for contiguous submatrix views; use <a href="#submat">span()</a> instead
</li>
<br>
<li>
See also:
<ul>
<li><a href="#linspace">linspace()</a></li>
<li><a href="#logspace">logspace()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="speye"></a>
<b>speye( n_rows, n_cols )</b>
<br>
<b>speye( size(X) )</b>
<ul>
<li>
Generate a sparse matrix with the elements along the main diagonal set to one
and off-diagonal elements set to zero
</li>
<br>
<li>
An identity matrix is generated when <i>n_rows</i> = <i>n_cols</i>
</li>
<br>
<li>
Usage:
<ul>
<li><i>sparse_matrix_type</i> X = speye&lt;<i>sparse_matrix_type</i>&gt;( n_rows, n_cols )</li>
<li><i>sparse_matrix_type</i> Y = speye&lt;<i>sparse_matrix_type</i>&gt;( size(X) )</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
sp_mat A = speye&lt;sp_mat&gt;(5,5);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#spones">spones()</a></li>
<li><a href="#sprandu_sprandn">sprandu() &amp; sprandn()</a></li>
<li><a href="#eye_standalone">eye()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="spones"></a>
<b>spones( A )</b>
<ul>
<li>
Generate a sparse matrix with the same structure as sparse matrix <i>A</i>,
but with the non-zero elements set to one
</li>
<br>
<li>
Examples:
<ul>
<pre>
sp_mat A = sprandu&lt;sp_mat&gt;(100, 200, 0.1);

sp_mat B = spones(A);
</pre>
</ul>
</li>
<li>See also:
<ul>
<li><a href="#speye">speye()</a></li>
<li><a href="#sprandu_sprandn">sprandu() &amp; sprandn()</a></li>
<li><a href="#ones_standalone">ones()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="sprandu_sprandn"></a>
<b>sprandu( n_rows, n_cols, density )</b>
<br><b>sprandn( n_rows, n_cols, density )</b>
<br>
<br><b>sprandu( size(X), density )</b>
<br><b>sprandn( size(X), density )</b>
<ul>
<li>
Generate a sparse matrix with the non-zero elements set to random values
</li>
<br>
<li>
The <i>density</i> argument specifies the percentage of non-zero elements; it must be in the [0,1] interval
</li>
<br>
<li><i>sprandu()</i> uses a uniform distribution in the [0,1] interval
</li>
<br>
<li><i>sprandn()</i> uses a normal/Gaussian distribution with zero mean and unit variance
</li>
<br>
<li>
Usage:
<ul>
<li><i>sparse_matrix_type</i> X = sprandu&lt;<i>sparse_matrix_type</i>&gt;( n_rows, n_cols, density )</li>
<li><i>sparse_matrix_type</i> Y = sprandu&lt;<i>sparse_matrix_type</i>&gt;( size(X), density )</li>
</ul>
</li>
<br>
<li>
To change the RNG seed, use <i>arma_rng::set_seed(value)</i> or <i>arma_rng::set_seed_random()</i> functions
</li>
<br>
<li>
Examples:
<ul>
<pre>
sp_mat A = sprandu&lt;sp_mat&gt;(100, 200, 0.1);
</pre>
</ul>
</li>
<li>See also:
<ul>
<li><a href="#speye">speye()</a></li>
<li><a href="#spones">spones()</a></li>
<li><a href="#randu_randn_standalone">randu() &amp; randn()</a></li>
<li><a href="#size">size()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)">uniform distribution in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Normal_distribution">normal distribution in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="toeplitz"></a>
<b>toeplitz( A )</b>
<br><b>toeplitz( A, B )</b>
<br><b>circ_toeplitz( A )</b>
<ul>
<li>
toeplitz(): generate a Toeplitz matrix, with the first column specified by <i>A</i>, and (optionally) the first row specified by <i>B</i>
</li>
<br>
<li>
circ_toeplitz(): generate a circulant Toeplitz matrix
</li>
<br>
<li>
A and B must be vectors
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec A = randu&lt;vec&gt;(5);
mat X = toeplitz(A);
mat Y = circ_toeplitz(A);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="http://mathworld.wolfram.com/ToeplitzMatrix.html">Toeplitz matrix in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz matrix in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="zeros_standalone"></a>
<b>zeros( n_elem )</b>
<br><b>zeros( n_rows, n_cols )</b>
<br><b>zeros( n_rows, n_cols, n_slices )</b>
<br><b>zeros( size(X) )</b>
<ul>
<li>
Generate a vector, matrix or cube with the elements set to zero
</li>
<br>
<li>
Usage:
<ul>
<li><i>vector_type</i> v = zeros&lt;<i>vector_type</i>&gt;( n_elem )</li>
<li><i>matrix_type</i> X = zeros&lt;<i>matrix_type</i>&gt;( n_rows, n_cols )</li>
<li><i>matrix_type</i> Y = zeros&lt;<i>matrix_type</i>&gt;( size(X) )</li>
<li><i>cube_type</i> Q = zeros&lt;<i>cube_type</i>&gt;( n_rows, n_cols, n_slices )</li>
<li><i>cube_type</i> R = zeros&lt;<i>cube_type</i>&gt;( size(Q) )</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec  v = zeros&lt;vec&gt;(10);
uvec u = zeros&lt;uvec&gt;(11);
mat  A = zeros&lt;mat&gt;(5,6);
cube Q = zeros&lt;cube&gt;(5,6,7);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#zeros_member">.zeros()</a> (member function of <i>Mat</i>, <i>Col</i>, <i>Row</i>, <i>SpMat</i> and <i>Cube</i>)</li>
<li><a href="#ones_member">.ones()</a> (member function of <i>Mat</i>, <i>Col</i>, <i>Row</i> and <i>Cube</i>)</li>
<li><a href="#ones_standalone">ones()</a></li>
<li><a href="#randu_randn_standalone">randu() &amp; randn()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Functions of Vectors/Matrices/Cubes</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="abs"></a>
<b>abs( X )</b>
<ul>
<li>
Obtain the magnitude of each element
</li>
<br>
<li>
Usage for non-complex X:
<ul>
<li>Y = abs(X)</li>
<li>X and Y must have the same matrix type or cube type, such as <i>mat</i> or <i>cube</i></li>
</ul>
</li>
<br>
<li>
Usage for complex X:
<ul>
<li><i>real_object_type</i> Y = abs(X)</li>
<li>The type of X must be a complex matrix or complex cube, such as <i>cx_mat</i> or <i>cx_cube</i></li>
<li>The type of Y must be the real counterpart to the type of X; if X has the type <i>cx_mat</i>, then the type of Y must be <i>mat</i>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = abs(A); 

cx_mat X = randu&lt;cx_mat&gt;(5,5);
   mat Y = abs(X);
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#arg">arg()</a></li>
<li><a href="#conj">conj()</a></li>
<li><a href="#imag_real">imag() / real()</a></li>
<li><a href="#conv_to">conv_to()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="accu"></a>
<b>accu( X )</b>
<ul>
<li>
Accumulate (sum) all elements of a vector, matrix or cube
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5, 6, fill::randu);
mat B(5, 6, fill::randu);

double x = accu(A);

double y = accu(A % B);

// accu(A % B) is a "multiply-and-accumulate" operation
// as operator % performs element-wise multiplication
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#sum">sum()</a></li>
<li><a href="#cumsum">cumsum()</a></li>
<li><a href="#trace">trace()</a></li>
<li><a href="#stats_fns">mean()</a></li>
<li><a href="#dot">dot()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="all"></a>
<b>all( V )</b>
<br><b>all( X )</b>
<br><b>all( X, dim )</b>
<ul>
<li>For vector <i>V</i>, return <i>true</i> if all elements of the vector are non-zero or satisfy a relational condition</li>
<br>
<li>
For matrix <i>X</i> and
<ul>
<li>
<i>dim=0</i>, return a row vector (of type <a href="#Row">urowvec</a> or <a href="#Mat">umat</a>),
with each element (0 or 1) indicating whether the corresponding column of <i>X</i> has all non-zero elements</li>
<li>
<i>dim=1</i>, return a column vector (of type <a href="#Col">ucolvec</a> or <a href="#Mat">umat</a>),
with each element (0 or 1) indicating whether the corresponding row of <i>X</i> has all non-zero elements</li>
</ul>
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Relational operators can be used instead of <i>V</i> or <i>X</i>, eg.&nbsp;<i>A&nbsp;&gt;&nbsp;0.5</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec V = randu&lt;vec&gt;(10);
mat X = randu&lt;mat&gt;(5,5);


// status1 will be set to true if vector V has all non-zero elements
bool status1 = all(V);

// status2 will be set to true if vector V has all elements greater than 0.5
bool status2 = all(V &gt; 0.5);

// status3 will be set to true if matrix X has all elements greater than 0.6;
// note the use of vectorise()
bool status3 = all(vectorise(X) &gt; 0.6);

// generate a row vector indicating which columns of X have all elements greater than 0.7
umat A = all(X &gt; 0.7);

</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#any">any()</a></li>
<li><a href="#approx_equal">approx_equal()</a></li>
<li><a href="#find">find()</a></li>
<li><a href="#conv_to">conv_to()</a> (convert between matrix/vector types)</li>
<li><a href="#vectorise">vectorise()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="any"></a>
<b>any( V )</b>
<br><b>any( X )</b>
<br><b>any( X, dim )</b>
<ul>
<li>For vector <i>V</i>, return <i>true</i> if any element of the vector is non-zero or satisfies a relational condition</li>
<br>
<li>
For matrix <i>X</i> and
<ul>
<li>
<i>dim=0</i>, return a row vector (of type <a href="#Row">urowvec</a> or <a href="#Mat">umat</a>),
with each element (0 or 1) indicating whether the corresponding column of <i>X</i> has any non-zero elements</li>
<li>
<i>dim=1</i>, return a column vector (of type <a href="#Col">ucolvec</a> or <a href="#Mat">umat</a>),
with each element (0 or 1) indicating whether the corresponding row of <i>X</i> has any non-zero elements</li>
</ul>
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Relational operators can be used instead of <i>V</i> or <i>X</i>, eg.&nbsp;<i>A&nbsp;&gt;&nbsp;0.9</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec V = randu&lt;vec&gt;(10);
mat X = randu&lt;mat&gt;(5,5);


// status1 will be set to true if vector V has any non-zero elements
bool status1 = any(V);

// status2 will be set to true if vector V has any elements greater than 0.5
bool status2 = any(V &gt; 0.5);

// status3 will be set to true if matrix X has any elements greater than 0.6;
// note the use of vectorise()
bool status3 = any(vectorise(X) &gt; 0.6);

// generate a row vector indicating which columns of X have elements greater than 0.7
umat A = any(X &gt; 0.7);

</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#all">all()</a></li>
<li><a href="#approx_equal">approx_equal()</a></li>
<li><a href="#find">find()</a></li>
<li><a href="#conv_to">conv_to()</a> (convert between matrix/vector types)</li>
<li><a href="#vectorise">vectorise()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="approx_equal"></a>
<b>approx_equal( A, B, method, tol )</b>
<br><b>approx_equal( A, B, method, abs_tol, rel_tol )</b>
<ul>
<li>
Return <i>true</i> if all corresponding elements in <i>A</i> and <i>B</i> are approximately equal
</li>
<br>
<li>
Return <i>false</i> if any of the corresponding elements in <i>A</i> and <i>B</i> are not approximately equal, or if <i>A</i> and <i>B</i> have different dimensions
</li>
<br>
<li>
The argument <i>method</i> controls how the approximate equality is determined; it is one of:
<ul>
<table>
<tbody>
<tr><td><code>"absdiff"</code></td><td>&nbsp;&mapsto;&nbsp;</td><td>scalars <i>x</i> and <i>y</i> are considered equal if <i>|x&nbsp;&minus;&nbsp;y|&nbsp;&le;&nbsp;tol</i></td></tr>
<tr><td><code>"reldiff"</code></td><td>&nbsp;&mapsto;&nbsp;</td><td>scalars <i>x</i> and <i>y</i> are considered equal if <i>|x&nbsp;&minus;&nbsp;y|&nbsp;/&nbsp;max(&nbsp;|x|,&nbsp;|y|&nbsp;)&nbsp;&le;&nbsp;tol</i></td></tr>
<tr><td><code>"both"</code></td><td>&nbsp;&mapsto;&nbsp;</td><td>scalars <i>x</i> and <i>y</i> are considered equal if <i>|x&nbsp;&minus;&nbsp;y|&nbsp;&le;&nbsp;abs_tol</i>&nbsp;&nbsp;<b>or</b>&nbsp;&nbsp;<i>|x&nbsp;&minus;&nbsp;y|&nbsp;/&nbsp;max(&nbsp;|x|,&nbsp;|y|&nbsp;)&nbsp;&le;&nbsp;rel_tol</i></td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = A + 0.001;

bool same1 = approx_equal(A, B, "absdiff", 0.002);


mat C = 1000 * randu&lt;mat&gt;(5,5);
mat D = C + 1;

bool same2 = approx_equal(C, D, "reldiff", 0.1);

bool same3 = approx_equal(C, D, "both", 2, 0.1);
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#all">all()</a></li>
<li><a href="#any">any()</a></li>
<li><a href="#find">find()</a></li>
<li><a href="#operators">relational operators</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="arg"></a>
<b>arg( X )</b>
<ul>
<li>
Obtain the phase angle (in radians) of each element
</li>
<br>
<li>
Usage for non-complex X:
<ul>
<li>Y = arg(X)</li>
<li>X and Y must have the same matrix type or cube type, such as <i>mat</i> or <i>cube</i></li>
<li>non-complex elements are treated as complex elements with zero imaginary component
</li>
</ul>
</li>
<br>
<li>
Usage for complex X:
<ul>
<li><i>real_object_type</i> Y = arg(X)</li>
<li>The type of X must be a complex matrix or complex cube, such as <i>cx_mat</i> or <i>cx_cube</i></li>
<li>The type of Y must be the real counterpart to the type of X; if X has the type <i>cx_mat</i>, then the type of Y must be <i>mat</i>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
cx_mat A = randu&lt;mat&gt;(5,5);
   mat B = arg(A); 
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#abs">abs()</a></li>
<li><a href="#trig_fns">atan2()</a></li>
<li><a href="https://en.wikipedia.org/wiki/Argument_(complex_analysis)">Argument (complex analysis) in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/ComplexArgument.html">Complex Argument in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="as_scalar"></a>
<b>as_scalar( expression )</b>
<ul>
<li>
Evaluate an expression that results in a 1x1 matrix,
followed by converting the 1x1 matrix to a pure scalar
</li>
<br>
<li>
Optimised expression evaluations are automatically used when a binary or trinary expression is given (ie. 2 or 3 terms)
</li>
<br>
<li>
Examples:
<ul>
<pre>
rowvec r = randu&lt;rowvec&gt;(5);
colvec q = randu&lt;colvec&gt;(5);
mat    X = randu&lt;mat&gt;(5,5);

// examples of expressions which have optimised implementations

double a = as_scalar(r*q);
double b = as_scalar(r*X*q);
double c = as_scalar(r*diagmat(X)*q);
double d = as_scalar(r*inv(diagmat(X))*q);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#vectorise">vectorise()</a></li>
<li><a href="#accu">accu()</a></li>
<li><a href="#trace">trace()</a></li>
<li><a href="#dot">dot()</a></li>
<li><a href="#norm">norm()</a></li>
<li><a href="#conv_to">conv_to()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="clamp"></a>
<b>clamp( X, min_val, max_val )</b>
<ul>
<li>
Create a copy of <i>X</i> with each element clamped to be between <i>min_val</i> and <i>max_val</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = clamp(A, 0.2, 0.8); 

mat C = clamp(A, A.min(), 0.8); 

mat D = clamp(A, 0.2, A.max()); 
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#min_and_max">min() &amp; max()</a></li>
<li><a href="#min_and_max_member">.min() &amp; .max()</a> (member functions of Mat)</li>
<li><a href="#find">find()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="cond"></a>
<b>cond( A )</b>
<ul>
<li>
Return the condition number of matrix <i>A</i> (the ratio of the largest singular value to the smallest)
</li>
<br>
<li>
Large condition numbers suggest that matrix <i>A</i> is nearly singular
</li>
<br>
<li>
The computation is based on singular value decomposition; if the decomposition fails, a <i>std::runtime_error</i> exception is thrown
</li>
<br>
<li>
<b>Caveat:</b> the <a href="#rcond">rcond()</a> function is faster for providing an estimate of the reciprocal of the condition number
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat    A = randu&lt;mat&gt;(5,5);
double c = cond(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#rcond">rcond()</a></li>
<li><a href="#rank">rank()</a></li>
<li><a href="#inv">inv()</a></li>
<li><a href="#solve">solve()</a></li>
<li><a href="http://mathworld.wolfram.com/ConditionNumber.html">condition number in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Condition_number">condition number in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="conj"></a>
<b>conj( X )</b>
<ul>
<li>
Obtain the complex conjugate of each element in a complex matrix or cube
</li>
<br>
<li>
Examples:
<ul>
<pre>
cx_mat X = randu&lt;cx_mat&gt;(5,5);
cx_mat Y = conj(X);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#abs">abs()</a></li>
<li><a href="#imag_real">imag() / real()</a></li>
<li><a href="#trans">trans()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="conv_to"></a>
<b>
conv_to&lt; <i>type</i> &gt;::from( X )
</b>
<ul>
<li>
Convert from one matrix type to another (eg. <i>mat</i> to <i>imat</i>), or one cube type to another (eg. <i>cube</i> to <i>icube</i>)
</li>
<br>
<li>
Conversion between <i>std::vector</i> and Armadillo matrices/vectors is also possible
</li>
<br>
<li>
Conversion of a <i>mat</i> object into <i>colvec</i>, <i>rowvec</i> or <i>std::vector</i> is possible if the object can be interpreted as a vector
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat  A = randu&lt;mat&gt;(5,5);
fmat B = conv_to&lt;fmat&gt;::from(A);

typedef std::vector&lt;double&gt; stdvec;

stdvec x(3);
x[0] = 0.0; x[1] = 1.0;  x[2] = 2.0;

colvec y = conv_to&lt; colvec &gt;::from(x);
stdvec z = conv_to&lt; stdvec &gt;::from(y); 
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#as_scalar">as_scalar()</a></li>
<li><a href="#abs">abs()</a></li>
<li><a href="#imag_real">imag() / real()</a></li>
<li><a href="#adv_constructors_mat">advanced constructors (matrices)</a></li>
<li><a href="#adv_constructors_cube">advanced constructors (cubes)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="cross"></a>
<b>cross( A, B )</b>
<ul>
<li>
Calculate the cross product between A and B, under the assumption that A and B are 3 dimensional vectors
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec a = randu&lt;vec&gt;(3);
vec b = randu&lt;vec&gt;(3);

vec c = cross(a,b);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#dot">dot()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Cross_product">Cross product in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/CrossProduct.html">Cross product in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="cumsum"></a>
<b>cumsum( V )</b>
<br><b>cumsum( X )</b>
<br><b>cumsum( X, dim )</b>
<ul>
<li>
For vector <i>V</i>, return a vector of the same orientation, containing the cumulative sum of elements
</li>
<br>
<li>
For matrix <i>X</i>, return a matrix containing the cumulative sum of elements in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = cumsum(A);
mat C = cumsum(A, 1);

vec x = randu&lt;vec&gt;(10);
vec y = cumsum(x);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#cumprod">cumprod()</a></li>
<li><a href="#accu">accu()</a></li>
<li><a href="#sum">sum()</a></li>
<li><a href="#diff">diff()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="cumprod"></a>
<b>cumprod( V )</b>
<br><b>cumprod( X )</b>
<br><b>cumprod( X, dim )</b>
<ul>
<li>
For vector <i>V</i>, return a vector of the same orientation, containing the cumulative product of elements
</li>
<br>
<li>
For matrix <i>X</i>, return a matrix containing the cumulative product of elements in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = cumprod(A);
mat C = cumprod(A, 1);

vec x = randu&lt;vec&gt;(10);
vec y = cumprod(x);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#cumsum">cumsum()</a></li>
<li><a href="#prod">prod()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="det"></a>
<b>det( A )</b>
<br><b>log_det( val, sign, A )</b>
<ul>
<li>
Determinant of square matrix <i>A</i>
</li>
<br>
<li>
If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>
<i>log_det():</i> find the log determinant, such that the determinant is equal to <i>exp(val)*sign</i>
</li>
<br>
<li>
<b>Caveat</b>: for large matrices <i>log_det()</i> is more precise than <i>det()</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat    A = randu&lt;mat&gt;(5,5);
double x = det(A);

double val;
double sign;

log_det(val, sign, A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="http://mathworld.wolfram.com/Determinant.html">determinant in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Determinant">determinant in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="diagmat"></a>
<b>diagmat( V )</b>
<br><b>diagmat( V, k )</b>
<br><b>diagmat( X )</b>
<br><b>diagmat( X, k )</b>
<ul>
<li>
Generate a diagonal matrix from vector <i>V</i> or matrix <i>X</i>
</li>
<br>
<li>
Given vector <i>V</i>, generate a square matrix with the <i>k</i>-th diagonal containing a copy of the vector; all other elements are set to zero
</li>
<br>
<li>
Given matrix <i>X</i>, generate a matrix with the <i>k</i>-th diagonal containing a copy of the <i>k</i>-th diagonal of <i>X</i>; all other elements are set to zero
</li>
<br>
<li>
The argument <i>k</i> is optional; by default the main diagonal is used (<i>k=0</i>)
</li>
<br>
<li>
For <i>k &gt; 0</i>, the <i>k</i>-th super-diagonal is used (top-right corner)
</li>
<br>
<li>
For <i>k &lt; 0</i>, the <i>k</i>-th sub-diagonal is used (bottom-left corner)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = diagmat(A);
mat C = diagmat(A,1);

vec v = randu&lt;vec&gt;(5);
mat D = diagmat(v);
mat E = diagmat(v,1);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#diag">.diag()</a></li>
<li><a href="#diagvec">diagvec()</a></li>
<li><a href="#trimat">trimatu() / trimatl()</a></li>
<li><a href="#symmat">symmatu() / symmatl()</a></li>
<li><a href="#reshape">reshape()</a></li>
<li><a href="#vectorise">vectorise()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="diagvec"></a>
<b>diagvec( A )</b>
<br><b>diagvec( A, k )</b>
<ul>
<li>
Extract the <i>k</i>-th diagonal from matrix <i>A</i>
</li>
<br>
<li>
The argument <i>k</i> is optional; by default the main diagonal is extracted (<i>k=0</i>)
</li>
<br>
<li>For <i>k &gt; 0</i>, the <i>k</i>-th super-diagonal is extracted (top-right corner)</li>
<br>
<li>For <i>k &lt; 0</i>, the <i>k</i>-th sub-diagonal is extracted (bottom-left corner)</li>
<br>
<li>
The extracted diagonal is interpreted as a column vector
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

vec d = diagvec(A);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#diag">.diag()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="#trace">trace()</a></li>
<li><a href="#vectorise">vectorise()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="diff"></a>
<b>diff( V )</b>
<br><b>diff( V, k )</b>
<br>
<br><b>diff( X )</b>
<br><b>diff( X, k )</b>
<br><b>diff( X, k, dim )</b>
<ul>
<li>
For vector <i>V</i>, return a vector of the same orientation, containing the differences between consecutive elements
</li>
<br>
<li>
For matrix <i>X</i>, return a matrix containing the differences between consecutive elements in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
The optional argument <i>k</i> indicates that the differences are calculated recursively <i>k</i> times; by default <i>k=1</i> is used
</li>
<br>
<li>
The resulting number of differences is <i>n - k</i>, where <i>n</i> is the number of elements; if <i>n &le; k</i>, the number of differences is zero (ie. an empty vector/matrix is returned)
</li>
<br>
<li>
The argument <i>dim</i> is optional; by default <i>dim=0</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec a = linspace&lt;vec&gt;(1,10,10);

vec b = diff(a);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#trapz">trapz()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Numerical_differentiation">numerical differentiation in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/NumericalDifferentiation.html">numerical differentiation in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="dot"></a>
<b>dot( A, B )</b>
<br><b>cdot( A, B )</b>
<br><b>norm_dot( A, B )</b>
<ul>
<li>
<i>dot(A,B)</i>: dot product of <i>A</i> and <i>B</i>, treating <i>A</i> and <i>B</i> as vectors
</li>
<br>
<li>
<i>cdot(A,B)</i>: as per <i>dot(A,B)</i>, but the complex conjugate of <i>A</i> is used
</li>
<br>
<li>
<i>norm_dot(A,B)</i>: normalised dot product; equivalent to <i>dot(A,B)&nbsp;/&nbsp;(&#8741;A&#8741;&#8226;&#8741;B&#8741;)</i>
</li>
<br>
<li>
<b>Caveat:</b> <a href="#norm">norm()</a> is more robust for calculating the norm, as it handles underflows and overflows
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec a = randu&lt;vec&gt;(10);
vec b = randu&lt;vec&gt;(10);

double x = dot(a,b);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#norm">norm()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
<li><a href="#cross">cross()</a></li>
<li><a href="#conj">conj()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eps"></a>
<b>eps( X )</b>
<ul>
<li>
Obtain the positive distance of the absolute value of each element of <i>X</i> to the next largest representable floating point number
</li>
<br>
<li>
<i>X</i> can be a scalar (eg. <i>double</i>), vector or matrix
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);
mat B = eps(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#constants">datum::eps</a></li>
<li><a href="http://mathworld.wolfram.com/Floating-PointArithmetic.html">Floating-Point Arithmetic in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/IEEE_754-2008">IEEE Standard for Floating-Point Arithmetic in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="expmat"></a>
<b>B = expmat(A)</b>
<br><b>expmat(B, A)</b>
<ul>
<li>Matrix exponential of general square matrix <i>A</i></li>
<br>
<li>If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the matrix exponential cannot be found:
<ul>
<li><i>B = expmat(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>expmat(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li><b>Caveat</b>: if matrix <i>A</i> is symmetric, using <a href="#expmat_sym">expmat_sym()</a> is faster</li>
<br>
<li><b>Caveat</b>: the matrix exponential operation is generally <b>not</b> the same as applying the <a href="#misc_fns">exp()</a> function to each element</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = expmat(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#expmat_sym">expmat_sym()</a></li>
<li><a href="#logmat">logmat()</a></li>
<li><a href="#sqrtmat">sqrtmat()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a></li>
<li><a href="http://en.wikipedia.org/wiki/Matrix_exponential">matrix exponential in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/MatrixExponential.html">matrix exponential in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="expmat_sym"></a>
<b>B = expmat_sym(A)</b>
<br><b>expmat_sym(B, A)</b>
<ul>
<li>Matrix exponential of symmetric/hermitian matrix <i>A</i></li>
<br>
<li>If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the matrix exponential cannot be found:
<ul>
<li><i>B = expmat_sym(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>expmat_sym(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li><b>Caveat</b>: there is currently no check whether <i>A</i> is symmetric</li>
<br>
<li><b>Caveat</b>: the matrix exponential operation is generally <b>not</b> the same as applying the <a href="#misc_fns">exp()</a> function to each element</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = A*A.t();   // make symmetric matrix

mat C = expmat_sym(B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#expmat">expmat()</a></li>
<li><a href="#logmat_sympd">logmat_sympd()</a></li>
<li><a href="#sqrtmat_sympd">sqrtmat_sympd()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a></li>
<li><a href="http://en.wikipedia.org/wiki/Matrix_exponential">matrix exponential in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/MatrixExponential.html">matrix exponential in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="find"></a>
<b>find( X )</b>
<br><b>find( X, k )</b>
<br><b>find( X, k, s )</b>
<ul>
<li>Return a column vector containing the indices of elements of <i>X</i> that are non-zero or satisfy a relational condition</li>
<br>
<li>The output vector must have the type <a href="#Col">uvec</a>
(ie. the indices are stored as unsigned integers of type <a href="#uword">uword</a>)
</li>
<br>
<li>
<i>X</i> is interpreted as a vector, with column-by-column ordering of the elements of <i>X</i>
</li>
<br>
<li>Relational operators can be used instead of <i>X</i>, eg.&nbsp;<i>A&nbsp;&gt;&nbsp;0.5</i>
</li>
<br>
<li>If <i>k=0</i> (default), return the indices of all non-zero elements, otherwise return at most <i>k</i> of their indices</li>
<br>
<li>If <i>s="first"</i> (default), return at most the first <i>k</i> indices of the non-zero elements
</li>
<br>
<li>If <i>s="last"</i>, return at most the last <i>k</i> indices of the non-zero elements
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat  A  = randu&lt;mat&gt;(5,5);
mat  B  = randu&lt;mat&gt;(5,5);

uvec q1 = find(A &gt; B);
uvec q2 = find(A &gt; 0.5);
uvec q3 = find(A &gt; 0.5, 3, "last");

// change elements of A greater than 0.5 to 1
A.elem( find(A &gt; 0.5) ).ones();
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#all">all()</a></li>
<li><a href="#any">any()</a></li>
<li><a href="#replace">.replace()</a></li>
<li><a href="#approx_equal">approx_equal()</a></li>
<li><a href="#clamp">clamp()</a></li>
<li><a href="#find_finite">find_finite()</a></li>
<li><a href="#find_nonfinite">find_nonfinite()</a></li>
<li><a href="#find_unique">find_unique()</a></li>
<li><a href="#nonzeros">nonzeros()</a></li>
<li><a href="#unique">unique()</a></li>
<li><a href="#sort_index">sort_index()</a></li>
<li><a href="#index_min_and_index_max_member">.index_min() &amp; .index_max()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#subcube">subcube views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="find_finite"></a>
<b>find_finite( X )</b>
<ul>
<li>Return a column vector containing the indices of elements of <i>X</i> that are finite (ie. not &plusmn;Inf and not NaN)</li>
<br>
<li>The output vector must have the type <a href="#Col">uvec</a>
(ie. the indices are stored as unsigned integers of type <a href="#uword">uword</a>)
</li>
<br>
<li>
<i>X</i> is interpreted as a vector, with column-by-column ordering of the elements of <i>X</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

A(1,1) = datum::inf;

// accumulate only finite elements
double val = accu( A.elem( find_finite(A) ) );
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#find">find()</a></li>
<li><a href="#find_nonfinite">find_nonfinite()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="find_nonfinite"></a>
<b>find_nonfinite( X )</b>
<ul>
<li>Return a column vector containing the indices of elements of <i>X</i> that are non-finite (ie. &plusmn;Inf or NaN)</li>
<br>
<li>The output vector must have the type <a href="#Col">uvec</a>
(ie. the indices are stored as unsigned integers of type <a href="#uword">uword</a>)
</li>
<br>
<li>
<i>X</i> is interpreted as a vector, with column-by-column ordering of the elements of <i>X</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

A(1,1) = datum::inf;

// change non-finite elements to zero
A.elem( find_nonfinite(A) ).zeros();
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#find">find()</a></li>
<li><a href="#find_finite">find_finite()</a></li>
<li><a href="#replace">.replace()</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="find_unique"></a>
<b>find_unique( X )</b>
<br><b>find_unique( X, ascending_indices )</b>
<ul>
<li>Return a column vector containing the indices of unique elements of <i>X</i></li>
<br>
<li>The output vector must have the type <a href="#Col">uvec</a>
(ie. the indices are stored as unsigned integers of type <a href="#uword">uword</a>)
</li>
<br>
<li>
<i>X</i> is interpreted as a vector, with column-by-column ordering of the elements of <i>X</i>
</li>
<br>
<li>The <i>ascending_indices</i> argument is optional; it is one of:
<ul>
<table>
<tbody>
<tr><td style="text-align: right;"><code>true</code></td><td>&nbsp;=&nbsp;</td><td>the returned indices are sorted to be ascending (<b>default setting</b>)</td></tr>
<tr><td style="text-align: right;"><code>false</code></td><td>&nbsp;=&nbsp;</td><td>the returned indices are in arbitrary order (faster operation)</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A;

A &lt;&lt; 2 &lt;&lt; 2 &lt;&lt; 4 &lt;&lt; endr
  &lt;&lt; 4 &lt;&lt; 6 &lt;&lt; 6 &lt;&lt; endr;

uvec indices = find_unique(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#find">find()</a></li>
<li><a href="#unique">unique()</a></li>
<li><a href="#submat">submatrix views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="flip"></a>
<b>fliplr( X )</b>
<br><b>flipud( X )</b>
<ul>
<li>
<i>fliplr():</i> generate a copy of matrix <i>X</i>, with the order of the columns reversed
</li>
<br>
<li>
<i>flipud():</i> generate a copy of matrix <i>X</i>, with the order of the rows reversed
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = fliplr(A);
mat C = flipud(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#shift">shift()</a></li>
<li><a href="#swap_rows">.swap_rows() &amp; .swap_cols()</a></li>
<li><a href="#t_st_members">.t()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="imag_real"></a>
<b>imag( X )</b>
<br><b>real( X )</b>
<ul>
<li>
Extract the imaginary/real part of a complex matrix or cube
</li>
<br>
<li>
Examples:
<ul>
<pre>
cx_mat C = randu&lt;cx_mat&gt;(5,5);

mat    A = imag(C);
mat    B = real(C);
</pre>
</ul>
</li>
<br>
<li><b>Caveat:</b> versions 4.4, 4.5 and 4.6 of the GCC C++ compiler have a bug when using the <i>-std=c++0x</i> compiler option (ie. experimental support for C++11);
to work around this bug, preface Armadillo's <i>imag()</i> and <i>real()</i> with the <i>arma</i> namespace qualification, eg. <i>arma::imag(C)</i>
</li>
<br>
<li>See also:
<ul>
<li><a href="#set_imag">.set_imag()&nbsp;/&nbsp;.set_real()</a></li>
<li><a href="#abs">abs()</a></li>
<li><a href="#conj">conj()</a></li>
<li><a href="#conv_to">conv_to()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="ind2sub"></a>
<table>
<tbody>
<tr><td>uvec</td><td>&nbsp;sub&nbsp;=&nbsp;<b>ind2sub(&nbsp;size(</b>X<b>),</b>&nbsp;index&nbsp;<b>)</b></td></tr>
<tr><td>umat</td><td>&nbsp;sub&nbsp;=&nbsp;<b>ind2sub(&nbsp;size(</b>X<b>),</b>&nbsp;vector_of_indices&nbsp;<b>)</b></td></tr>
</tbody>
</table>
<ul>
<li>
Convert a linear index, or a vector of indices, to subscript notation
</li>
<br>
<li>
The argument <i><b>size(</b>X<b>)</b></i> can be replaced with <i><b>size(</b>n_rows<b>,</b>&nbsp;n_cols<b>)</b></i> or <i><b>size(</b>n_rows<b>,</b>&nbsp;n_cols<b>,</b>&nbsp;n_slices<b>)</b></i>
</li>
<br>
<li>
A&nbsp;<i>std::logic_error</i> exception is thrown if an index is out of range
</li>
<br>
<li>
When only one index is given, the subscripts are returned in a vector of type <a href="#Col">uvec</a>
</li>
<br>
<li>
When a vector of indices (of type <a href="#Col">uvec</a>) is given, the corresponding subscripts are returned in each column of an <i>m&nbsp;x&nbsp;n</i> matrix of type <a href="#Mat">umat</a>;
<i>m=2</i> for matrix subscripts, while <i>m=3</i> for cube subscripts
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat M(4, 5, fill::randu);

uvec s = ind2sub( size(M), 6 );

cout &lt;&lt; "row: " &lt;&lt; s(0) &lt;&lt; endl;
cout &lt;&lt; "col: " &lt;&lt; s(1) &lt;&lt; endl;


uvec indices = find(M &gt; 0.5);
umat t       = ind2sub( size(M), indices );


cube Q(2,3,4);

uvec u = ind2sub( size(Q), 8 );

cout &lt;&lt; "row:   " &lt;&lt; u(0) &lt;&lt; endl;
cout &lt;&lt; "col:   " &lt;&lt; u(1) &lt;&lt; endl;
cout &lt;&lt; "slice: " &lt;&lt; u(2) &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#size">size()</a></li>
<li><a href="#sub2ind">sub2ind()</a></li>
<li><a href="#element_access">element access</a></li>
<li><a href="#find">find()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="index_min_and_index_max_standalone"></a>
<table>
<tbody>
<tr>
<td>
<b>index_min( V )</b>
<br><b>index_min( M )</b>
<br><b>index_min( M, dim )</b>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<b>index_max( V )</b>
<br><b>index_max( M )</b>
<br><b>index_max( M, dim )</b>
</td>
</tr>
</tbody>
</table>
<ul>
<li>
For vector <i>V</i>, return the linear index of the extremum value; the returned index is of type <a href="#uword">uword</a>
</li>
<br>
<li>
For matrix <i>M</i> and:
<ul>
<li>
<i>dim=0</i>, return a row vector (of type <a href="#Row">urowvec</a> or <a href="#Mat">umat</a>), 
with each column containing the index of the extremum value in the corresponding column of <i>M</i>
</li>
<li>
<i>dim=1</i>, return a column vector (of type <a href="#Col">uvec</a> or <a href="#Mat">umat</a>),
with each row containing the index of the extremum value in the corresponding row of <i>M</i>
</li>
</ul>
</li>
<br>
<li>
For each column and row, the index starts at zero
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
For objects with complex numbers, absolute values are used for comparison
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec   v = randu&lt;vec&gt;(10);
uword i = index_max(v);

double max_val_in_v = v(i);


mat     M  = randu&lt;mat&gt;(5,6);

urowvec ii = index_max(M);
ucolvec jj = index_max(M,1);

double max_val_in_col_2 = M( ii(2), 2 );

double max_val_in_row_4 = M( 4, jj(4) );
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#min_and_max">min() &amp; max()</a></li>
<li><a href="#index_min_and_index_max_member">.index_min() &amp; .index_max()</a> (member functions)</li>
<li><a href="#sort_index">sort_index()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="inplace_trans"></a>
<b>inplace_trans( X )
<br>inplace_trans( X, method )
<br>
<br>inplace_strans( X )
<br>inplace_strans( X, method )
</b>
<ul>
<li>
In-place / in-situ transpose of matrix <i>X</i>
</li>
<br>
<li>
For real (non-complex) matrix:
<ul>
<li><i>inplace_trans()</i> performs a normal transpose</li>
<li><i>inplace_strans()</i> not applicable</li>
</ul>
</li>
<br>
<li>
For complex matrix:
<ul>
<li><i>inplace_trans()</i> performs a Hermitian transpose (ie. the conjugate of the elements is taken during the transpose)</li>
<li><i>inplace_strans()</i> provides a transposed copy without taking the conjugate of the elements</li>
</ul>
</li>
<br>
<li>
The argument <i>method</i> is optional
</li>
<br>
<li>
By default, a greedy transposition algorithm is used; a low-memory algorithm can be used instead by explicitly setting <i>method</i> to <code>"lowmem"</code>
</li>
<br>
<li>
The low-memory algorithm is considerably slower than the greedy algorithm;
using the low-memory algorithm is only recommended for cases where <i>X</i> takes up more than half of available memory (ie. very large <i>X</i>)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(4,5);
mat Y = randu&lt;mat&gt;(20000,30000);

inplace_trans(X);            // use greedy algorithm by default

inplace_trans(Y, "lowmem");  // use low-memory (and slow) algorithm
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#t_st_members">.t()</a></li>
<li><a href="#trans">trans()</a></li>
<li><a href="https://en.wikipedia.org/wiki/In-place_matrix_transposition">inplace matrix transpose in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="is_finite"></a>
<b>is_finite( X )</b>
<ul>
<li>
Returns <i>true</i> if all elements in <i>X</i> are finite
</li>
<br>
<li>
Returns <i>false</i> if at least one element in <i>X</i> is non-finite (&plusmn;infinity or NaN)
</li>
<br>
<li>
<i>X</i> can be a scalar (eg. <i>double</i>), vector, matrix or cube
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = randu&lt;mat&gt;(5,5);

B(1,1) = datum::inf;

cout &lt;&lt; is_finite(A) &lt;&lt; endl;
cout &lt;&lt; is_finite(B) &lt;&lt; endl;

cout &lt;&lt; is_finite( 0.123456789 ) &lt;&lt; endl;
cout &lt;&lt; is_finite( datum::nan  ) &lt;&lt; endl;
cout &lt;&lt; is_finite( datum::inf  ) &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#is_finite_member">.is_finite()</a> (member function of <i>Mat</i> and <i>Cube</i>)</li>
<li><a href="#find_finite">find_finite()</a></li>
<li><a href="#find_nonfinite">find_nonfinite()</a></li>
<li><a href="#constants">constants (pi, nan, inf, ...)</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="join"></a>
<b>join_rows( A, B )</b>
<br><b>join_horiz( A,  B )</b>
<br>
<br><b>join_cols( A, B )</b>
<br><b>join_vert( A, B )</b>
<ul>
<li>
<i>join_rows()</i> and <i>join_horiz()</i>: horizontal concatenation;
for two matrices <i>A</i> and <i>B</i>, join each row of <i>A</i> with the corresponding row of <i>B</i>;
matrices <i>A</i> and <i>B</i> must have the same number of rows
</li>
<br>
<li>
<i>join_cols()</i> and <i>join_vert()</i>: vertical concatenation;
for two matrices <i>A</i> and <i>B</i>, join each column of <i>A</i> with the corresponding column of <i>B</i>;
matrices <i>A</i> and <i>B</i> must have the same number of columns
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);
mat B = randu&lt;mat&gt;(4,6);
mat C = randu&lt;mat&gt;(6,5);

mat X = join_rows(A,B);
mat Y = join_cols(A,C);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#shed">.shed_rows/cols/slices</a></li>
<li><a href="#insert">.insert_rows/cols/slices</a></li>
<li><a href="#submat">submatrix views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="join_slices"></a>
<b>join_slices( cube C, cube D )</b>
<br><b>join_slices( mat  M, mat  N )</b>
<br>
<br><b>join_slices( mat  M, cube C )</b>
<br><b>join_slices( cube C, mat  M )</b>
<ul>
<li>
for two cubes <i>C</i> and <i>D</i>: join the slices of <i>C</i> with the slices of <i>D</i>;
cubes <i>C</i> and <i>D</i> must have the same number of rows and columns (ie. all slices must have the same size)
</li>
<br>
<li>
for two matrices <i>M</i> and <i>N</i>: treat <i>M</i> and <i>N</i> as cube slices and join them to form a cube with 2 slices;
matrices <i>M</i> and <i>N</i> must have the same number of rows and columns
</li>
<br>
<li>
for matrix <i>M</i> and cube <i>C</i>: treat <i>M</i> as a cube slice and join it with the slices of <i>C</i>;
matrix <i>M</i> and cube <i>C</i> must have the same number of rows and columns
</li>
<br>
<li>
Examples:
<ul>
<pre>
cube C(5, 10, 3, fill::randu);
cube D(5, 10, 4, fill::randu);

cube E = join_slices(C,D);

mat M(10, 20, fill::randu);
mat N(10, 20, fill::randu);

cube Q = join_slices(M,N);

cube R = join_slices(Q,M);

cube S = join_slices(M,Q);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#shed">.shed_rows/cols/slices</a></li>
<li><a href="#insert">.insert_rows/cols/slices</a></li>
<li><a href="#subcube">subcube views</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="kron"></a>
<b>kron( A, B )</b>

<ul>
<li>Kronecker tensor product</li>
<br>
<li>Given matrix <i>A</i> (with <i>n</i> rows and <i>p</i> columns) and matrix <i>B</i> (with <i>m</i> rows and <i>q</i> columns),
generate a matrix (with <i>nm</i> rows and <i>pq</i> columns) that denotes the tensor product of <i>A</i> and <i>B</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);
mat B = randu&lt;mat&gt;(5,4);

mat K = kron(A,B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="http://mathworld.wolfram.com/KroneckerProduct.html">Kronecker product in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Kronecker_product">Kronecker product in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="logmat"></a>
<b>B = logmat(A)</b>
<br><b>logmat(B, A)</b>
<ul>
<li>Complex matrix logarithm of general square matrix <i>A</i></li>
<br>
<li>If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the matrix logarithm cannot be found:
<ul>
<li><i>B = logmat(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>logmat(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li><b>Caveat</b>: if matrix <i>A</i> is symmetric positive definite, using <a href="#logmat_sympd">logmat_sympd()</a> is faster</li>
<br>
<li><b>Caveat</b>: the matrix logarithm operation is generally <b>not</b> the same as applying the <a href="#misc_fns">log()</a> function to each element</li>
<br>
<li>
Examples:
<ul>
<pre>
   mat A = randu&lt;mat&gt;(5,5);

cx_mat B = logmat(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#logmat_sympd">logmat_sympd()</a></li>
<li><a href="#expmat">expmat()</a></li>
<li><a href="#sqrtmat">sqrtmat()</a></li>
<li><a href="#imag_real">real()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a></li>
<li><a href="https://en.wikipedia.org/wiki/Logarithm_of_a_matrix">matrix logarithm in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="logmat_sympd"></a>
<b>B = logmat_sympd(A)</b>
<br><b>logmat_sympd(B, A)</b>
<ul>
<li>Matrix logarithm of symmetric/hermitian positive definite matrix <i>A</i></li>
<br>
<li>If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the matrix logarithm cannot be found:
<ul>
<li><i>B = logmat_sympd(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>logmat_sympd(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li><b>Caveat</b>: there is currently no check whether <i>A</i> is symmetric</li>
<br>
<li><b>Caveat</b>: the matrix logarithm operation is generally <b>not</b> the same as applying the <a href="#misc_fns">log()</a> function to each element</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = A*A.t();   // make symmetric matrix

mat C = logmat_sympd(B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#logmat">logmat()</a></li>
<li><a href="#expmat_sym">expmat_sym()</a></li>
<li><a href="#sqrtmat_sympd">sqrtmat_sympd()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a></li>
<li><a href="https://en.wikipedia.org/wiki/Logarithm_of_a_matrix">matrix logarithm in Wikipedia</a></li>
<li><a href="https://en.wikipedia.org/wiki/Positive-definite_matrix">positive definite matrix in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html">positive definite matrix in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="min_and_max"></a>
<table>
<tbody>
<tr>
<td>
<b>min( V )</b>
<br><b>min( M )</b>
<br><b>min( M, dim )</b>
<br><b>min( Q )</b>
<br><b>min( Q, dim )</b>
<br><b>min( A, B )</b>
</td>
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
<td>
<b>max( V )</b>
<br><b>max( M )</b>
<br><b>max( M, dim )</b>
<br><b>max( Q )</b>
<br><b>max( Q, dim )</b>
<br><b>max( A, B )</b>
</td>
</tr>
</tbody>
</table>
<ul>
<li>
For vector <i>V</i>, return the extremum value
</li>
<br>
<li>
For matrix <i>M</i>, return the extremum value for each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
For cube <i>Q</i>, return the extremum values of elements along dimension <i>dim</i>, where <i>dim</i> &isin; {&nbsp;0,&nbsp;1,&nbsp;2&nbsp;}
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
For two matrices/cubes <i>A</i> and <i>B</i>, return a matrix/cube containing element-wise extremum values
</li>
<br>
<li>
For objects with complex numbers, absolute values are used for comparison
</li>
<br>
<li>
Examples:
<ul>
<pre>
colvec v = randu&lt;colvec&gt;(10,1);
double x = max(v);

mat    M = randu&lt;mat&gt;(10,10);

rowvec a = max(M);
rowvec b = max(M,0); 
colvec c = max(M,1);

// element-wise maximum
mat X = randu&lt;mat&gt;(5,6);
mat Y = randu&lt;mat&gt;(5,6);
mat Z = arma::max(X,Y);  // use arma:: prefix to distinguish from std::max()
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#min_and_max_member">.min() &amp; .max()</a> (member functions)</li>
<li><a href="#index_min_and_index_max_standalone">index_min() &amp; index_max()</a>
<li><a href="#clamp">clamp()</a>
<li><a href="#running_stat">running_stat</a></li>
<li><a href="#running_stat_vec">running_stat_vec</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="nonzeros"></a>
<b>nonzeros(X)</b>
<ul>
<li>
Return a column vector containing the non-zero values of <i>X</i>
</li>
<br>
<li>
<i>X</i> can be a sparse or dense matrix
</li>
<br>
<li>
Examples:
<ul>
<pre>
sp_mat A = sprandu&lt;sp_mat&gt;(100, 100, 0.1);
   vec a = nonzeros(A);

mat B(100, 100, fill::eye);
vec b = nonzeros(B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#find">find()</a>
<li><a href="#unique">unique()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="norm"></a>
<b>norm( X )</b>
<br><b>norm( X, p )</b>
<ul>
<li>
Compute the <i>p</i>-norm of <i>X</i>, where <i>X</i> can be a vector or matrix
</li>
<br>
<li>
For vectors, <i>p</i> is an integer &ge;1, or one of: <code>"-inf"</code>, <code>"inf"</code>, <code>"fro"</code>
</li>
<br>
<li>
For matrices, <i>p</i> is one of: 1, 2, <code>"inf"</code>, <code>"fro"</code>; the calculated norm is the <i>induced norm</i> (not entrywise norm)
</li>
<br>
<li>
<code>"-inf"</code> is the minimum norm, <code>"inf"</code> is the maximum norm, while <code>"fro"</code> is the Frobenius norm
</li>
<br>
<li>
The argument <i>p</i> is optional; by default <i>p=2</i> is used
</li>
<br>
<li>
For vector norm with <i>p=2</i> and matrix norm with <i>p="fro"</i>, a robust algorithm is used to reduce the likelihood of underflows and overflows
</li>
<br>
<li>
To obtain the zero norm or Hamming norm (ie. the number of non-zero elements),
use this expression: <code><a href="#accu">accu</a>(X&nbsp;!=&nbsp;0)</code>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec    q = randu&lt;vec&gt;(5);

double x = norm(q, 2);
double y = norm(q, "inf");
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#normalise">normalise()</a></li>
<li><a href="#vectorise">vectorise()</a></li>
<li><a href="#dot">dot()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Norm_(mathematics)">vector norm in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/VectorNorm.html">vector norm in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Matrix_norm">matrix norm in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/MatrixNorm.html">matrix norm in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="normalise"></a>
<b>normalise( V )</b>
<br><b>normalise( V, p )</b>
<br>
<br><b>normalise( X )</b>
<br><b>normalise( X, p )</b>
<br><b>normalise( X, p, dim )</b>
<ul>
<li>For vector <i>V</i>, return its normalised version (ie. having unit <i>p</i>-norm)</li>
<br>
<li>
For matrix <i>X</i>, return its normalised version, where each column (<i>dim=0</i>) or row (<i>dim=1</i>) has been normalised to have unit <i>p</i>-norm</li>
<br>
<li>
The <i>p</i> argument is optional; by default <i>p=2</i> is used
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec A = randu&lt;vec&gt;(10);
vec B = normalise(A);
vec C = normalise(A, 1);

mat X = randu&lt;mat&gt;(5,6);
mat Y = normalise(X);
mat Z = normalise(X, 2, 1);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#norm">norm()</a></li>
<li><a href="#dot">norm_dot()</a></li>
<li><a href="http://mathworld.wolfram.com/NormalizedVector.html">Normalised vector in MathWorld</a>
<li><a href="http://en.wikipedia.org/wiki/Unit_vector">Unit vector in Wikipedia</a>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="prod"></a>
<b>prod( V )</b>
<br><b>prod( M )</b>
<br><b>prod( M, dim )</b>
<ul>
<li>
For vector <i>V</i>, return the product of all elements
</li>
<br>
<li>
For matrix <i>M</i>, return the product of elements in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Examples:
<ul>
<pre>
colvec v = randu&lt;colvec&gt;(10,1);
double x = prod(v);

mat    M = randu&lt;mat&gt;(10,10);

rowvec a = prod(M);
rowvec b = prod(M,0);
colvec c = prod(M,1);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#cumprod">cumprod()</a></li>
<li><a href="#schur_product">Schur product</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="rank"></a>
<b>rank( X )</b>
<br>
<b>rank( X, tolerance )</b>
<ul>
<li>Return the rank of matrix <i>X</i></li>
<br>
<li>Any singular values less than <i>tolerance</i> are treated as zero</li>
<br>
<li>
The <i>tolerance</i> argument is optional; by default <i>tolerance</i> is <i>max(m,n)*max_sv*datum::eps</i>, where:
<ul>
<li><i>m</i> = number of rows and <i>n</i> = number of columns in <i>X</i></li>
<li><i>max_sv</i> = maximal singular value of <i>X</i></li>
<li><i>datum::eps</i> = difference between 1 and the least value greater than 1 that is representable</li>
</ul>
</li>
<br>
<li>The computation is based on singular value decomposition;
if the decomposition fails, a <i>std::runtime_error</i> exception is thrown</li>
<br>
<li>
<b>Caveat:</b> to confirm whether a matrix is singular, use <a href="#rcond">rcond()</a> or <a href="#cond">cond()</a>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat   A = randu&lt;mat&gt;(4,5);

uword r = rank(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#cond">cond()</a></li>
<li><a href="#rcond">rcond()</a></li>
<li><a href="#svd">svd()</a></li>
<li><a href="#orth">orth()</a></li>
<li><a href="#constants">datum::eps</a></li>
<li><a href="http://mathworld.wolfram.com/MatrixRank.html">Rank in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Rank_(linear_algebra)">Rank in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="rcond"></a>
<b>rcond( A )</b>
<ul>
<li>
Return an estimate of the reciprocal of the condition number of square matrix <i>A</i>
</li>
<br>
<li>
Values close to 1 suggest that <i>A</i> is well-conditioned
</li>
<br>
<li>
Values close to 0 suggest that <i>A</i> is badly conditioned
</li>
<br>
<li>
If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat    A = randu&lt;mat&gt;(5,5);
double r = rcond(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#cond">cond()</a></li>
<li><a href="#rank">rank()</a></li>
<li><a href="#inv">inv()</a></li>
<li><a href="#solve">solve()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="repmat"></a>
<b>repmat( A, num_copies_per_row, num_copies_per_col )</b>
<ul>
<li>Generate a matrix by replicating matrix <i>A</i> in a block-like fashion</li>
<br>
<li>The generated matrix has the following size:
<ul>
<table border="0">
<tbody>
<tr><td>rows</td><td>&nbsp;=&nbsp;</td><td>num_copies_per_row</td><td>*</td><td>A.n_rows</td></tr>
<tr><td>cols</td><td>&nbsp;=&nbsp;</td><td>num_copies_per_col</td><td>*</td><td>A.n_cols</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(2, 3);

mat B = repmat(A, 4, 5);
</pre>
</ul>
</li>
<li>See also:
<ul>
<li><a href="#each_colrow">.each_col() &amp; .each_row()</a> &nbsp; (vector operations repeated on each column or row)</li>
<li><a href="#copy_size">.copy_size()</a></li>
<li><a href="#reshape">reshape()</a></li>
<li><a href="#resize">resize()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="reshape"></a>
<b>reshape( mat, n_rows, n_cols )</b>
<br><b>reshape( mat, size(X) )</b>
<br>
<br><b>reshape( cube, n_rows, n_cols, n_slices )</b>
<br><b>reshape( cube, size(X) )</b>
<ul>
<li>
Generate a matrix or cube sized according to given size specifications,
whose elements are taken from the given matrix/cube in a column-wise manner;
the elements in the generated object are placed column-wise (ie. the first column is filled up before filling the second column)
</li>
<br>
<li>
The layout of the elements in the generated object will be different to the layout in the given object
</li>
<br>
<li>
If the total number of elements in the given object is less than the specified size,
the remaining elements in the generated object are set to zero
</li>
<br>
<li>
If the total number of elements in the given object is greater than the specified size,
only a subset of elements is taken from the given object
</li>
<br>
<li>
<b>Caveats:</b>
<ul>
<li>
do not use <i>reshape()</i> if you simply want to change the size without preserving data;
use <a href="#set_size">.set_size()</a> instead, which is much faster
</li>
<li>
to grow/shrink a matrix while preserving the elements <b>as well as</b> the layout of the elements,
use <a href="#resize">resize()</a> instead
</li>
<li>
to create a vector representation of a matrix (ie. concatenate all the columns or rows),
use <a href="#vectorise">vectorise()</a> instead
</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(10, 5);
mat B = reshape(A, 5, 10);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#reshape_member">.reshape()</a></li>
<li><a href="#set_size">.set_size()</a></li>
<li><a href="#resize">resize()</a></li>
<li><a href="#vectorise">vectorise()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
<li><a href="#conv_to">conv_to()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="#repmat">repmat()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="resize"></a>
<b>resize( mat, n_rows, n_cols )</b>
<br><b>resize( mat, size(X) )</b>
<br>
<br><b>resize( cube, n_rows, n_cols, n_slices )</b>
<br><b>resize( cube, size(X) )</b>
<ul>
<li>
Generate a matrix or cube sized according to given size specifications,
whose elements as well as the layout of the elements are taken from the given matrix/cube
</li>
<br>
<li>
<b>Caveat:</b>
do not use <i>resize()</i> if you simply want to change the size without preserving data;
use <a href="#set_size">.set_size()</a> instead, which is much faster
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4, 5);
mat B = resize(A, 7, 6);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#resize_member">.resize()</a> (member function of Mat and Cube)</li>
<li><a href="#set_size">.set_size()</a> (member function of Mat and Cube)</li>
<li><a href="#reshape">reshape()</a></li>
<li><a href="#vectorise">vectorise()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
<li><a href="#conv_to">conv_to()</a></li>
<li><a href="#repmat">repmat()</a></li>
<li><a href="#size">size()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="shift"></a>
<b>shift( V, N )</b>
<br><b>shift( X, N )</b>
<br><b>shift( X, N, dim )</b>
<ul>
<li>
For vector <i>V</i>, generate a copy of the vector with the elements shifted by <i>N</i> positions in a circular manner
</li>
<br>
<li>
For matrix <i>X</i>, generate a copy of the matrix with the elements shifted by <i>N</i> positions in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
<i>N</i> can be positive of negative
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);
mat B = shift(A, -1);
mat C = shift(A, +1);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#shuffle">shuffle()</a></li>
<li><a href="#flip">fliplr() &amp; flipud()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="shuffle"></a>
<b>shuffle( V )</b>
<br><b>shuffle( X )</b>
<br><b>shuffle( X, dim )</b>
<ul>
<li>
For vector <i>V</i>, generate a copy of the vector with the elements shuffled
</li>
<br>
<li>
For matrix <i>X</i>, generate a copy of the matrix with the elements shuffled in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);
mat B = shuffle(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#shift">shift()</a></li>
<li><a href="#sort">sort()</a></li>
<li><a href="#randu_randn_standalone">randu() / randn()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="size"></a>
<b>size(&nbsp;X&nbsp;)</b>
<br>
<b>size(&nbsp;n_rows,&nbsp;n_cols&nbsp;)</b>
<br>
<b>size(&nbsp;n_rows,&nbsp;n_cols,&nbsp;n_slices&nbsp;)</b>
<ul>
<li>
Obtain the dimensions of object <i>X</i>, or explicitly specify the dimensions
</li>
<br>
<li>
The dimensions can be used in conjunction with:
<ul>
<li><a href="#part_a">object constructors</a></li>
<li><a href="#part_b">functions for changing size</a></li>
<li><a href="#part_c">generator functions</a></li>
<li><a href="#submat">submatrix views</a></li>
<li><a href="#subcube">subcube views</a></li>
<li><a href="#subfield">subfield views</a></li>
<li><a href="#ind2sub">ind2sub()</a> and <a href="#sub2ind">sub2ind()</a></li>
</ul>
</li>
<br>
<li>
The dimensions support simple arithmetic operations; they can also be printed and compared for equality/inequality
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(5,6);

mat B(size(A), fill::zeros);

mat C; C.randu(size(A));

mat D = ones&lt;mat&gt;(size(A));

mat E(10,20, fill::ones);
E(3,4,size(C)) = C;    // access submatrix of E

mat F( size(A) + size(E) );

mat G( size(A) * 2 );

cout &lt;&lt; "size of A: " &lt;&lt; size(A) &lt;&lt; endl;

bool is_same_size = (size(A) == size(E));
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#attributes">attributes</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="sort"></a>
<b>sort( V )</b>
<br><b>sort( V, sort_direction )</b>
<br>
<br><b>sort( X )</b>
<br><b>sort( X, sort_direction )</b>
<br><b>sort( X, sort_direction, dim )</b>
<br>
<ul>
<li>For vector <i>V</i>, return a vector which is a sorted version of the input vector</li>
<br>
<li>For matrix <i>X</i>, return a matrix with the elements of the input matrix sorted in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>The <i>sort_direction</i> argument is optional; <i>sort_direction</i> is either <code>"ascend"</code> or <code>"descend"</code>; by default <code>"ascend"</code> is used</li>
<br>
<li>For matrices and vectors with complex numbers, sorting is via absolute values</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(10,10);
mat B = sort(A);
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#sort_index">sort_index()</a></li>
<li><a href="#is_sorted">.is_sorted()</a></li>
<li><a href="#shuffle">shuffle()</a></li>
<li><a href="#randu_randn_standalone">randu() / randn()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="sort_index"></a>
<b>sort_index( X )</b>
<br><b>sort_index( X, sort_direction )</b>
<br>
<br><b>stable_sort_index( X )</b>
<br><b>stable_sort_index( X, sort_direction )</b>
<ul>
<li>Return a vector which describes the sorted order of the elements of <i>X</i>
(ie. it contains the indices of the elements of <i>X</i>)
</li>
<br>
<li>The output vector must have the type <a href="#Col">uvec</a>
(ie. the indices are stored as unsigned integers of type <a href="#uword">uword</a>)
</li>
<br>
<li>
<i>X</i> is interpreted as a vector, with column-by-column ordering of the elements of <i>X</i>
</li>
<br>
<li>The <i>sort_direction</i> argument is optional; <i>sort_direction</i> is either <code>"ascend"</code> or <code>"descend"</code>; by default <code>"ascend"</code> is used</li>
<br>
<li>The <i>stable_sort_index()</i> variant preserves the relative order of elements with equivalent values</li>
<br>
<li>For matrices and vectors with complex numbers, sorting is via absolute values</li>
<br>
<li>
Examples:
<ul>
<pre>
vec  q       = randu&lt;vec&gt;(10);
uvec indices = sort_index(q);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#sort">sort()</a></li>
<li><a href="#find">find()</a></li>
<li><a href="#index_min_and_index_max_member">.index_min() &amp; .index_max()</a></li>
<li><a href="#ind2sub">ind2sub()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="sqrtmat"></a>
<b>B = sqrtmat(A)</b>
<br><b>sqrtmat(B,A)</b>
<ul>
<li>Complex square root of general square matrix <i>A</i></li>
<br>
<li>If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>
If matrix <i>A</i> appears to be singular, an approximate square root is attempted; additionally, <i>sqrtmat(B,A)</i> returns a bool set to false
</li>
<br>
<li><b>Caveat</b>: if matrix <i>A</i> is symmetric positive definite, using <a href="#sqrtmat_sympd">sqrtmat_sympd()</a> is faster</li>
<br>
<li><b>Caveat</b>: the square root of a matrix is generally <b>not</b> the same as applying the <a href="#misc_fns">sqrt()</a> function to each element</li>
<br>
<li>
Examples:
<ul>
<pre>
   mat A = randu&lt;mat&gt;(5,5);

cx_mat B = sqrtmat(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#sqrtmat_sympd">sqrtmat_sympd()</a></li>
<li><a href="#expmat">expmat()</a></li>
<li><a href="#logmat">logmat()</a></li>
<li><a href="#chol">chol()</a></li>
<li><a href="#imag_real">real()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a></li>
<li><a href="https://en.wikipedia.org/wiki/Square_root_of_a_matrix">square root of a matrix in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="sqrtmat_sympd"></a>
<b>B = sqrtmat_sympd(A)</b>
<br><b>sqrtmat_sympd(B,A)</b>
<ul>
<li>Square root of symmetric/hermitian positive definite matrix <i>A</i></li>
<br>
<li>If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the square root cannot be found:
<ul>
<li><i>B = sqrtmat_sympd(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>sqrtmat_sympd(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li><b>Caveat</b>: there is currently no check whether <i>A</i> is symmetric</li>
<br>
<li><b>Caveat</b>: the matrix square root operation is generally <b>not</b> the same as applying the <a href="#misc_fns">sqrt()</a> function to each element</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = A*A.t();   // make symmetric matrix

mat C = sqrtmat_sympd(B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#sqrtmat">sqrtmat()</a></li>
<li><a href="#expmat_sym">expmat_sym()</a></li>
<li><a href="#logmat_sympd">logmat_sympd()</a></li>
<li><a href="#chol">chol()</a></li>
<li><a href="#misc_fns">miscellaneous element-wise functions</a></li>
<li><a href="https://en.wikipedia.org/wiki/Square_root_of_a_matrix">square root of a matrix in Wikipedia</a></li>
<li><a href="https://en.wikipedia.org/wiki/Positive-definite_matrix">positive definite matrix in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html">positive definite matrix in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="sum"></a>
<b>sum( V )</b>
<br><b>sum( M )</b>
<br><b>sum( M, dim )</b>
<br><b>sum( Q )</b>
<br><b>sum( Q, dim )</b>
<ul>
<li>
For vector <i>V</i>, return the sum of all elements
</li>
<br>
<li>
For matrix <i>M</i>, return the sum of elements in each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
For cube <i>Q</i>, return the sums of elements along dimension <i>dim</i>, where <i>dim</i> &isin; {&nbsp;0,&nbsp;1,&nbsp;2&nbsp;};
for example, <i>dim=0</i> indicates the sum of elements in each column within each slice
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
<b>Caveat:</b> to get a sum of all the elements regardless of the object type (ie. vector, or matrix, or cube), use <a href="#accu">accu()</a> instead
</li>
<br>
<li>
Examples:
<ul>
<pre>
colvec v = randu&lt;colvec&gt;(10,1);
double x = sum(v);

mat    M = randu&lt;mat&gt;(10,10);

rowvec a = sum(M);
rowvec b = sum(M,0);
colvec c = sum(M,1);

double y = accu(M);   // find the overall sum regardless of object type
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#accu">accu()</a></li>
<li><a href="#cumsum">cumsum()</a></li>
<li><a href="#trace">trace()</a></li>
<li><a href="#trapz">trapz()</a></li>
<li><a href="#stats_fns">mean()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="sub2ind"></a>
<table>
<tbody>
<tr><td>uword&nbsp;</td><td>index</td><td>&nbsp;=&nbsp;<b>sub2ind(&nbsp;size(</b>M<b>),</b>&nbsp;row<b>,</b>&nbsp;col&nbsp;<b>)</b></td><td>(<i>M</i> is a matrix)</td></tr>
<tr><td>uvec&nbsp;</td><td>indices</td><td>&nbsp;=&nbsp;<b>sub2ind(&nbsp;size(</b>M<b>),</b>&nbsp;matrix_of_subscripts&nbsp;<b>)</b></td><td></td></tr>
<tr><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>uword&nbsp;</td><td>index</td><td>&nbsp;=&nbsp;<b>sub2ind(&nbsp;size(</b>Q<b>),</b>&nbsp;row<b>,</b>&nbsp;col<b>,</b>&nbsp;slice&nbsp;<b>)</b></td><td>(<i>Q</i> is a cube)</td></tr>
<tr><td>uvec&nbsp;</td><td>indices</td><td>&nbsp;=&nbsp;<b>sub2ind(&nbsp;size(</b>Q<b>),</b>&nbsp;matrix_of_subscripts&nbsp;<b>)</b></td><td></td></tr>
</tbody>
</table>
<ul>
<li>
Convert subscripts to a linear index
</li>
<br>
<li>
The argument <i><b>size(</b>X<b>)</b></i> can be replaced with <i><b>size(</b>n_rows<b>,</b>&nbsp;n_cols<b>)</b></i> or <i><b>size(</b>n_rows<b>,</b>&nbsp;n_cols<b>,</b>&nbsp;n_slices<b>)</b></i>
</li>
<br>
<li>
For the <i>matrix_of_subscripts</i> argument, the subscripts must be stored in each column of an <i>m&nbsp;x&nbsp;n</i> matrix of type <a href="#Mat">umat</a>;
<i>m=2</i> for matrix subscripts, while <i>m=3</i> for cube subscripts
</li>
<br>
<li>
A&nbsp;<i>std::logic_error</i> exception is thrown if a subscript is out of range
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat  M(4,5);
cube Q(4,5,6);

uword i = sub2ind( size(M), 2, 3 );
uword j = sub2ind( size(Q), 2, 3, 4 );
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#size">size()</a></li>
<li><a href="#ind2sub">ind2sub()</a></li>
<li><a href="#element_access">element access</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="symmat"></a>
<b>symmatu( A )</b>
<br><b>symmatu( A, do_conj )</b>
<br>
<br><b>symmatl( A )</b>
<br><b>symmatl( A, do_conj )</b>
<ul>
<li>
<i>symmatu(A)</i>: generate symmetric matrix from square matrix <i>A</i>, by reflecting the upper triangle to the lower triangle 
</li>
<br>
<li>
<i>symmatl(A)</i>: generate symmetric matrix from square matrix <i>A</i>, by reflecting the lower triangle to the upper triangle
</li>
<br>
<li>
If <i>A</i> is a complex matrix, the reflection uses the complex conjugate of the elements;
to disable the complex conjugate, set <i>do_conj</i> to <i>false</i>
</li>
<br>
<li>
If <i>A</i> is non-square, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = symmatu(A);
mat C = symmatl(A);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="#trimat">trimatu() / trimatl()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="trace"></a>
<b>trace( X )</b>
<ul>
<li>
Sum of the elements on the main diagonal of matrix <i>X</i>
</li>
<br>
<li>
If <i>X</i> is an expression,
the function will try to use optimised expression evaluations to calculate only the diagonal elements
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat    A = randu&lt;mat&gt;(5,5);

double x = trace(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#accu">accu()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
<li><a href="#diag">.diag()</a></li>
<li><a href="#diagvec">diagvec()</a></li>
<li><a href="#sum">sum()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="trans"></a>
<b>trans( A )</b>
<br><b>strans( A )</b>
<ul>
<li>
For real (non-complex) matrix:
<ul>
<li><i>trans()</i> provides a transposed copy of the matrix</li>
<li><i>strans()</i> not applicable</li>
</ul>
</li>
<br>
<li>
For complex matrix:
<ul>
<li><i>trans()</i> provides a Hermitian transpose (ie. the conjugate of the elements is taken during the transpose)</li>
<li><i>strans()</i> provides a transposed copy without taking the conjugate of the elements</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,10);

mat B = trans(A);
mat C = A.t();    // equivalent to trans(A), but more compact
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#t_st_members">.t()</a></li>
<li><a href="#inplace_trans">inplace_trans</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="trapz"></a>
<b>trapz( X, Y )</b>
<br><b>trapz( X, Y, dim )</b>
<br>
<br><b>trapz( Y )</b>
<br><b>trapz( Y, dim )</b>
<ul>
<li>
Compute the trapezoidal integral of <i>Y</i> with respect to spacing in <i>X</i>, in each column (<i>dim=0</i>) or each row (<i>dim=1</i>) of <i>Y</i>
</li>
<br>
<li>
<i>X</i> must be a vector; its length must equal either the number of rows in <i>Y</i> (when <i>dim=0</i>), or the number of columns in <i>Y</i> (when <i>dim=1</i>)
</li>
<br>
<li>
If <i>X</i> is not specified, unit spacing is used
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec X = linspace&lt;vec&gt;(0, datum::pi, 1000);
vec Y = sin(X);

mat Z = trapz(X,Y);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#sum">sum()</a></li>
<li><a href="#diff">diff()</a></li>
<li><a href="#linspace">linspace()</a></li>
<li><a href="https://en.wikipedia.org/wiki/Numerical_integration">numerical integration in Wikipedia</a>
<li><a href="http://mathworld.wolfram.com/NumericalIntegration.html">numerical integration in MathWorld</a>
<li><a href="https://en.wikipedia.org/wiki/Trapezoidal_rule">trapezoidal rule in Wikipedia</a>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="trimat"></a>
<b>trimatu( A )</b>
<br><b>trimatl( A )</b>
<ul>
<li>
<i>trimatu(A)</i>: generate upper triangular matrix from square matrix <i>A</i>
</li>
<br>
<li>
<i>trimatl(A)</i>: generate lower triangular matrix from square matrix <i>A</i>
</li>
<br>
<li>
If <i>A</i> is non-square, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat U = trimatu(A);
mat L = trimatl(A);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#symmat">symmatu() / symmatl()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="http://mathworld.wolfram.com/TriangularMatrix.html">Triangular matrix in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Triangular_matrix">Triangular matrix in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="unique"></a>
<b>unique( A )</b>
<br>
<ul>
<li>
Return the unique elements of <i>A</i>, sorted in ascending order
</li>
<br>
<li>
If <i>A</i> is a vector, the output is also a vector with the same orientation (row or column) as <i>A</i>;
if <i>A</i> is a matrix, the output is always a column vector
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X;
X &lt;&lt; 1 &lt;&lt; 2 &lt;&lt; endr
  &lt;&lt; 2 &lt;&lt; 3 &lt;&lt; endr;

mat Y = unique(X);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#find">find()</a></li>
<li><a href="#find_unique">find_unique()</a></li>
<li><a href="#sort">sort()</a></li>
<li><a href="#nonzeros">nonzeros()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="vectorise"></a>
<b>vectorise( X )</b>
<br><b>vectorise( X, dim )</b>
<ul>
<li>
Generate a column vector (<i>dim=0</i>) or row vector (<i>dim=1</i>) from matrix <i>X</i>
</li>
<br>
<li>
The argument <i>dim</i> is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
For <i>dim=0</i>, the elements are copied from <i>X</i> column-wise; equivalent to concatenating all the columns of <i>X</i>
</li>
<br>
<li>
For <i>dim=1</i>, the elements are copied from <i>X</i> row-wise; equivalent to concatenating all the rows of <i>X</i>
</li>
<br>
<li>
Concatenating columns is faster than concatenating rows
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(4, 5);

vec v = vectorise(X);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#reshape">reshape()</a></li>
<li><a href="#resize">resize()</a></li>
<li><a href="#as_scalar">as_scalar()</a></li>
<li><a href="#diagvec">diagvec()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="misc_fns"></a>
<b>miscellaneous element-wise functions:</b>
<ul>
<table border="0" cellpadding="0" cellspacing="0">
<tr><td><b>exp   </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>exp2  </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>exp10 </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>trunc_exp</b></td></tr>
<tr><td><b>log   </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>log2  </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>log10 </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>trunc_log</b></td></tr>
<tr><td><b>pow   </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>square</b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>sqrt  </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;   </b></td></tr>
<tr><td><b>floor </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>ceil  </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>round </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>trunc    </b></td></tr>
<tr><td><b>erf   </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>erfc  </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;</b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;   </b></td></tr>
<tr><td><b>lgamma</b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;</b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;</b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;   </b></td></tr>
<tr><td><b>sign  </b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;</b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;</b></td><td>&nbsp;&nbsp;&nbsp;</td><td><b>&nbsp;   </b></td></tr>
</table>
</ul>
<ul>
<li>
Apply a function to each element
</li>
<br>
<li>
Usage:
<ul>
<li>B = fn(A)</li>
<li>A and B must have the same matrix type or cube type, such as <i>mat</i> or <i>cube</i></li>
<li>
fn(A) is one of:
<br>
<br>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
<tbody>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
       <code>exp(A)</code><sup>&nbsp;</sup>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      base-e exponential: <i>e<sup>&nbsp;x</sup></i>
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
       <code>exp2(A)</code><sup>&nbsp;</sup>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      base-2 exponential: <i>2<sup>&nbsp;x</sup></i>
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
       <code>exp10(A)</code><sup>&nbsp;</sup>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      base-10 exponential: <i>10<sup>&nbsp;x</sup></i>
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
      <code>trunc_exp(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      base-e exponential,
      truncated to avoid infinity &nbsp; <font size=-1>(only for <i>float</i> and <i>double</i> elements)</font>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
       <code>log(A)</code><sub>&nbsp;</sub>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      natural log: <i>log<sub>e</sub>&nbsp;x</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
       <code>log2(A)</code><sub>&nbsp;</sub>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      base-2 log: <i>log<sub><small>2</small></sub>&nbsp;x</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
       <code>log10(A)</code><sub>&nbsp;</sub>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      base-10 log: <i>log<sub><small>10</small></sub>&nbsp;x</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
      <code>trunc_log(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      natural log,
      truncated to avoid &plusmn;infinity &nbsp; <font size=-1>(only for <i>float</i> and <i>double</i> elements)</font>
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
       <code>pow(A, p)</code><sup>&nbsp;</sup>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      raise to the power of p: <i>x<sup><small>&nbsp;p</small></sup></i>
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
      <code>square(A)</code><sup>&nbsp;</sup>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      square: <i>x<sup><small>&nbsp;2</small></sup></i>
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
       <code>sqrt(A)</code><sup>&nbsp;</sup>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      square root: <i>x<sup>&nbsp;&frac12;</sup></i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
      <code>floor(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      largest integral value that is not greater than the input value
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
      <code>ceil(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      smallest integral value that is not less than the input value
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
      <code>round(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      round to nearest integer, with halfway cases rounded away from zero
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
      <code>trunc(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      round to nearest integer, towards zero
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
      <code>erf(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      error function
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
      <code>erfc(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      complementary error function
    </td>
  </tr>
  <tr style="background-color: #F5F5F5;">
    <td style="vertical-align: top; text-align: right;">
      <code>lgamma(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      natural log of the gamma function
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top; text-align: right;">
      <code>sign(A)</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      signum function;
      for each element <i>a</i> in <i>A</i>, the corresponding element <i>b</i> in <i>B</i> is:
      <ul>
      <table border="0" cellpadding="0" cellspacing="0">
      <tbody>
      <tr><td>              </td><td>&#9127;&nbsp;</td><td style="text-align: right;">&minus;1</td><td>&nbsp;if <i>a</i> &lt; 0</td></tr>
      <tr><td>b&nbsp;=&nbsp;</td><td>&#9128;&nbsp;</td><td style="text-align: right;">       0</td><td>&nbsp;if <i>a</i> =    0</td></tr>
      <tr><td>              </td><td>&#9129;&nbsp;</td><td style="text-align: right;">      +1</td><td>&nbsp;if <i>a</i> &gt; 0</td></tr>
      </tbody>
      </table>
      </ul>
      if <i>a</i> is complex and non-zero, then <i>b</i> = <i>a</i> / abs(<i>a</i>)
    </td>
  </tr>
</tbody>
</table>
</li>

</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = exp(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#abs">abs()</a></li>
<li><a href="#clamp">clamp()</a></li>
<li><a href="#conj">conj()</a></li>
<li><a href="#imag_real">imag() / real()</a></li>
<li><a href="#transform">.transform()</a> &nbsp; (apply user-defined function to each element)</li>
<li><a href="#expmat">expmat()</a></li>
<li><a href="#logmat">logmat()</a></li>
<li><a href="#sqrtmat">sqrtmat()</a></li>
<li><a href="#trig_fns">trigonometric functions</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#constants">miscellaneous constants</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="trig_fns"></a>
<b>trigonometric element-wise functions (cos, sin, tan, ...)</b>
<ul>
<li>
For single argument functions, <i>B = trig_fn(A)</i>, where <i>trig_fn</i> is applied to each element in <i>A</i>, with <i>trig_fn</i> as one of:
<ul>
<li>
<i>cos</i>, <i>acos</i>, <i>cosh</i>, <i>acosh</i>
</li>
<li>
<i>sin</i>, <i>asin</i>, <i>sinh</i>, <i>asinh</i>
</li>
<li>
<i>tan</i>, <i>atan</i>, <i>tanh</i>, <i>atanh</i>
</li>
</ul>
</li>
<br>
<li>
For dual argument functions, apply the function to each tuple of two corresponding elements in <i>X</i> and <i>Y</i>:
<ul>
<li><i>Z = atan2(Y, X)</i></li>
<li><i>Z = hypot(X, Y)</i></li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(5,5);
mat Y = cos(X);
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#misc_fns">miscellaneous functions</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="https://en.wikipedia.org/wiki/Trigonometric_functions">trigonometric functions in Wikipedia</a></li>
<li><a href="https://en.wikipedia.org/wiki/Atan2">atan2 function in Wikipedia</a></li>
<li><a href="https://en.wikipedia.org/wiki/Hypot">hypot function in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Decompositions, Factorisations, Inverses and Equation Solvers (Dense Matrices)</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="chol"></a>
<b>R = chol( X )</b>
<br><b>R = chol( X, layout )</b>
<br>
<br><b>chol( R, X )</b>
<br><b>chol( R, X, layout )</b>
<ul>
<li>
Cholesky decomposition of matrix <i>X</i>
</li>
<br>
<li>
Matrix <i>X</i> must be symmetric and positive-definite
</li>
<br>
<li>
By default, <i>R</i> is upper triangular, such that <i>R.t()*R = X</i>
</li>
<br>
<li>
The argument <i>layout</i> is optional; <i>layout</i> is either <code>"upper"</code> or <code>"lower"</code>, which specifies whether <i>R</i> is upper or lower triangular
</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>R = chol(X)</i> resets <i>R</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>chol(R,X)</i> resets <i>R</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(5,5);
mat Y = X.t()*X;

mat R1 = chol(Y);
mat R2 = chol(Y, "lower");
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#sqrtmat">sqrtmat()</a></li>
<li><a href="#lu">lu()</a></li>
<li><a href="#qr">qr()</a></li>
<li><a href="http://mathworld.wolfram.com/CholeskyDecomposition.html">Cholesky decomposition in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Cholesky decomposition in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eig_sym"></a>
<b>vec eigval = eig_sym( X )</b>
<br>
<br><b>eig_sym( eigval,  X )</b>
<br>
<br><b>eig_sym( eigval, eigvec, X )</b>
<br><b>eig_sym( eigval, eigvec, X, method )</b>
<ul>
<li>Eigen decomposition of <b>dense</b> symmetric/hermitian matrix <i>X</i></li>
<br>
<li>The eigenvalues and corresponding eigenvectors are stored in <i>eigval</i> and <i>eigvec</i>, respectively</li>
<br>
<li>The eigenvalues are in ascending order</li>
<br>
<li>If <i>X</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>
The <i>method</i> argument is optional; <i>method</i> is either <code>"dc"</code> or <code>"std"</code>
<ul>
<li>
<code>"dc"</code> indicates divide-and-conquer method (default setting)
</li>
<li>
<code>"std"</code> indicates standard method
</li>
<li>
the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
</li>
</ul>
</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>eigval = eig_sym(X)</i> resets <i>eigval</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>eig_sym(eigval,X)</i> resets <i>eigval</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>eig_sym(eigval,eigvec,X)</i> resets <i>eigval</i> &amp; <i>eigvec</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li><b>Caveat:</b> there is currently no check whether <i>X</i> is symmetric</li>
<br>
<li>
Examples:
<ul>
<pre>
// for matrices with real elements

mat A = randu&lt;mat&gt;(50,50);
mat B = A.t()*A;  // generate a symmetric matrix

vec eigval;
mat eigvec;

eig_sym(eigval, eigvec, B);


// for matrices with complex elements

cx_mat C = randu&lt;cx_mat&gt;(50,50);
cx_mat D = C.t()*C;

   vec eigval2;
cx_mat eigvec2;

eig_sym(eigval2, eigvec2, D);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#eig_gen">eig_gen()</a></li>
<li><a href="#eig_pair">eig_pair()</a></li>
<li><a href="#svd">svd()</a></li>
<li><a href="#svd_econ">svd_econ()</a></li>
<li><a href="#eigs_sym">eigs_sym()</a></li>
<li><a href="http://mathworld.wolfram.com/EigenDecomposition.html">eigen decomposition in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors">eigenvalues &amp; eigenvectors in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Divide-and-conquer_eigenvalue_algorithm">divide &amp; conquer eigenvalue algorithm in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eig_gen"></a>
<b>cx_vec eigval = eig_gen( X )</b>
<br>
<br><b>eig_gen( eigval, X )</b>
<br>
<br><b>eig_gen( eigval, eigvec, X )</b>
<ul>
<li>
Eigen decomposition of <b>dense</b> general (non-symmetric/non-hermitian) square matrix <i>X</i></li>
<br>
<li>The eigenvalues and corresponding eigenvectors are stored in <i>eigval</i> and <i>eigvec</i>, respectively</li>
<br>
<li>If <i>X</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>eigval = eig_gen(X)</i> resets <i>eigval</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>eig_gen(eigval,X)</i> resets <i>eigval</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>eig_gen(eigval,eigvec,X)</i> resets <i>eigval</i> &amp; <i>eigvec</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(10,10);

cx_vec eigval;
cx_mat eigvec;

eig_gen(eigval, eigvec, A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#eig_pair">eig_pair()</a></li>
<li><a href="#eig_sym">eig_sym()</a></li>
<li><a href="#svd">svd()</a></li>
<li><a href="#svd_econ">svd_econ()</a></li>
<li><a href="#schur">schur()</a></li>
<li><a href="#eigs_gen">eigs_gen()</a></li>
<li><a href="http://mathworld.wolfram.com/EigenDecomposition.html">eigen decomposition in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors">eigenvalues &amp; eigenvectors in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eig_pair"></a>
<b>cx_vec eigval = eig_pair( A, B )</b>
<br>
<br><b>eig_pair( eigval, A, B )</b>
<br>
<br><b>eig_pair( eigval, eigvec, A, B )</b>
<ul>
<li>
Eigen decomposition for pair of general <b>dense</b> square matrices <i>A</i> and <i>B</i> of the same size,
such that <i>A*eigvec&nbsp;=&nbsp;B*eigvec*diagmat(eigval)</i>
</li>
<br>
<li>The eigenvalues and corresponding eigenvectors are stored in <i>eigval</i> and <i>eigvec</i>, respectively</li>
<br>
<li>If <i>A</i> or <i>B</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>eigval = eig_pair(A,B)</i> resets <i>eigval</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>eig_pair(eigval,A,B)</i> resets <i>eigval</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>eig_pair(eigval,eigvec,A,B)</i> resets <i>eigval</i> &amp; <i>eigvec</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(10,10);
mat B = randu&lt;mat&gt;(10,10);

cx_vec eigval;
cx_mat eigvec;

eig_pair(eigval, eigvec, A, B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#eig_gen">eig_gen()</a></li>
<li><a href="#eig_sym">eig_sym()</a></li>
<li><a href="#qz">qz()</a></li>
<li><a href="http://mathworld.wolfram.com/EigenDecomposition.html">eigen decomposition in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors">eigenvalues &amp; eigenvectors in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="inv"></a>
<b>B = inv( A )</b>
<br><b>inv( B, A )</b>
<ul>
<li>
Inverse of general square matrix <i>A</i>
</li>
<br>
<li>
If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>If <i>A</i> appears to be singular:
<ul>
<li><i>B = inv(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>inv(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
<b>Caveats:</b>
<ul>
<li>
if matrix <i>A</i> is know to be symmetric positive definite, using <a href="#inv_sympd">inv_sympd()</a> is faster
</li>
<li>
if matrix <i>A</i> is know to be diagonal, use <i>inv(&nbsp;diagmat(A)&nbsp;)</i>
</li>
<li>
to solve a system of linear equations, such as <i>Z&nbsp;=&nbsp;inv(X)*Y</i>,
using <a href="#solve">solve()</a> is faster and more accurate
</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat B = inv(A);
</pre>
</ul>
</li>
<br>
<li>
See also: 
<ul>
<li><a href="#i_member">.i()</a>
<li><a href="#inv_sympd">inv_sympd()</a>
<li><a href="#rcond">rcond()</a>
<li><a href="#pinv">pinv()</a>
<li><a href="#solve">solve()</a></li>
<li><a href="#spsolve">spsolve()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="http://mathworld.wolfram.com/MatrixInverse.html">matrix inverse in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Invertible_matrix">invertible matrix in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="inv_sympd"></a>
<b>B = inv_sympd( A )</b>
<br><b>inv_sympd( B, A )</b>
<ul>
<li>
Inverse of symmetric/hermitian positive definite matrix <i>A</i>
</li>
<br>
<li>
If <i>A</i> is not square sized, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>
There is currently no explicit check whether <i>A</i> is symmetric or positive definite
</li>
<br>
<li>If <i>A</i> appears to be singular:
<ul>
<li><i>B = inv_sympd(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>inv_sympd(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
<b>Caveat:</b> 
to solve a system of linear equations, such as <i>Z&nbsp;=&nbsp;inv(X)*Y</i>,
using <a href="#solve">solve()</a> is faster and more accurate
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A   = randu&lt;mat&gt;(5,5);

mat AtA = A.t() * A;

mat B   = inv_sympd(AtA);
</pre>
</ul>
</li>
<br>
<li>
See also: 
<ul>
<li><a href="#inv">inv()</a>
<li><a href="#rcond">rcond()</a>
<li><a href="#pinv">pinv()</a>
<li><a href="#solve">solve()</a></li>
<li><a href="#eig_sym">eig_sym()</a></li>
<li><a href="http://mathworld.wolfram.com/MatrixInverse.html">matrix inverse in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Invertible_matrix">invertible matrix in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html">positive definite matrix in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Positive-definite_matrix">positive definite matrix in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="lu"></a>
<b>lu( L, U, P, X )</b>
<br><b>lu( L, U, X )</b>
<ul>
<li>
Lower-upper decomposition (with partial pivoting) of matrix <i>X</i>
</li>
<br>
<li>
The first form provides 
a lower-triangular matrix <i>L</i>,
an upper-triangular matrix <i>U</i>,
and a permutation matrix <i>P</i>,
such that <i>P.t()*L*U&nbsp;=&nbsp;X</i>
</li>
<br>
<li>
The second form provides permuted <i>L</i> and <i>U</i>, such that <i>L*U = X</i>;
note that in this case <i>L</i> is generally not lower-triangular
</li>
<br>
<li>
If the decomposition fails:
<ul>
<li><i>lu(L,U,P,X)</i> resets <i>L</i>, <i>U</i>, <i>P</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>lu(L,U,X)</i> resets <i>L</i>, <i>U</i>  and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);

mat L, U, P;

lu(L, U, P, A);

mat B = P.t()*L*U;
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#chol">chol()</a></li>
<li><a href="http://en.wikipedia.org/wiki/LU_decomposition">LU decomposition in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/LUDecomposition.html">LU decomposition in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="null"></a>
<b>B = null( A )</b>
<br><b>B = null( A, tolerance )</b>
<br>
<br><b>null( B, A )</b>
<br><b>null( B, A, tolerance )</b>
<ul>
<li>
Find the orthonormal basis of the null space of matrix <i>A</i>
</li>
<br>
<li>
The dimension of the range space is the number of singular values of <i>A</i> not greater than <i>tolerance</i>
</li>
<br>
<li>
The <i>tolerance</i> argument is optional; by default <i>tolerance</i> is <i>max(m,n)*max_sv*datum::eps</i>, where:
<ul>
<li><i>m</i> = number of rows and <i>n</i> = number of columns in <i>A</i></li>
<li><i>max_sv</i> = maximal singular value of <i>A</i></li>
<li><i>datum::eps</i> = difference between 1 and the least value greater than 1 that is representable</li>
</ul>
</li>
<br>
<li>
The computation is based on singular value decomposition; if the decomposition fails:
<ul>
<li><i>B = null(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>null(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,6);

A.row(0).zeros();
A.col(0).zeros();

mat B = null(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#orth">orth()</a></li>
<li><a href="#qr">qr()</a></li>
<li><a href="#svd">svd()</a></li>
<li><a href="#rank">rank()</a></li>
<li><a href="#constants">datum::eps</a></li>
<li><a href="http://en.wikipedia.org/wiki/Orthonormal_basis">Orthonormal basis in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="orth"></a>
<b>B = orth( A )</b>
<br><b>B = orth( A, tolerance )</b>
<br>
<br><b>orth( B, A )</b>
<br><b>orth( B, A, tolerance )</b>
<ul>
<li>
Find the orthonormal basis of the range space of matrix <i>A</i>,
so that <i>B.t()*B&nbsp;&asymp;&nbsp;eye(r,r)</i>, where <i>r&nbsp;=&nbsp;rank(A)</i>
</li>
<br>
<li>
The dimension of the range space is the number of singular values of <i>A</i> greater than <i>tolerance</i>
</li>
<br>
<li>
The <i>tolerance</i> argument is optional; by default <i>tolerance</i> is <i>max(m,n)*max_sv*datum::eps</i>, where:
<ul>
<li><i>m</i> = number of rows and <i>n</i> = number of columns in <i>A</i></li>
<li><i>max_sv</i> = maximal singular value of <i>A</i></li>
<li><i>datum::eps</i> = difference between 1 and the least value greater than 1 that is representable</li>
</ul>
</li>
<br>
<li>
The computation is based on singular value decomposition; if the decomposition fails:
<ul>
<li><i>B = orth(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>orth(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,6);

mat B = orth(A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#null">null()</a></li>
<li><a href="#qr">qr()</a></li>
<li><a href="#svd">svd()</a></li>
<li><a href="#rank">rank()</a></li>
<li><a href="#constants">datum::eps</a></li>
<li><a href="http://en.wikipedia.org/wiki/Orthonormal_basis">Orthonormal basis in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="pinv"></a>
<b>B = pinv( A )</b>
<br><b>B = pinv( A, tolerance )</b>
<br><b>B = pinv( A, tolerance, method )</b>
<br>
<br><b>pinv( B, A )</b>
<br><b>pinv( B, A, tolerance )</b>
<br><b>pinv( B, A, tolerance, method )</b>
<ul>
<li>Moore-Penrose pseudo-inverse of matrix <i>A</i></li>
<br>
<li>
The computation is based on singular value decomposition
</li>
<br>
<li>The <i>tolerance</i> argument is optional</li>
<br>
<li>The default tolerance is <i>max(m,n)*max_sv*datum::eps</i>, where:
<ul>
<li><i>m</i> = number of rows and <i>n</i> = number of columns in <i>A</i></li>
<li><i>max_sv</i> = maximal singular value of <i>A</i></li>
<li><i>datum::eps</i> = difference between 1 and the least value greater than 1 that is representable</li>
</ul>
</li>
<br>
<li>Any singular values less than <i>tolerance</i> are treated as zero</li>
<br>
<li>
The <i>method</i> argument is optional; <i>method</i> is either <code>"dc"</code> or <code>"std"</code>
<ul>
<li>
<code>"dc"</code> indicates divide-and-conquer method (default setting)
</li>
<li>
<code>"std"</code> indicates standard method
</li>
<li>
the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
</li>
</ul>
</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>B = pinv(A)</i> resets <i>B</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>pinv(B,A)</i> resets <i>B</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(4,5);

mat B = pinv(A);        // use default tolerance

mat C = pinv(A, 0.01);  // set tolerance to 0.01
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#i_member">.i()</a></li>
<li><a href="#inv">inv()</a></li>
<li><a href="#solve">solve()</a></li>
<li><a href="#spsolve">spsolve()</a></li>
<li><a href="#constants">datum::eps</a></li>
<li><a href="http://mathworld.wolfram.com/Pseudoinverse.html">Pseudoinverse in MathWorld</a></li>
<li><a href="http://mathworld.wolfram.com/Moore-PenroseMatrixInverse.html">Moore-Penrose Matrix Inverse in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse">Moore-Penrose pseudoinverse in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="qr"></a>
<b>qr( Q, R, X )</b>
<ul>
<li>
Decomposition of <i>X</i> into an orthogonal matrix <i>Q</i> and a right triangular matrix <i>R</i>, such that <i>Q*R = X</i>
</li>
<br>
<li>
If the decomposition fails, <i>Q</i> and <i>R</i> are reset and the function returns a bool set to <i>false</i> (exception is not thrown)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(5,5);
mat Q, R;

qr(Q,R,X);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#qr_econ">qr_econ()</a></li>
<li><a href="#chol">chol()</a></li>
<li><a href="#orth">orth()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Orthogonal_matrix">orthogonal matrix in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/QR_decomposition">QR decomposition in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/QRDecomposition.html">QR decomposition in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="qr_econ"></a>
<b>qr_econ( Q, R, X )</b>
<ul>
<li>
Economical decomposition of <i>X</i> (with size <i>m</i> x <i>n</i>) into an orthogonal matrix <i>Q</i> and a right triangular matrix <i>R</i>, such that <i>Q*R = X</i>
</li>
<br>
<li>
If <i>m</i> &gt; <i>n</i>, only the first <i>n</i> rows of <i>R</i> and the first <i>n</i> columns of <i>Q</i> are calculated 
(ie. the zero rows of <i>R</i> and the corresponding columns of <i>Q</i> are omitted)
</li>
<br>
<li>
If the decomposition fails, <i>Q</i> and <i>R</i> are reset and the function returns a bool set to <i>false</i> (exception is not thrown)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(6,5);
mat Q, R;

qr_econ(Q,R,X);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#qr">qr()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Orthogonal_matrix">orthogonal matrix in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/QR_decomposition">QR decomposition in Wikipedia</a></li>
<li><a href="http://octave.sourceforge.net/octave/function/qr.html">QR decomposition in Octave</a></li>
<li><a href="http://mathworld.wolfram.com/QRDecomposition.html">QR decomposition in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="qz"></a>
<b>qz( AA, BB, Q, Z, A, B )</b>
<br><b>qz( AA, BB, Q, Z, A, B, select )</b>
<ul>
<li>
Generalised Schur decomposition for pair of general square matrices <i>A</i> and <i>B</i> of the same size,
<br>such that <i>A&nbsp;=&nbsp;Q.t()*AA*Z.t()</i> and <i>B&nbsp;=&nbsp;Q.t()*BB*Z.t()</i>
</li>
<br>
<li>The <i>select</i> argument is optional and specifies the ordering of the top left of the Schur form; it is one of the following:
<br>
<ul>
<table>
<tbody>
<tr><td><code>"none"</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>no ordering (default operation)</td></tr>
<tr><td><code>"lhp"</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>left-half-plane: eigenvalues with real part &lt; 0</td></tr>
<tr><td><code>"rhp"</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>right-half-plane: eigenvalues with real part &gt; 0</td></tr>
<tr><td><code>"iuc"</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>inside-unit-circle: eigenvalues with absolute value &lt; 1</td></tr>
<tr><td><code>"ouc"</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>outside-unit-circle: eigenvalues with absolute value &gt; 1</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>The left and right Schur vectors are stored in <i>Q</i> and <i>Z</i>, respectively</li>
<br>
<li>In the complex-valued problem, the generalised eigenvalues are found in <i>diagvec(AA) / diagvec(BB)</i></li>
<br>
<li>If <i>A</i> or <i>B</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the decomposition fails, <i>AA</i>, <i>BB</i>, <i>Q</i> and <i>Z</i> are reset, and the function returns a bool set to <i>false</i> (exception is not thrown)</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(10,10);
mat B = randu&lt;mat&gt;(10,10);

mat AA;
mat BB;
mat Q;
mat Z; 

qz(AA, BB, Q, Z, A, B);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#schur">schur()</a></li>
<li><a href="#eig_pair">eig_pair()</a></li>
<li><a href="https://en.wikipedia.org/wiki/Schur_decomposition#Generalized_Schur_decomposition">generalised Schur decomposition in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="schur"></a>
<b>S = schur( X )</b>
<br>
<br><b>schur( S,  X )</b>
<br>
<br><b>schur( U, S, X )</b>
<ul>
<li>Schur decomposition of square matrix <i>X</i>, such that <i>X = U*S*U.t()</i></li>
<br>
<li><i>U</i> is a unitary matrix containing the Schur vectors</li>
<br>
<li><i>S</i> is an upper triangular matrix, called the Schur form of <i>X</i></li>
<br>
<li>If <i>X</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>S = schur(X)</i> resets <i>S</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>schur(S,X)</i> resets <i>S</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>schur(U,S,X)</i> resets <i>U</i> &amp; <i>S</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
<b>Caveat:</b> in general, Schur decomposition is not unique
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X(20,20, fill::randu);

mat U;
mat S;

schur(U, S, X);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#qz">qz()</a></li>
<li><a href="#eig_gen">eig_gen()</a></li>
<li><a href="http://mathworld.wolfram.com/SchurDecomposition.html">Schur decomposition in MathWorld</a></li>
<li><a href="https://en.wikipedia.org/wiki/Schur_decomposition">Schur decomposition in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="solve"></a>
<b>X = solve( A, B )</b>
<br><b>X = solve( A, B, settings )</b>
<br>
<br><b>solve( X, A, B )</b>
<br><b>solve( X, A, B, settings )</b>
<ul>
<li>Solve a <b>dense</b> system of linear equations, <i>A*X = B</i>, where <i>X</i> is unknown;
similar functionality to the \ operator in Matlab/Octave, ie. <i>X&nbsp;=&nbsp;A&nbsp;\&nbsp;B</i>
</li>
<br>
<li><i>A</i> can be square (critically determined system), or non-square (under/over-determined system)
</li>
<br>
<li>
<i>B</i> can be a vector or matrix
</li>
<br>
<li>
The number of rows in <i>A</i> and <i>B</i> must be the same
</li>
<br>
<li>The <i>settings</i> argument is optional; it is one of the following, or a combination thereof:
<br>
<br>
<table>
<tbody>
<tr><td><code>solve_opts::fast</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>fast mode: do not apply iterative refinement and/or equilibration</td></tr>
<tr><td><code>solve_opts::equilibrate</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>equilibrate the system before solving &nbsp; (matrix <i>A</i> must be square)</td></tr>
<tr><td><code>solve_opts::no_approx</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>do not find approximate solutions for rank deficient systems</td></tr>
</tbody>
</table>
<br>
the above settings can be combined using the <code>+</code> operator; for example: <code>solve_opts::fast&nbsp;+&nbsp;solve_opts::no_approx</code>
</li>
<br>
<li>
<b>Caveat:</b> using <i><code>solve_opts::fast</code></i> will speed up finding the solution, but for poorly conditioned systems the solution may have lower quality 
</li>
<br>
<li>
If <i>A</i> is known to be a triangular matrix,
the solution can be computed faster by explicitly indicating that <i>A</i> is triangular through <a href="#trimat">trimatu()</a> or <a href="#trimat">trimatl()</a>;
<br>indicating a triangular matrix also implies that <i><code>solve_opts::fast</code></i> is enabled
</li>
<br>
<li>
If no solution is found:
<ul>
<li><i>X = solve(A,B)</i> resets <i>X</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>solve(X,A,B)</i> resets <i>X</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
vec b = randu&lt;vec&gt;(5);
mat B = randu&lt;mat&gt;(5,5);

vec x1 = solve(A, b);

vec x2;
bool status = solve(x2, A, b);

mat X1 = solve(A, B);

mat X2 = solve(A, B, solve_opts::fast);  // enable fast mode

mat X3 = solve(trimatu(A), B);  // indicate that A is triangular
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#inv">inv()</a></li>
<li><a href="#pinv">pinv()</a></li>
<li><a href="#rcond">rcond()</a></li>
<li><a href="#syl">syl()</a></li>
<li><a href="#spsolve">spsolve()</a></li>
<li><a href="http://mathworld.wolfram.com/LinearSystemofEquations.html">linear system of equations in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Linear_system_of_equations">system of linear equations in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="svd"></a>
<b>vec s = svd( X )</b>
<br>
<br><b>svd( vec s, X )</b>
<br>
<br><b>svd( mat U, vec s, mat V, mat X )</b>
<br><b>svd( mat U, vec s, mat V, mat X, method )</b>
<br>
<br><b>svd( cx_mat U, vec s, cx_mat V, cx_mat X )</b>
<br><b>svd( cx_mat U, vec s, cx_mat V, cx_mat X, method )</b>
<ul>
<li>
Singular value decomposition of <b>dense</b> matrix <i>X</i>
</li>
<br>
<li>If <i>X</i> is square, it can be reconstructed using <i>X = U*diagmat(s)*V.t()</i>
</li>
<br>
<li>
The singular values are in descending order
</li>
<br>
<li>
The <i>method</i> argument is optional; <i>method</i> is either <code>"dc"</code> or <code>"std"</code>
<ul>
<li>
<code>"dc"</code> indicates divide-and-conquer method (default setting)
</li>
<li>
<code>"std"</code> indicates standard method
</li>
<li>
the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
</li>
</ul>
</li>
<br>
<li>
If the decomposition fails, the output objects are reset and:
<ul>
<li><i>s = svd(X)</i> resets <i>s</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>svd(s,X)</i> resets <i>s</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>svd(U,s,V,X)</i> resets <i>U</i>, <i>s</i>, <i>V</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(5,5);

mat U;
vec s;
mat V;

svd(U,s,V,X);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#svd_econ">svd_econ()</a></li>
<li><a href="#eig_gen">eig_gen()</a></li>
<li><a href="#eig_sym">eig_sym()</a></li>
<li><a href="#svds">svds()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">singular value decomposition in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">singular value decomposition in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="svd_econ"></a>
<b>svd_econ( mat U, vec s, mat V, mat X )</b>
<br><b>svd_econ( mat U, vec s, mat V, mat X, mode )</b>
<br><b>svd_econ( mat U, vec s, mat V, mat X, mode, method )</b>
<br>
<br><b>svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X )</b>
<br><b>svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X, mode )</b>
<br><b>svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X, mode, method )</b>
<ul>
<li>
Economical singular value decomposition of <b>dense</b> matrix <i>X</i>
</li>
<br>
<li>
The singular values are in descending order
</li>
<br>
<li>
The <i>mode</i> argument is optional; <i>mode</i> is one of:
<ul>
<table>
<tbody>
<tr><td></td><td style="text-align: left;"><code>"both"</code></td><td>&nbsp;=&nbsp;</td><td>compute both left and right singular vectors (default operation)</td></tr>
<tr><td></td><td style="text-align: left;"><code>"left"</code></td><td>&nbsp;=&nbsp;</td><td>compute only left singular vectors</td></tr>
<tr><td></td><td style="text-align: left;"><code>"right"</code></td><td>&nbsp;=&nbsp;</td><td>compute only right singular vectors</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
The <i>method</i> argument is optional; <i>method</i> is either <code>"dc"</code> or <code>"std"</code>
<ul>
<li>
<code>"dc"</code> indicates divide-and-conquer method (default setting)
</li>
<li>
<code>"std"</code> indicates standard method
</li>
<li>
the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices
</li>
</ul>
</li>
<br>
<li>
If the decomposition fails, <i>U</i>, <i>s</i>, <i>V</i> are reset and a bool set to <i>false</i> is returned (exception is not thrown)
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(4,5);

mat U;
vec s;
mat V;

svd_econ(U, s, V, X);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#svd">svd()</a></li>
<li><a href="#eig_gen">eig_gen()</a></li>
<li><a href="#eig_sym">eig_sym()</a></li>
<li><a href="#svds">svds()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">singular value decomposition in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">singular value decomposition in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="syl"></a>
<b>X = syl( A, B, C )</b>
<br><b>syl( X, A, B, C )</b>
<ul>
<li>Solve the Sylvester equation, ie. <i>AX + XB + C = 0</i>, where <i>X</i> is unknown</li>
<br>
<li>Matrices <i>A</i>, <i>B</i> and <i>C</i> must be square sized</li>
<br>
<li>
If no solution is found:
<ul>
<li><i>syl(A,B,C)</i> resets <i>X</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>syl(X,A,B,C)</i> resets <i>X</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,5);
mat B = randu&lt;mat&gt;(5,5);
mat C = randu&lt;mat&gt;(5,5);

mat X1 = syl(A, B, C);

mat X2;
syl(X2, A, B, C);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#solve">solve()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sylvester_equation">Sylvester equation in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Decompositions, Factorisations and Equation Solvers (Sparse Matrices)</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eigs_sym"></a>
<b>vec eigval = eigs_sym( X, k )</b>
<br><b>vec eigval = eigs_sym( X, k, form )</b>
<br><b>vec eigval = eigs_sym( X, k, form, tol )</b>
<br>
<br><b>eigs_sym( eigval, X, k )</b>
<br><b>eigs_sym( eigval, X, k, form )</b>
<br><b>eigs_sym( eigval, X, k, form, tol )</b>
<br>
<br><b>eigs_sym( eigval, eigvec, X, k )</b>
<br><b>eigs_sym( eigval, eigvec, X, k, form )</b>
<br><b>eigs_sym( eigval, eigvec, X, k, form, tol )</b>
<ul>
<li>Obtain a limited number of eigenvalues and eigenvectors of <b>sparse</b> symmetric real matrix <i>X</i></li>
<br>
<li>
<i>k</i> specifies the number of eigenvalues and eigenvectors
</li>
<br>
<li>The argument <i>form</i> is optional; <i>form</i> is one of:
<ul>
<table>
<tbody>
<tr><td><code>"lm"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with largest magnitude (default operation)</td></tr>
<tr><td><code>"sm"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with smallest magnitude</td></tr>
<tr><td><code>"la"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with largest algebraic value</td></tr>
<tr><td><code>"sa"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with smallest algebraic value</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
The argument <i>tol</i> is optional; it specifies the tolerance for convergence
</li>
<br>
<li>The eigenvalues and corresponding eigenvectors are stored in <i>eigval</i> and <i>eigvec</i>, respectively</li>
<br>
<li>If <i>X</i> is not square sized, a <i>std::logic_error</i> exception is thrown</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>eigval = eigs_sym(X,k)</i> resets <i>eigval</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>eigs_sym(eigval,X,k)</i> resets <i>eigval</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>eigs_sym(eigval,eigvec,X,k)</i> resets <i>eigval</i> &amp; <i>eigvec</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li><b>Caveat:</b> there is currently no check whether <i>X</i> is symmetric</li>
<br>
<li>
Examples:
<ul>
<pre>
// generate sparse matrix
sp_mat A = sprandu&lt;sp_mat&gt;(1000, 1000, 0.1);
sp_mat B = A.t()*A;

vec eigval;
mat eigvec;

eigs_sym(eigval, eigvec, B, 5);  // find 5 eigenvalues/eigenvectors
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#eigs_gen">eigs_gen()</a></li>
<li><a href="#eig_sym">eig_sym()</a></li>
<li><a href="#svds">svds()</a></li>
<li><a href="http://mathworld.wolfram.com/EigenDecomposition.html">eigen decomposition in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors">eigenvalues &amp; eigenvectors in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="eigs_gen"></a>
<b>cx_vec eigval = eigs_gen( X, k )</b>
<br><b>cx_vec eigval = eigs_gen( X, k, form )</b>
<br><b>cx_vec eigval = eigs_gen( X, k, form, tol )</b>
<br>
<br><b>eigs_gen( eigval, X, k )</b>
<br><b>eigs_gen( eigval, X, k, form )</b>
<br><b>eigs_gen( eigval, X, k, form, tol )</b>
<br>
<br><b>eigs_gen( eigval, eigvec, X, k )</b>
<br><b>eigs_gen( eigval, eigvec, X, k, form )</b>
<br><b>eigs_gen( eigval, eigvec, X, k, form, tol )</b>
<ul>
<li>
Obtain a limited number of eigenvalues and eigenvectors of <b>sparse</b> general (non-symmetric/non-hermitian) square matrix <i>X</i>
</li>
<br>
<li>
<i>k</i> specifies the number of eigenvalues and eigenvectors
</li>
<br>
<li>The argument <i>form</i> is optional; <i>form</i> is one of:
<ul>
<table>
<tbody>
<tr><td><code>"lm"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with largest magnitude (default operation)</td></tr>
<tr><td><code>"sm"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with smallest magnitude</td></tr>
<tr><td><code>"lr"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with largest real part</td></tr>
<tr><td><code>"sr"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with smallest real part</td></tr>
<tr><td><code>"li"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with largest imaginary part</td></tr>
<tr><td><code>"si"</code></td><td>&nbsp;=&nbsp;</td><td>obtain eigenvalues with smallest imaginary part</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
The argument <i>tol</i> is optional; it specifies the tolerance for convergence
</li>
<br>
<li>
The eigenvalues and corresponding eigenvectors are stored in <i>eigval</i> and <i>eigvec</i>, respectively
</li>
<br>
<li>
If <i>X</i> is not square sized, a <i>std::logic_error</i> exception is thrown
</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>eigval = eigs_gen(X,k)</i> resets <i>eigval</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>eigs_gen(eigval,X,k)</i> resets <i>eigval</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>eigs_gen(eigval,eigvec,X,k)</i> resets <i>eigval</i> &amp; <i>eigvec</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
// generate sparse matrix
sp_mat A = sprandu&lt;sp_mat&gt;(1000, 1000, 0.1);  

cx_vec eigval;
cx_mat eigvec;

eigs_gen(eigval, eigvec, A, 5);  // find 5 eigenvalues/eigenvectors
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#eigs_sym">eigs_sym()</a></li>
<li><a href="#eig_gen">eig_gen()</a></li>
<li><a href="#svds">svds()</a></li>
<li><a href="http://mathworld.wolfram.com/EigenDecomposition.html">eigen decomposition in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors">eigenvalues &amp; eigenvectors in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="spsolve"></a>
<b>X = spsolve( A, B )</b>
<br><b>X = spsolve( A, B, solver )</b>
<br><b>X = spsolve( A, B, solver, settings )</b>
<br>
<br><b>spsolve( X, A, B )</b>
<br><b>spsolve( X, A, B, solver )</b>
<br><b>spsolve( X, A, B, solver, settings )</b>
<br>
<ul>
<li>
Solve a <b>sparse</b> system of linear equations, <i>A*X = B</i>, where <i>A</i> is a sparse matrix, <i>B</i> is a dense matrix or vector, and <i>X</i> is unknown
</li>
<br>
<li>
The number of rows in <i>A</i> and <i>B</i> must be the same
</li>
<br>
<li>
If no solution is found:
<ul>
<li><i>X = spsolve(A, B)</i> resets <i>X</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>spsolve(X, A, B)</i> resets <i>X</i> and returns a bool set to <i>false</i> (no exception is thrown)</li>
</ul>
</li>
<br>
<li>
The <i>solver</i> argument is optional; <i>solver</i> is either <code>"superlu"</code> or <code>"lapack"</code>; by default <code>"superlu"</code> is used
<ul>
<li>
For <code>"superlu"</code>, <i>ARMA_USE_SUPERLU</i> must be enabled in <a href="#config_hpp">config.hpp</a>
</li>
<li>
For <code>"lapack"</code>, sparse matrix <i>A</i> is converted to a dense matrix before using the LAPACK solver; this considerably increases memory usage
</li>
</ul>
</li>
<br>
<li>
The <i>settings</i> argument is optional 
<br>
<br>
<ul>
<li>
when <i>solver</i> is "superlu", <i>settings</i> is an instance of the <i>superlu_opts</i> structure:
<pre>
struct superlu_opts
  {
  bool             equilibrate;  // default: false
  bool             symmetric;    // default: false
  double           pivot_thresh; // default: 1.0
  permutation_type permutation;  // default: superlu_opts::COLAMD
  refine_type      refine;       // default: superlu_opts::REF_DOUBLE
  };
</pre>
</li>
<li>
<i>equilibrate</i> is either <i>true</i> or <i>false</i>; it indicates whether to equilibrate the system (scale the rows and columns of <i>A</i> to have unit norm)
</li>
<br>
<li>
<i>symmetric</i> is either <i>true</i> or <i>false</i>; it indicates whether to use SuperLU symmetric mode, which gives preference to diagonal pivots
</li>
<br>
<li>
<i>pivot_threshold</i> is in the range [0.0,&nbsp;1.0], used for determining whether a diagonal entry is an acceptable pivot (details in SuperLU documentation)
</li>
<br>
<li>
<i>permutation</i> specifies the type of column permutation; it is one of:
<br>
<br>
<table style="text-align: left;" border="0" cellpadding="0" cellspacing="0">
<tr><td><code>superlu_opts::NATURAL</code></td><td>&nbsp;&nbsp;</td><td>natural ordering</td></tr>
<tr><td><code>superlu_opts::MMD_ATA</code></td><td>&nbsp;&nbsp;</td><td>minimum degree ordering on structure of <code>A.t()&nbsp;*&nbsp;A</code></td></tr>
<tr><td><code>superlu_opts::MMD_AT_PLUS_A</code></td><td>&nbsp;&nbsp;</td><td>minimum degree ordering on structure of <code>A.t()&nbsp;+&nbsp;A</code></td></tr>
<tr><td><code>superlu_opts::COLAMD</code></td><td>&nbsp;&nbsp;</td><td>approximate minimum degree column ordering</td></tr>
</table>
</li>
<br>
<li>  
<i>refine</i> specifies the type of iterative refinement; it is one of:
<br>
<br>
<table style="text-align: left;" border="0" cellpadding="0" cellspacing="0">
<tr><td><code>superlu_opts::REF_NONE</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>no refinement</td></tr>
<tr><td><code>superlu_opts::REF_SINGLE</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>iterative refinement in single precision</td></tr>
<tr><td><code>superlu_opts::REF_DOUBLE</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>iterative refinement in double precision</td></tr>
<tr><td><code>superlu_opts::REF_EXTRA</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>iterative refinement in extra precision</td></tr>
</table>
</li>
</ul>
<br>
<li>
Examples:
<ul>
<pre>
sp_mat A = sprandu&lt;sp_mat&gt;(1000, 1000, 0.1);

vec b = randu&lt;vec&gt;(1000);
mat B = randu&lt;mat&gt;(1000, 5);

vec x = spsolve(A, b);  // solve one system
mat X = spsolve(A, B);  // solve several systems

bool status = spsolve(x, A, b);  // use default solver
if(status == false)  { cout &lt;&lt; "no solution" &lt;&lt; endl; }

spsolve(x, A, b, "lapack");   // use LAPACK  solver
spsolve(x, A, b, "superlu");  // use SuperLU solver

superlu_opts settings;

settings.permutation = superlu_opts::NATURAL;
settings.refine      = superlu_opts::REF_NONE;

spsolve(x, A, b, "superlu", settings);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#solve">solve()</a></li>
<li><a href="http://crd-legacy.lbl.gov/~xiaoye/SuperLU/">SuperLU home page</a>
<li><a href="http://mathworld.wolfram.com/LinearSystemofEquations.html">linear system of equations in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Linear_system_of_equations">system of linear equations in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="svds"></a>
<b>vec s = svds( X, k )</b>
<br><b>vec s = svds( X, k, tol )</b>
<br>
<br><b>svds( vec s, X, k )</b> 
<br><b>svds( vec s, X, k, tol )</b> 
<br>
<br><b>svds( mat U, vec s, mat V, sp_mat X, k )</b>
<br><b>svds( mat U, vec s, mat V, sp_mat X, k, tol )</b>
<br>
<br><b>svds( cx_mat U, vec s, cx_mat V, sp_cx_mat X, k )</b>
<br><b>svds( cx_mat U, vec s, cx_mat V, sp_cx_mat X, k, tol )</b>
<ul>
<li>
Obtain a limited number of singular values and singular vectors of <b>sparse</b> matrix <i>X</i>
</li>
<br>
<li>
<i>k</i> specifies the number of singular values and singular vectors 
</li>
<br>
<li>
The argument <i>tol</i> is optional; it specifies the tolerance for convergence
</li>
<br>
<li>
The singular values are calculated via sparse eigen decomposition of:
<code>
<table style="text-align: left;" border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr><td>&#9121;&nbsp;</td><td>zeros(X.n_rows, X.n_rows)</td><td>&nbsp;</td><td>                          X</td><td>&nbsp;&#9124;</td></tr>
<tr><td>&#9123;&nbsp;</td><td style="text-align: right;">X.t()</td><td>&nbsp;</td><td>zeros(X.n_cols, X.n_cols)</td><td>&nbsp;&#9126;</td></tr> 
</tbody>
</table>
</code>
</li>
<br>
<li>
The singular values are in descending order
</li>
<br>
<li>
If the decomposition fails, the output objects are reset and:
<ul>
<li><i>s = svds(X,k)</i> resets <i>s</i> and throws a <i>std::runtime_error</i> exception</li>
<li><i>svds(s,X,k)</i> resets <i>s</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
<li><i>svds(U,s,V,X,k)</i> resets <i>U</i>, <i>s</i>, <i>V</i> and returns a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
<b>Caveat:</b> <i>svds()</i> is intended only for finding a few singular values from a large sparse matrix;
to find all singular values, use <a href="#svd">svd()</a> instead
</li>
<br>
<li>
Examples:
<ul>
<pre>
sp_mat X = sprandu&lt;sp_mat&gt;(100, 200, 0.1);

mat U;
vec s;
mat V;

svds(U, s, V, X, 10);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#eigs_gen">eigs_gen()</a></li>
<li><a href="#eigs_sym">eigs_sym()</a></li>
<li><a href="#svd">svd()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">singular value decomposition in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">singular value decomposition in MathWorld</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Signal &amp; Image Processing</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="conv"></a>
<b>conv( A, B )</b>
<br><b>conv( A, B, shape )</b>
<ul>
<li>
1D convolution of vectors <i>A</i> and <i>B</i>
</li>
<br>
<li>
The orientation of the result vector is the same as the orientation of <i>A</i> (ie. either column or row vector)
</li>
<br>
<li>
The <i>shape</i> argument is optional; it is one of:
<ul>
<ul>
<table>
<tbody>
<tr><td style="text-align: right;"><code>"full"</code></td><td>&nbsp;=&nbsp;</td><td>return the full convolution (<b>default setting</b>), with the size equal to <i>A.n_elem&nbsp;+&nbsp;B.n_elem&nbsp;-&nbsp;1</i></td></tr>
<tr><td style="text-align: right;"><code>"same"</code></td><td>&nbsp;=&nbsp;</td><td>return the central part of the convolution, with the same size as vector <i>A</i></td></tr>
</tbody>
</table>
</ul>
</ul>
</li>
<br>
<li>
The convolution operation is also equivalent to FIR filtering
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec A(256, fill::randu);

vec B(16, fill::randu);

vec C = conv(A, B);

vec D = conv(A, B, "same");
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#conv2">conv2()</a></li>
<li><a href="#fft">fft()</a></li>
<li><a href="#cor">cor()</a></li>
<li><a href="#interp1">interp1()</a></li>
<li><a href="http://mathworld.wolfram.com/Convolution.html">Convolution in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Convolution">Convolution in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Finite_impulse_response">FIR filter in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="conv2"></a>
<b>conv2( A, B )</b>
<br><b>conv2( A, B, shape )</b>
<ul>
<li>
2D convolution of matrices <i>A</i> and <i>B</i>
</li>
<br>
<li>
The <i>shape</i> argument is optional; it is one of:
<ul>
<ul>
<table>
<tbody>
<tr><td style="text-align: right;"><code>"full"</code></td><td>&nbsp;=&nbsp;</td><td>return the full convolution (<b>default setting</b>), with the size equal to <i>size(A)&nbsp;+&nbsp;size(B)&nbsp;-&nbsp;1</i></td></tr>
<tr><td style="text-align: right;"><code>"same"</code></td><td>&nbsp;=&nbsp;</td><td>return the central part of the convolution, with the same size as matrix <i>A</i></td></tr>
</tbody>
</table>
</ul>
</ul>
</li>
<br>
<li>The implementation of 2D convolution in this version is preliminary; it is not yet fully optimised</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A(256, 256, fill::randu);

mat B(16, 16, fill::randu);

mat C = conv2(A, B);

mat D = conv2(A, B, "same");
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#conv">conv()</a></li>
<li><a href="#fft2">fft2()</a></li>
<li><a href="http://mathworld.wolfram.com/Convolution.html">Convolution in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Convolution">Convolution in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Kernel_%28image_processing%29">Kernel (image processing) in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="fft"></a>
<b>cx_mat Y = &nbsp;fft( X )</b><br>
<b>cx_mat Y = &nbsp;fft( X, n )</b><br>
<br>
<b>cx_mat Z = ifft( cx_mat Y )</b><br>
<b>cx_mat Z = ifft( cx_mat Y, n )</b><br>
<ul>
<li><i>fft():</i> fast Fourier transform of a vector or matrix (real or complex)</li>
<br>
<li><i>ifft():</i> inverse fast Fourier transform of a vector or matrix (complex only)</li>
<br>
<li>If given a matrix, the transform is done on each column vector of the matrix</li>
<br>
<li>
The optional <i>n</i> argument specifies the transform length:
<ul>
<li>if <i>n</i> is larger than the length of the input vector, a zero-padded version of the vector is used</li>
<li>if <i>n</i> is smaller than the length of the input vector, only the first <i>n</i> elements of the vector are used</li>
</ul>
</li>
<br>
<li>
If <i>n</i> is not specified, the transform length is the same as the length of the input vector
</li>
<br>
<li><b>Caveat:</b> the transform is fastest when the transform length is a power of 2, eg. 64, 128, 256, 512, 1024, ...</li>
<br>
<li>The implementation of the transform in this version is preliminary; it is not yet fully optimised</li>
<br>
<li>
Examples:
<ul>
<pre>
   vec X = randu&lt;vec&gt;(100);
cx_vec Y = fft(X, 128);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#fft2">fft2()</a></li>
<li><a href="#conv">conv()</a></li>
<li><a href="#imag_real">real()</a></li>
<li><a href="http://mathworld.wolfram.com/FastFourierTransform.html">fast Fourier transform in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fast_Fourier_transform">fast Fourier transform in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="fft2"></a>
<b>cx_mat Y = &nbsp;fft2( X )</b><br>
<b>cx_mat Y = &nbsp;fft2( X, n_rows, n_cols )</b><br>
<br>
<b>cx_mat Z = ifft2( cx_mat Y )</b><br>
<b>cx_mat Z = ifft2( cx_mat Y, n_rows, n_cols )</b><br>
<ul>
<li><i>fft2():</i> 2D fast Fourier transform of a matrix (real or complex)</li>
<br>
<li><i>ifft2():</i> 2D inverse fast Fourier transform of a matrix (complex only)</li>
<br>
<li>
The optional arguments <i>n_rows</i> and <i>n_cols</i> specify the size of the transform;
a truncated and/or zero-padded version of the input matrix is used
</li>
<br>
<li><b>Caveat:</b> the transform is fastest when both <i>n_rows</i> and <i>n_cols</i> are a power of 2, eg. 64, 128, 256, 512, 1024, ...</li>
<br>
<li>The implementation of the transform in this version is preliminary; it is not yet fully optimised</li>
<br>
<li>
Examples:
<ul>
<pre>
   mat A = randu&lt;mat&gt;(100,100);
cx_mat B = fft2(A);
cx_mat C = fft2(A, 128, 128);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#fft">fft()</a></li>
<li><a href="#conv2">conv2()</a></li>
<li><a href="#imag_real">real()</a></li>
<li><a href="http://mathworld.wolfram.com/FastFourierTransform.html">fast Fourier transform in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fast_Fourier_transform">fast Fourier transform in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="interp1"></a>
<b>interp1( X, Y, XI, YI )</b>
<br><b>interp1( X, Y, XI, YI, method )</b>
<br><b>interp1( X, Y, XI, YI, method, extrapolation_value )</b>
<ul>
<li>
1D data interpolation
</li>
<br>
<li>
Given a 1D function specified in vectors <i>X</i> and <i>Y</i>
(where <i>X</i> specifies locations and <i>Y</i> specifies the corresponding values),
<br>
generate vector <i>YI</i> which contains interpolated values at locations <i>XI</i>
</li>
<br>
<li>
The <i>method</i> argument is optional; it is one of:
<ul>
<table>
<tbody>
<tr><td style="text-align: right;"><code>"nearest"</code></td><td>&nbsp;=&nbsp;</td><td>interpolate using single nearest neighbour</td></tr>
<tr><td style="text-align: right;"><code>"linear"</code></td><td>&nbsp;=&nbsp;</td><td>linear interpolation between two nearest neighbours (<b>default setting</b>)</td></tr>
<tr><td style="text-align: right;"><code>"*nearest"</code></td><td>&nbsp;=&nbsp;</td><td>as per <code>"nearest"</code>, but faster by assuming that <i>X</i> and <i>XI</i> are monotonically increasing</td></tr>
<tr><td style="text-align: right;"><code>"*linear"</code></td><td>&nbsp;=&nbsp;</td><td>as per <code>"linear"</code>, but faster by assuming that <i>X</i> and <i>XI</i> are monotonically increasing</td></tr>
</tbody>
</table>
</ul>
</li>
<br>
<li>
If a location in <i>XI</i> is outside the domain of <i>X</i>, the corresponding value in <i>YI</i> is set to <i>extrapolation_value</i>
</li>
<br>
<li>
The <i>extrapolation_value</i> argument is optional; by default it is <a href="#constants">datum::nan</a> (not-a-number)
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec x = linspace&lt;vec&gt;(0, 3, 20);
vec y = square(x);

vec xx = linspace&lt;vec&gt;(0, 3, 100);

vec yy;

interp1(x, y, xx, yy);  // use linear interpolation by default

interp1(x, y, xx, yy, "*linear");  // faster than "linear"

interp1(x, y, xx, yy, "nearest");
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#linspace">linspace()</a></li>
<li><a href="#regspace">regspace()</a></li>
<li><a href="#conv">conv()</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Statistics &amp; Clustering</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="stats_fns"></a>
<b>mean, median, stddev, var</b>
<ul>
<table style="text-align: left;" border="0" cellpadding="0" cellspacing="0">
<tbody>
  <tr>
    <td style="vertical-align: top;">
       <b>mean( V )</b>
       <br><b>mean( M )</b>
       <br><b>mean( M, dim )</b>
       <br><b>mean( Q )</b>
       <br><b>mean( Q, dim )</b>
       <br>
       <br>
    </td>
    <td style="vertical-align: top;">
      &nbsp;&nbsp;&nbsp;
    </td>
    <td style="vertical-align: top;">
      &#9131;&nbsp;<br>
      &#9130;&nbsp;<br>
      &#9132;&nbsp;&nbsp;mean (average value)<br>
      &#9130;&nbsp;<br>
      &#9133;&nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
       <b>median( V )</b>
       <br><b>median( M )</b>
       <br><b>median( M, dim )</b>
       <br>
       <br>
    </td>
    <td style="vertical-align: top;">
      &nbsp;&nbsp;&nbsp;
    </td>
    <td style="vertical-align: top;">
      &#9131;&nbsp;<br>
      &#9132;&nbsp;&nbsp;median<br>
      &#9133;&nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
       <b>stddev( V )</b>
       <br><b>stddev( V, norm_type )</b>
       <br><b>stddev( M )</b>
       <br><b>stddev( M, norm_type )</b>
       <br><b>stddev( M, norm_type, dim )</b>
       <br>
       <br>
    </td>
    <td style="vertical-align: top;">
      &nbsp;&nbsp;&nbsp;
    </td>
    <td style="vertical-align: top;">
      &#9131;&nbsp;<br>
      &#9130;&nbsp;<br>
      &#9132;&nbsp;&nbsp;standard deviation<br>
      &#9130;&nbsp;<br>
      &#9133;&nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
       <b>var( V )</b>
       <br><b>var( V, norm_type )</b>
       <br><b>var( M )</b>
       <br><b>var( M, norm_type )</b>
       <br><b>var( M, norm_type, dim )</b>
       <br>
       <br>
    </td>
    <td style="vertical-align: top;">
      &nbsp;&nbsp;&nbsp;
    </td>
    <td style="vertical-align: top;">
      &#9131;&nbsp;<br>
      &#9130;&nbsp;<br>
      &#9132;&nbsp;&nbsp;variance<br>
      &#9130;&nbsp;<br>
      &#9133;&nbsp;
    </td>
  </tr>
</tbody>
</table>
<li>
For vector <i>V</i>, return the statistic calculated using all the elements of the vector
</li>
<br>
<li>
For matrix <i>M</i>, find the statistic for each column (<i>dim=0</i>), or each row (<i>dim=1</i>)
</li>
<br>
<li>
For cube <i>Q</i>, find the statistics of elements along dimension <i>dim</i>, where <i>dim</i> &isin; {&nbsp;0,&nbsp;1,&nbsp;2&nbsp;}
</li>
<br>
<li>
The <i>dim</i> argument is optional; by default <i>dim=0</i> is used
</li>
<br>
<li>
The <i>norm_type</i> argument is optional; by default <i>norm_type=0</i> is used
</li>
<br>
<li>
For the <i>var()</i> and <i>stddev()</i> functions:
<ul>
<li>the default <i>norm_type=0</i> performs normalisation using <i>N-1</i> (where <i>N</i> is the number of samples), providing the best unbiased estimator</li>
<li>using <i>norm_type=1</i> performs normalisation using <i>N</i>, which provides the second moment around the mean</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A    = randu&lt;mat&gt;(5,5);
mat B    = mean(A);
mat C    = var(A);
double m = mean(mean(A));

vec    v = randu&lt;vec&gt;(5);
double x = var(v);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#cov">cov()</a></li>
<li><a href="#cor">cor()</a></li>
<li><a href="#diff">diff()</a></li>
<li><a href="#hist">hist()</a></li>
<li><a href="#histc">histc()</a></li>
<li><a href="#running_stat">running_stat</a> - class for running statistics of scalars</li>
<li><a href="#running_stat_vec">running_stat_vec</a> - class for running statistics of vectors</li>
<li><a href="#gmm_diag">gmm_diag</a> - class for modelling data as a Gaussian mixture model</li>
<li><a href="#kmeans">kmeans()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="cov"></a>
<b>cov( X, Y )</b>
<br><b>cov( X, Y, norm_type )</b>
<br>
<br><b>cov( X )</b>
<br><b>cov( X, norm_type )</b>
<ul>
<li>
For two matrix arguments <i>X</i> and <i>Y</i>,
if each row of <i>X</i> and <i>Y</i>  is an observation and each column is a variable,
the <i>(i,j)</i>-th entry of <i>cov(X,Y)</i> is the covariance between the <i>i</i>-th variable in <i>X</i> and the <i>j</i>-th variable in <i>Y</i>
</li>
<br>
<li>
For vector arguments, the type of vector is ignored and each element in the vector is treated as an observation
</li>
<br>
<li>
For matrices, <i>X</i> and <i>Y</i> must have the same dimensions
</li>
<br>
<li>
For vectors, <i>X</i> and <i>Y</i> must have the same number of elements
</li>
<br>
<li>
<i>cov(X)</i> is equivalent to <i>cov(X, X)</i>
</li>
<br>
<li>
The default <i>norm_type=0</i> performs normalisation using <i>N-1</i> (where <i>N</i> is the number of observations),
providing the best unbiased estimation of the covariance matrix (if the observations are from a normal distribution).
Using <i>norm_type=1</i> causes normalisation to be done using <i>N</i>, which provides the second moment matrix of the observations about their mean
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(4,5);
mat Y = randu&lt;mat&gt;(4,5);

mat C = cov(X,Y);
mat D = cov(X,Y, 1);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="#cor">cor()</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#running_stat_vec">running_stat_vec</a></li>
<li><a href="http://mathworld.wolfram.com/Covariance.html">Covariance in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="cor"></a>
<b>cor( X, Y )</b>
<br><b>cor( X, Y, norm_type )</b>
<br>
<br><b>cor( X )</b>
<br><b>cor( X, norm_type )</b>
<ul>
<li>
For two matrix arguments <i>X</i> and <i>Y</i>,
if each row of <i>X</i> and <i>Y</i>  is an observation and each column is a variable,
the <i>(i,j)</i>-th entry of <i>cor(X,Y)</i> is the correlation coefficient between the <i>i</i>-th variable in <i>X</i> and the <i>j</i>-th variable in <i>Y</i>
</li>
<br>
<li>
For vector arguments, the type of vector is ignored and each element in the vector is treated as an observation 
</li>
<br>
<li>
For matrices, <i>X</i> and <i>Y</i> must have the same dimensions
</li>
<br>
<li>
For vectors, <i>X</i> and <i>Y</i> must have the same number of elements
</li>
<br>
<li>
<i>cor(X)</i> is equivalent to <i>cor(X, X)</i>
</li>
<br>
<li>
The default <i>norm_type=0</i> performs normalisation of the correlation matrix using <i>N-1</i> (where <i>N</i> is the number of observations).
Using <i>norm_type=1</i> causes normalisation to be done using <i>N</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat X = randu&lt;mat&gt;(4,5);
mat Y = randu&lt;mat&gt;(4,5);

mat R = cor(X,Y);
mat S = cor(X,Y, 1);
</pre>
</ul>
</li>
<br>
<li>
See also: 
<ul>
<li><a href="#cov">cov()</a></li>
<li><a href="#conv">conv()</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#running_stat_vec">running_stat_vec</a></li>
<li><a href="http://mathworld.wolfram.com/Correlation.html">Correlation in MathWorld</a></li>
<li><a href="http://mathworld.wolfram.com/Autocorrelation.html">Autocorrelation in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="hist"></a>
<b>hist( V )</b>
<br><b>hist( V, n_bins )</b>
<br><b>hist( V, centers )</b>
<br>
<br><b>hist( X, centers )</b>
<br><b>hist( X, centers, dim )</b>
<ul>
<li>
For vector <i>V</i>,
produce an unsigned vector of the same orientation as <i>V</i> (ie. either <a href="#Col">uvec</a> or <a href="#Row">urowvec</a>)
that represents a histogram of counts
</li>
<br>
<li>
For matrix <i>X</i>,
produce a <a href="#Mat">umat</a> matrix containing either
column histogram counts (for <i>dim=0</i>, default),
or
row histogram counts (for <i>dim=1</i>)
</li>
<br>
<li>
The bin centers can be automatically determined from the data, with the number of bins specified via <i>n_bins</i> (default is 10);
the range of the bins is determined by the range of the data
</li>
<br>
<li>
The bin centers can also be explicitly specified via the <i>centers</i> vector;
the vector must contain monotonically increasing values (eg. 0.1, 0.2, 0.3, ...)
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec  v  = randn&lt;vec&gt;(1000); // Gaussian distribution

uvec h1 = hist(v, 11);
uvec h2 = hist(v, linspace&lt;vec&gt;(-2,2,11));
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#histc">histc()</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#conv_to">conv_to()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="histc"></a>
<b>histc( V, edges )</b>
<br><b>histc( X, edges )</b>
<br><b>histc( X, edges, dim )</b>
<ul>
<li>
For vector <i>V</i>,
produce an unsigned vector of the same orientation as <i>V</i> (ie. either <a href="#Col">uvec</a> or <a href="#Row">urowvec</a>)
that contains the counts of the number of values that fall between the elements in the <i>edges</i> vector
</li>
<br>
<li>
For matrix <i>X</i>,
produce a <i>umat</i> matrix containing either
column histogram counts (for <i>dim=0</i>, default),
or
row histogram counts (for <i>dim=1</i>)
</li>
<br>
<li>
The <i>edges</i> vector must contain monotonically increasing values (eg. 0.1, 0.2, 0.3, ...)
</li>
<br>
<li>
Examples:
<ul>
<pre>
vec  v = randn&lt;vec&gt;(1000);  // Gaussian distribution

uvec h = histc(v, linspace&lt;vec&gt;(-2,2,11));
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#hist">hist()</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#conv_to">conv_to()</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="princomp"></a>
<b>mat coeff = princomp( mat X )</b>
<br><b>cx_mat coeff = princomp( cx_mat X )</b><br>

<br><b>princomp( mat coeff, mat X )</b>
<br><b>princomp( cx_mat coeff, cx_mat X )</b><br>

<br><b>princomp( mat coeff, mat score, mat X )</b>
<br><b>princomp( cx_mat coeff, cx_mat score, cx_mat X )</b><br>

<br><b>princomp( mat coeff, mat score, vec latent, mat X )</b>
<br><b>princomp( cx_mat coeff, cx_mat score, vec latent, cx_mat X )</b><br>

<br><b>princomp( mat coeff, mat score, vec latent, vec tsquared, mat X )</b>
<br><b>princomp( cx_mat coeff, cx_mat score, vec latent, cx_vec tsquared, cx_mat X )</b><br>
<ul>
<li>Principal component analysis of matrix <i>X</i></li><br>
<li>Each row of <i>X</i> is an observation and each column is a variable</li><br>
<li>output objects:
<ul>
<li><i>coeff</i>: principal component coefficients</li>
<li><i>score</i>: projected data</li>
<li><i>latent</i>: eigenvalues of the covariance matrix of <i>X</i></li>
<li><i>tsquared</i>: Hotteling's statistic for each sample</li>
</ul>
</li>
<br>
<li>
The computation is based on singular value decomposition
</li>
<br>
<li>If the decomposition fails:
<ul>
<li><i>coeff = princomp(X)</i> resets <i>coeff</i> and throws a <i>std::runtime_error</i> exception</li>
<li>remaining forms of <i>princomp()</i> reset all output matrices and return a bool set to <i>false</i> (exception is not thrown)</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
mat A = randu&lt;mat&gt;(5,4);

mat coeff;
mat score;
vec latent;
vec tsquared;

princomp(coeff, score, latent, tsquared, A);
</pre>
</ul>
</li>
<br>
<li>
See also:
<ul>
<li><a href="http://en.wikipedia.org/wiki/Principal_component_analysis">principal components analysis in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/PrincipalComponentAnalysis.html">principal components analysis in MathWorld</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="running_stat"></a>
<b>running_stat&lt;</b><i>type</i><b>&gt;</b>
<ul>
<li>
Class for keeping the running statistics of a continuously sampled one dimensional process/signal
</li>
<br>
<li>
Useful if the storage of individual samples (scalars) is not required, or the number of samples is not known beforehand
</li>
<br>
<li>
<i>type</i> is one of: <i>float</i>, <i>double</i>, <i><a href="#cx_double">cx_float</a></i>, <i><a href="#cx_double">cx_double</a></i>
</li>
<br>
<li>
For an instance of <i>running_stat</i> named as <i>X</i>, the member functions are:
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>X(</b>scalar<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      update the statistics so far using the given scalar
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.min()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the minimum value so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.max()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the maximum value so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.mean()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the mean or average value so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.var()</b> &nbsp;and&nbsp; <b>X.var(</b>norm_type<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the variance so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.stddev()</b> &nbsp;and&nbsp; <b>X.stddev(</b>norm_type<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the standard deviation so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.reset()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      reset all statistics and set the number of samples to zero
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.count()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the number of samples so far
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
The <i>norm_type</i> argument is optional; by default <i>norm_type=0</i> is used
</li>
<br>
<li>
For the <i>.var()</i> and <i>.stddev()</i> functions, the default <i>norm_type=0</i> performs normalisation using <i>N-1</i>
(where <i>N</i> is the number of samples so far),
providing the best unbiased estimator;
using <i>norm_type=1</i> causes normalisation to be done using <i>N</i>, which provides the second moment around the mean
</li>
<br>
<li>
The return type of <i>.count()</i> depends on the underlying form of <i>type</i>: it is either <i>float</i> or <i>double</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
running_stat&lt;double&gt; stats;

for(uword i=0; i&lt;10000; ++i)
  {
  double sample = randn();
  stats(sample);
  }

cout &lt;&lt; "mean = " &lt;&lt; stats.mean() &lt;&lt; endl;
cout &lt;&lt; "var  = " &lt;&lt; stats.var()  &lt;&lt; endl;
cout &lt;&lt; "min  = " &lt;&lt; stats.min()  &lt;&lt; endl;
cout &lt;&lt; "max  = " &lt;&lt; stats.max()  &lt;&lt; endl;
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#running_stat_vec">running_stat_vec</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="running_stat_vec"></a>
<b>running_stat_vec&lt;</b><i>vec_type</i><b>&gt;</b>
<br><b>running_stat_vec&lt;</b><i>vec_type</i><b>&gt;(calc_cov)</b>
<ul>
<li>
Class for keeping the running statistics of a continuously sampled multi-dimensional process/signal
</li>
<br>
<li>
Useful if the storage of individual samples (vectors) is not required, or if the number of samples is not known beforehand
</li>
<br>
<li>
This class is similar to <i>running_stat</i>, with the difference that vectors are processed instead of scalars
</li>
<br>
<li>
<i>vec_type</i> is the vector type of the samples; for example: <i><a href="#Col">vec</a></i>, <i><a href="#Col">cx_vec</a></i>, <i><a href="#Row">rowvec</a></i>, ...
</li>
<br>
<li>
For an instance of <i>running_stat_vec</i> named as <i>X</i>, the member functions are:
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>X(</b>vector<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      update the statistics so far using the given vector
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.min()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the vector of minimum values so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.max()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the vector of maximum values so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.mean()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the mean vector so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.var()</b> &nbsp;and&nbsp; <b>X.var(</b>norm_type<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the vector of variances so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.stddev()</b> &nbsp;and&nbsp; <b>X.stddev(</b>norm_type<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the vector of standard deviations so far
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.cov()</b> &nbsp;and&nbsp; <b>X.cov(</b>norm_type<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the covariance matrix so far;
      valid if <i>calc_cov=true</i> during construction of <i>running_stat_vec</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.reset()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      reset all statistics and set the number of samples to zero
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>X.count()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      get the number of samples so far
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
The <i>calc_cov</i> argument is optional; by default <i>calc_cov=false</i>, indicating that the covariance matrix will not be calculated;
to enable the covariance matrix, use <i>calc_cov=true</i> during construction; for example: <i>running_stat_vec&lt;vec&gt;&nbsp;X(true);</i>
</li>
<br>
<li>
The <i>norm_type</i> argument is optional; by default <i>norm_type=0</i> is used
</li>
<br>
<li>
For the <i>.var()</i> and <i>.stddev()</i> functions, the default <i>norm_type=0</i> performs normalisation using <i>N-1</i>
(where <i>N</i> is the number of samples so far),
providing the best unbiased estimator;
using <i>norm_type=1</i> causes normalisation to be done using <i>N</i>, which provides the second moment around the mean
</li>
<br>
<li>
The return type of <i>.count()</i> depends on the underlying form of <i>vec_type</i>: it is either <i>float</i> or <i>double</i>
</li>
<br>
<li>
Examples:
<ul>
<pre>
running_stat_vec&lt;vec&gt; stats;

vec sample;

for(uword i=0; i&lt;10000; ++i)
  {
  sample = randu&lt;vec&gt;(5);
  stats(sample);
  }

cout &lt;&lt; "mean = " &lt;&lt; endl &lt;&lt; stats.mean() &lt;&lt; endl;
cout &lt;&lt; "var  = " &lt;&lt; endl &lt;&lt; stats.var()  &lt;&lt; endl;
cout &lt;&lt; "max  = " &lt;&lt; endl &lt;&lt; stats.max()  &lt;&lt; endl;

//
//

running_stat_vec&lt;rowvec&gt; more_stats(true);

for(uword i=0; i&lt;20; ++i)
  {
  sample = randu&lt;rowvec&gt;(3);
  
  sample(1) -= sample(0);
  sample(2) += sample(1);
  
  more_stats(sample);
  }

cout &lt;&lt; "covariance matrix = " &lt;&lt; endl;
cout &lt;&lt; more_stats.cov() &lt;&lt; endl;

rowvec sd = more_stats.stddev();

cout &lt;&lt; "correlations = " &lt;&lt; endl;
cout &lt;&lt; more_stats.cov() / (sd.t() * sd);
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#cov">cov()</a></li>
<li><a href="#cor">cor()</a></li>
<li><a href="#running_stat">running_stat</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#gmm_diag">gmm_diag</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="kmeans"></a>
<b>kmeans(</b> means<b>,</b> data<b>,</b> k<b>,</b> seed_mode<b>,</b> n_iter<b>,</b> print_mode <b>)</b>
<ul>
<li>
Cluster given data into <i>k</i> disjoint sets
</li>
<br>
<li>
The <i>means</i> parameter is the output matrix for storing the resulting centroids of the sets, with each centroid stored as a column vector
</li>
<br>
<li>
The <i>data</i> parameter is the input data matrix, with each sample stored as a column vector
</li>
<br>
<li>
The <i>k</i> parameter indicates the number of centroids
</li>
<br>
<li>
The <i>seed_mode</i> parameter specifies how the initial centroids are seeded; it is one of:
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
  <tr><td><code>keep_existing</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>use the centroids specified in the <i>means</i> matrix as the starting point</td></tr>
  <tr><td><code>static_subset</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>use a subset of the data vectors (repeatable)</td></tr>
  <tr><td><code>random_subset</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>use a subset of the data vectors (random)</td></tr>
  <tr><td><code>static_spread</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>use a maximally spread subset of data vectors (repeatable)</td></tr>
  <tr><td><code>random_spread</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>use a maximally spread subset of data vectors (random start)</td></tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
The <i>n_iter</i> parameter specifies the number of clustering iterations; about 10 is typically sufficient
</li>
<br>
<li>
The <i>print_mode</i> parameter is either <i>true</i> or <i>false</i>, indicating whether progress is printed during clustering
</li>
<br>
<li>
If the clustering fails, the <i>means</i> matrix is reset and a bool set to <i>false</i> is returned
</li>
<br>
<li>
The clustering will run faster on multi-core machines when OpenMP is enabled in your compiler (eg. <i>-fopenmp</i> in GCC)
</li>
<br>
<li>
<b>Caveats</b>:
<ul>
<br>
<li>
the number of data vectors (number of columns in the <i>data</i> matrix) should be much larger than <i>k</i>&nbsp; (number of centroids)
</li>
<br>
<li>
seeding the initial centroids with <code>static_spread</code> and <code>random_spread</code>
can be much more time consuming than with <code>static_subset</code> and <code>random_subset</code>
</li>
</ul>
</li>
<br>


<li>
Examples:
<ul>
<pre>
uword d = 5;       // dimensionality
uword N = 10000;   // number of vectors

mat data(d, N, fill::randu);

mat means;

bool status = kmeans(means, data, 2, random_subset, 10, true);

if(status == false)
  {
  cout &lt;&lt; "clustering failed" &lt;&lt; endl;
  }

means.print("means:");
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#gmm_diag">gmm_diag</a></li>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#running_stat_vec">running_stat_vec</a></li>
<li><a href="http://en.wikipedia.org/wiki/K-means_clustering">k-means clustering in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/K-MeansClusteringAlgorithm.html">k-means clustering in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/OpenMP">OpenMP in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="gmm_diag"></a>
<b>gmm_diag</b>
<ul>
<li>
Class for modelling data as a Gaussian Mixture Model (GMM), under the assumption of diagonal covariance matrices
</li>
<br>
<li>
Can also be used for clustering and vector quantisation (VQ)
</li>
<br>
<li>
Provides associated parameter estimation algorithms: k-means clustering and Expectation-Maximisation (EM)
</li>
<br>
<li>
The k-means and EM algorithms are parallelised (multi-threaded) when compiling with OpenMP enabled (eg. the <i>-fopenmp</i> option in gcc)
</li>
<br>
<li>
Data is modelled as:
<table style="text-align: left;" border="0" cellpadding="0" cellspacing="0">
  <tbody>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top;"><i><sub>n_gaus-1</sub></i></td><td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>p<font size=+1>(</font>x<font size=+1>)</font></i></td><td style="vertical-align: top;">&nbsp;<font size=+1>=</font>&nbsp;</td><td style="vertical-align: top;"><i><font size=+1><b>&sum;</b></font></i></td>
      <td style="vertical-align: top;"><i>h<sub>g</sub></i>&nbsp;&nbsp;<font size=+1>N(</font><small>&nbsp;</small><i>x</i><small>&nbsp;</small>|<small>&nbsp;</small><i>m<sub>g</sub></i><small>&nbsp;</small>,&nbsp;<i>C<sub>g</sub></i><small>&nbsp;</small><font size=+1>)</font></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top;">&nbsp;</td><td style="vertical-align: top;"><i><sup>g=0</sup></i></td><td style="vertical-align: top;">&nbsp;</td>
    </tr>
  </tbody>
</table>
where:
<ul>
<li><i>n_gaus</i> is the number of Gaussians</li>
<li>N(<small>&nbsp;</small><i>x</i><small>&nbsp;</small>|<small>&nbsp;</small><i>m<sub>g</sub></i><small>&nbsp;</small>,&nbsp;<i>C<sub>g</sub></i><small>&nbsp;</small>) represents a Gaussian (normal) distribution</li>
<li>for each Gaussian <i>g</i>:
<ul>
<li><i>h<sub>g</sub></i> is the heft (weight), with <i>h<sub>g</sub></i> &ge; 0 and &sum;<i>h<sub>g</sub></i> = 1</li>
<li><i>m<sub>g</sub></i> is the mean (centroid) vector with dimensionality <i>n_dims</i></li>
<li><i>C<sub>g</sub></i> is the diagonal covariance matrix</li>
</ul>
</li>
</ul>
</li>
<br>
<li>
For an instance of <i>gmm_diag</i> named as <i>M</i>, the member functions and variables are:
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">
      <b>M.log_p(</b>V<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a scalar representing the log-likelihood of vector <i>V</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.log_p(</b>V, g<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a scalar representing the log-likelihood of vector <i>V</i>, according to Gaussian <i>g</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.log_p(</b>X<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a row vector (of type <code>rowvec</code>) containing log-likelihoods of each column vector in matrix <i>X</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.log_p(</b>X, g<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a row vector (of type <code>rowvec</code>) containing log-likelihoods of each column vector in matrix <i>X</i>, according to Gaussian <i>g</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.avg_log_p(</b>X<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a scalar representing the average log-likelihood of all column vectors in matrix <i>X</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.avg_log_p(</b>X, g<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a scalar representing the average log-likelihood of all column vectors in matrix <i>X</i>, according to Gaussian <i>g</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.assign(</b>V,&nbsp;dist_mode<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return the index of the closest mean (or Gaussian) to vector <i>V</i>;<br>
      <i>dist_mode</i> is one of:
      <table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
        <tbody>
        <tr><td style="vertical-align: top;"><code>eucl_dist</code></td><td>&nbsp;&nbsp;&nbsp;</td><td style="vertical-align: top;">Euclidean distance (takes only means into account)</td></tr>
        <tr><td style="vertical-align: top;"><code>prob_dist</code></td><td>&nbsp;&nbsp;&nbsp;</td><td style="vertical-align: top;">probabilistic "distance", defined as the inverse likelihood (takes into account means, covariances and hefts)</td></tr>
        </tbody>
      </table>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.assign(</b>X,&nbsp;dist_mode<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a row vector containing the indices of the closest means (or Gaussians) to each column vector in matrix <i>X</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.raw_hist(</b>X,&nbsp;dist_mode<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a row vector (of type <code>urowvec</code>) representing the raw histogram of counts;
      each entry is the number of counts corresponding to a Gaussian;
      each count is the number times the corresponding Gaussian was the closest to each column vector in matrix <i>X</i>;
      parameter <i>dist_mode</i> is either <code>eucl_dist</code> or <code>prob_dist</code> (as per the <i>.assign()</i> function above)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.norm_hist(</b>X,&nbsp;dist_mode<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a row vector (of type <code>rowvec</code>) containing normalised counts;
      the vector sums to one
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.generate()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a column vector representing a random sample generated according to the model's parameters
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.generate(</b>N<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return a matrix containing <i>N</i> column vectors, with each vector representing a random sample generated according to the model's parameters
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.save(</b>filename<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      save the model to the given <i>filename</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.load(</b>filename<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      load the model from the given <i>filename</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.n_gaus()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return the number of means/Gaussians in the model
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.n_dims()</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      return the dimensionality of the means/Gaussians in the model
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.reset(</b>n_dims, n_gaus<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      set the model to have dimensionality <i>n_dims</i>, with <i>n_gaus</i> number of Gaussians;<br>
      all the means are set to zero, all diagonal covariances are set to one, and all the hefts (weights) are set to be uniform
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.means</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      read-only matrix containing the means (centroids), stored as column vectors
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.dcovs</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      read-only matrix containing the diagonal covariance matrices,
      stored as column vectors
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.hefts</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      read-only row vector containing the hefts (weights)
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.set_means(</b>X<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      set the means to be as specified in matrix <i>X</i>;<br>
      the number of means and their dimensionality must match the existing model
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.set_dcovs(</b>X<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      set the diagonal covariances matrices to be as specified in matrix <i>X</i>;<br>
      the number of diagonal covariance matrices and their dimensionality must match the existing model
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.set_hefts(</b>V<b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      set the hefts (weights) of the model to be as specified in row vector <i>V</i>;<br>
      the number of hefts must match the existing model
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <b>M.set_params(</b><small>means</small>,&nbsp;<small>dcovs</small>,&nbsp;<small>hefts</small><b>)</b>
      </td>
      <td style="vertical-align: top;">&nbsp;<br>
      </td>
      <td style="vertical-align: top;">
      set all the parameters in one hit;<br>
      the number of Gaussians and dimensionality can be different from the existing model
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
    </tr>
    <tr>
      <td style="vertical-align: top;" colspan=3>
      <b>M.learn(</b>data,&nbsp;n_gaus,&nbsp;dist_mode,&nbsp;seed_mode,&nbsp;km_iter,&nbsp;em_iter,&nbsp;var_floor,&nbsp;print_mode<b>)</b>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">&nbsp;<br></td>
      <td style="vertical-align: top;">
      learn the model parameters via the k-means and/or EM algorithms,
      <br>and return a <code>bool</code> variable, with <i>true</i> indicating success, and <i>false</i> indicating failure
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<li>
Parameters for the <code>.learn()</code> member function:
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
  <tbody>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>data</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      matrix containing training samples; each sample is stored as a column vector
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>n_gaus</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      set the number of Gaussians to <i>n_gaus</i>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>dist_mode</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      specifies the distance used during the seeding of initial means and k-means clustering:
      <table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
        <tbody>
        <tr><td><code>eucl_dist</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>Euclidean distance</td></tr>
        <tr><td><code>maha_dist</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>Mahalanobis distance, which uses a global diagonal covariance matrix estimated from the training samples</td></tr>
        </tbody>
      </table>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>seed_mode</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      specifies how the initial means are seeded prior to running k-means and/or EM algorithms:
      <table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
        <tbody>
        <tr><td><code>keep_existing</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>keep the existing model (do not modify the means, covariances and hefts)</td></tr>
        <tr><td><code>static_subset</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>a subset of the training samples (repeatable)</td></tr>
        <tr><td><code>random_subset</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>a subset of the training samples (random)</td></tr>
        <tr><td><code>static_spread</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>a maximally spread subset of training samples (repeatable)</td></tr>
        <tr><td><code>random_spread</code></td><td>&nbsp;&nbsp;&nbsp;</td><td>a maximally spread subset of training samples (random start)</td></tr>
        </tbody>
      </table>
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>km_iter</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      the number of iterations of the k-means algorithm
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>em_iter</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      the number of iterations of the EM algorithm
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>var_floor</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      the variance floor (smallest allowed value) for the diagonal covariances
      </td>
    </tr>
    <tr>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">&nbsp;</td>
    </tr>
    <tr>
      <td style="vertical-align: top;"><i>print_mode</i></td>
      <td style="vertical-align: top;">&nbsp;</td>
      <td style="vertical-align: top;">
      either <code>true</code> or <code>false</code>;
      enable or disable printing of progress during the k-means and EM algorithms
      </td>
    </tr>
  </tbody>
</table>
</ul>
</li>
<br>
<br>
<li>
<b>Notes</b>:
<ul>
<br>
<li>
for probabilistic applications,
better model parameters are typically learned with <i>dist_mode</i> set to <code>maha_dist</code>
</li>
<br>
<li>
for vector quantisation applications, 
model parameters should be learned with <i>dist_mode</i> set to <code>eucl_dist</code>,
and the number of EM iterations set to zero
</li>
<br>
<li>
in general, a sufficient number of k-means and EM iterations is typically about 10
</li>
<br>
<li>
the number of training samples should be much larger than the number of Gaussians
</li>
<br>
<li>
seeding the initial means with <code>static_spread</code> and <code>random_spread</code>
can be much more time consuming than with <code>static_subset</code> and <code>random_subset</code>
</li>
<br>
<li>
the k-means and EM algorithms will run faster on multi-core machines when OpenMP is enabled in your compiler (eg. <i>-fopenmp</i> in GCC)
</li>
</ul>
</li>
<br>
<li>
Examples:
<ul>
<pre>
// create synthetic data with 2 Gaussians

uword d = 5;       // dimensionality
uword N = 10000;   // number of vectors

mat data(d, N, fill::zeros);

vec mean0 = linspace&lt;vec&gt;(1,d,d);
vec mean1 = mean0 + 2;

uword i = 0;

while(i &lt; N)
  {
  if(i &lt; N)  { data.col(i) = mean0 + randn&lt;vec&gt;(d); ++i; }
  if(i &lt; N)  { data.col(i) = mean0 + randn&lt;vec&gt;(d); ++i; }
  if(i &lt; N)  { data.col(i) = mean1 + randn&lt;vec&gt;(d); ++i; }
  }


// model the data as a GMM with 2 Gaussians

gmm_diag model;

bool status = model.learn(data, 2, maha_dist, random_subset, 10, 5, 1e-10, true);

if(status == false)
  {
  cout &lt;&lt; "learning failed" &lt;&lt; endl;
  }

model.means.print("means:");

double  scalar_likelihood = model.log_p( data.col(0)    );
rowvec     set_likelihood = model.log_p( data.cols(0,9) );

double overall_likelihood = model.avg_log_p(data);

uword   gaus_id  = model.assign( data.col(0),    eucl_dist );
urowvec gaus_ids = model.assign( data.cols(0,9), prob_dist );

urowvec hist1 = model.raw_hist (data, prob_dist);
 rowvec hist2 = model.norm_hist(data, eucl_dist);

model.save("my_model.gmm");
</pre>
</ul>
</li>
<br>
<li>See also:
<ul>
<li><a href="#stats_fns">statistics functions</a></li>
<li><a href="#running_stat_vec">running_stat_vec</a></li>
<li><a href="#kmeans">kmeans()</a></li>
<li><a href="#diagmat">diagmat()</a></li>
<li><a href="http://en.wikipedia.org/wiki/Mahalanobis_distance">Mahalanobis distance in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Normal_distribution">normal distribution in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/Mixture_model">mixture model in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/K-means_clustering">k-means clustering in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/K-MeansClusteringAlgorithm.html">k-means clustering in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Expectation-maximization_algorithm">expectation maximisation algorithm in Wikipedia</a></li>
<li><a href="http://mathworld.wolfram.com/MaximumLikelihood.html">maximum likelihood in MathWorld</a></li>
<li><a href="http://en.wikipedia.org/wiki/Vector_quantization">vector quantisation in Wikipedia</a></li>
<li><a href="http://en.wikipedia.org/wiki/OpenMP">OpenMP in Wikipedia</a></li>
</ul>
</li>
<br>
</ul>



<div class="pagebreak"></div>
<hr class="greyline">
<hr class="greyline">
<br>
<br>
<font size=+1><b>Miscellaneous</b></font>
<br>
<br>



<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="constants"></a>
<b>constants (pi, inf, speed of light, ...)</b>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
<tbody>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::pi</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &pi;, the ratio of any circle's circumference to its diameter
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::inf</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &infin;, infinity
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::nan</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &ldquo;not a number&rdquo; (NaN); <b>caveat:</b> NaN is not equal to anything, even itself
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::e</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      base of the natural logarithm
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::sqrt2</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      square root of 2
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::eps</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      the difference between 1 and the least value greater than 1 that is representable  (type and machine dependent)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::log_min</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      log of minimum non-zero value (type and machine dependent)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::log_max</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      log of maximum value  (type and machine dependent)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::euler</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Euler's constant, aka Euler-Mascheroni constant
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::gratio</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      golden ratio
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::m_u</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      atomic mass constant (in kg)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::N_A</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Avogadro constant
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::k</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Boltzmann constant (in joules per kelvin)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::k_evk</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Boltzmann constant (in eV/K)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::a_0</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Bohr radius (in meters)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::mu_B</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Bohr magneton
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::Z_0</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      characteristic impedance of vacuum (in ohms)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::G_0</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      conductance quantum (in siemens)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::k_e</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Coulomb's constant (in meters per farad)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::eps_0</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      electric constant (in farads per meter)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::m_e</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      electron mass (in kg)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::eV</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      electron volt (in joules)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::ec</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      elementary charge (in coulombs)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::F</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Faraday constant (in coulombs)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::alpha</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      fine-structure constant
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::alpha_inv</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      inverse fine-structure constant
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::K_J</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Josephson constant
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::mu_0</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      magnetic constant (in henries per meter)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::phi_0</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      magnetic flux quantum (in webers)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::R</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      molar gas constant (in joules per mole kelvin)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::G</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Newtonian constant of gravitation (in newton square meters per kilogram squared)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::h</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Planck constant (in joule seconds)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::h_bar</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Planck constant over 2 pi, aka reduced Planck constant (in joule seconds)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::m_p</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      proton mass (in kg)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::R_inf</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Rydberg constant (in reciprocal meters)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::c_0</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      speed of light in vacuum (in meters per second)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
    <td style="vertical-align: top;">&nbsp;</td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::sigma</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Stefan-Boltzmann constant
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::R_k</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      von Klitzing constant (in ohms)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      <code>datum::b</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Wien wavelength displacement law constant
    </td>
  </tr>
</tbody>
</table>
<br>
<li>
The constants are stored in the <i>Datum&lt;type&gt;</i> class,
where <i>type</i> is either <i>float</i> or <i>double</i>;
<br>
for convenience,
<i>Datum&lt;double&gt;</i> is typedefed as <i>datum</i>,
and
<i>Datum&lt;float&gt;</i> is typedefed as <i>fdatum</i>
</li>
<br>
<li>
<b>Caveat:</b>
<code>datum::nan</code> is not equal to anything, even itself;
to check whether a given number <i>x</i> is finite,
use <a href="#is_finite">is_finite</a>(<i>x</i>)
</li>
<br>
<li>
The physical constants were mainly taken from
<a href="http://physics.nist.gov/cuu/Constants">NIST 2014 CODATA values</a>,
and some from
<a href="http://www.wolframalpha.com">WolframAlpha</a> (as of 2009-06-23)
</li>
<br>
<li>
Examples:
<ul>
<pre>
cout &lt;&lt; "2.0 * pi = " &lt;&lt; 2.0 * datum::pi &lt;&lt; endl;

cout &lt;&lt; "speed of light = " &lt;&lt; datum::c_0 &lt;&lt; endl;

cout &lt;&lt; "log_max for floats = ";
cout &lt;&lt; fdatum::log_max &lt;&lt; endl;

cout &lt;&lt; "log_max for doubles = ";
cout &lt;&lt; datum::log_max &lt;&lt; endl;
</pre>
</ul>
</li>
<li>
See also:
<ul>
<li><a href="#is_finite">is_finite()</a></li>
<li><a href="#fill">.fill()</a></li>
<li><a href="http://en.wikipedia.org/wiki/NaN">NaN</a> in Wikipedia</li>
<li><a href="http://en.wikipedia.org/wiki/Physical_constant">physical constant</a> in Wikipedia</li>
<li><a href="http://cplusplus.com/reference/std/limits/numeric_limits/">std::numeric_limits</a></li>
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="wall_clock"></a>
<b>wall_clock</b>
<ul>
<li>
Simple wall clock timer class for measuring the number of elapsed seconds
</li>
<br>
<li>
Examples:
<ul>
<pre>
wall_clock timer;

mat A = randu&lt;mat&gt;(100,100);
mat B = randu&lt;mat&gt;(100,100);
mat C;

timer.tic();

for(uword i=0; i&lt;100000; ++i)
  {
  C = A + B + A + B;
  }

double n = timer.toc();

cout &lt;&lt; "number of seconds: " &lt;&lt; n &lt;&lt; endl;
</pre>
</ul>
</li>
</ul>
<br>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="logging"></a>
<b>logging of warnings and errors</b>
<br>
<br>
<b>set_stream_err1( user_stream )</b><br>
<b>set_stream_err2( user_stream )</b><br>
<br>
<b>std::ostream&amp; x = get_stream_err1()</b>
<br>
<b>std::ostream&amp; x = get_stream_err2()</b>
<br>
<ul>
<li>
By default, warnings and messages associated with exceptions are printed to the <i>std::cout</i> stream
<ul>
<li>the printing can be disabled by defining <a href="#config_hpp">ARMA_DONT_PRINT_ERRORS</a> before including the armadillo header</li>
<li>detailed information about errors is always reported via the base <i>std::exception</i> class</li>
</ul>
</li>
<br>
<li><b>set_stream_err1()</b>: change the stream for messages associated with <i>std::logic_error</i> exceptions (eg. out of bounds accesses)</li>
<br>
<li><b>set_stream_err2()</b>: change the stream for warnings and messages associated with <i>std::runtime_error</i> and <i>std::bad_alloc</i> exceptions (eg. failed decompositions, out of memory)</li>
<br>
<li><b>get_stream_err1()</b>: get a reference to the stream for messages associated with <i>std::logic_error</i> exceptions</li>
<br>
<li><b>get_stream_err2()</b>: get a reference to the stream for warnings and messages associated with <i>std::runtime_error</i> exceptions</li>
<br>
<li>
Examples:
<ul>
<pre>
// print "hello" to the current err1 stream
get_stream_err1() &lt;&lt; "hello" &lt;&lt; endl;

// change the err2 stream to be a file
ofstream f("my_log.txt");
set_stream_err2(f);

// trying to invert a singular matrix
// will print a message to the err2 stream
// and throw an exception
mat X = zeros&lt;mat&gt;(5,5);
mat Y = inv(X);

// disable messages being printed to the err2 stream
std::ostream nullstream(0);
set_stream_err2(nullstream);
</pre>
</ul>
</li>
<br>
<li>
<b>Caveat</b>: set_stream_err1() and set_stream_err2() will not change the stream used by .print()
</li>
<br>
<li>
See also:
<ul>
<li><a href="#config_hpp">config.hpp</a></li>
<li><a href="#print">.print()</a></li>
<li><a href="http://cplusplus.com/reference/iostream/cout/">std::cout</a></li>
<li><a href="http://cplusplus.com/reference/iostream/ostream/">std::ostream</a></li>
<li><a href="http://www.cplusplus.com/reference/exception/exception/">std::exception</a></li>
<li><a href="http://cplusplus.com/reference/std/stdexcept/logic_error/">std::logic_error</a></li>
<li><a href="http://cplusplus.com/reference/std/stdexcept/runtime_error/">std::runtime_error</a></li>
<li><a href="http://cplusplus.com/doc/tutorial/exceptions/">tutorial on exceptions</a></li>
</ul>
</li>
<br>
</ul>

<!--
<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="log_add"></a>
<b>log_add(log_a, log_b)</b>
<ul>
<li>
Safe replacement for log(exp(log_a) + exp(log_b))
</li>
<br>
<li>
Usage:
<ul>
<li>
<i>scalar_type</i> log_c = log_add(log_a, log_b)
</li>
<li>
<i>scalar_type</i> is either <i>float</i> or <i>double</i>
</li>
<li>
log_a, log_b and log_c must have the same type
</li>
</ul>
</li>
</ul>
<br>
-->

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="uword"></a>
<b>uword</b>, <b>sword</b>
<ul>
<li><i>uword</i> is a typedef for an unsigned integer type; it is used for matrix <a href="#element_access">indices</a> as well as all internal counters and loops</li>
<br>
<li><i>sword</i> is a typedef for a signed integer type</li>
<br>
<li>The minimum width of both <i>uword</i> and <i>sword</i> is either 32 or 64 bits:
<ul>
<li>when using the old C++98 / C++03 standards, the default width is 32 bits</li>
<li>when using the new C++11 / C++14 standards, the default width is 64 bits on 64-bit platforms</li>
</ul>
</li>
<br>
<li>The width can also be forcefully set to 64 bits by enabling <a href="#config_hpp_arma_64bit_word"><i>ARMA_64BIT_WORD</i></a> via editing <i>include/armadillo_bits/config.hpp</i></li>
<br>
<li>See also:
<ul>
<li><a href="http://cplusplus.com/doc/tutorial/variables/">C++ variable types</a></li>
<li><a href="http://www.cplusplus.com/doc/tutorial/other_data_types/">explanation of <i>typedef</i></a></li>
<li><a href="#Mat">imat &amp; umat</a> matrix types
<li><a href="#Col">ivec &amp; uvec</a> vector types
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="cx_double"></a>
<b>cx_double</b>, <b>cx_float</b> 
<ul>
<li>
<i>cx_double</i> is a shorthand / typedef for <i>std::complex&lt;double&gt;</i>
</li>
<br>
<li>
<i>cx_float</i> is a shorthand / typedef for <i>std::complex&lt;float&gt;</i>
</li>
<br>
<li>
Example:
<ul>
<pre>
cx_mat X = randu&lt;cx_mat&gt;(5,5);

X(1,2) = cx_double(2.0, 3.0);
</pre>
</ul>
</li>
<li>See also:
<ul>
<li><a href="http://cplusplus.com/reference/std/complex/">complex numbers in the standard C++ library</a></li>
<li><a href="http://www.cplusplus.com/doc/tutorial/other_data_types/">explanation of <i>typedef</i></a></li>
<li><a href="#Mat">cx_mat</a> matrix type
<li><a href="#Col">cx_vec</a> vector type
</ul>
</li>
<br>
</ul>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="syntax"></a>
<b>Examples of Matlab/Octave syntax and conceptually corresponding Armadillo syntax</b>
<br>
<br>
<ul>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
<tbody>
  <tr>
    <td style="vertical-align: top;">
      <b>Matlab/Octave</b>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      <b>Armadillo</b>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      <b>Notes</b>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A(1, 1)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A(0, 0)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      indexing in Armadillo starts at 0
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A(k, k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A(k-1, k-1)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      size(A,1)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#attributes">.n_rows</a>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      read only
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      size(A,2)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#attributes">.n_cols</a>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      size(Q,3)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Q<a href="#attributes">.n_slices</a>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Q is a <a href="#Cube">cube</a> (3D array)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      numel(A)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#attributes">.n_elem</a>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A(:, k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#submat">.col</a>(k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
this is a conceptual example only; 
exact conversion from Matlab/Octave to Armadillo syntax
will require taking into account that indexing starts at 0
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A(k, :)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#submat">.row</a>(k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A(:, p:q)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#submat">.cols</a>(p, q)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A(p:q, :)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#submat">.rows</a>(p, q)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A(p:q, r:s)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A(&nbsp;<a href="#submat">span</a>(p,q),&nbsp;<a href="#submat">span</a>(r,s)&nbsp;)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A(&nbsp;span(first_row,&nbsp;last_row), span(first_col,&nbsp;last_col)&nbsp;)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      Q(:, :, k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Q<a href="#subcube">.slice</a>(k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Q is a <a href="#Cube">cube</a> (3D array)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      Q(:, :, t:u)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Q<a href="#subcube">.slices</a>(t, u)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      Q(p:q, r:s, t:u)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      <font size=-1>
      Q(&nbsp;<a href="#subcube">span</a>(p,q),&nbsp;<a href="#subcube">span</a>(r,s),&nbsp;<a href="#subcube">span</a>(t,u)&nbsp;)
      </font>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A'
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#t_st_members">.t()</a> or <a href="#trans">trans</a>(A)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      matrix transpose / Hermitian transpose
      <br>
      (for complex matrices, the conjugate of each element is taken)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A&nbsp;=&nbsp;zeros(size(A))
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#zeros_member">.zeros()</a>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A&nbsp;=&nbsp;ones(size(A))
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A.<a href="#ones_member">ones()</a>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A = zeros(k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A = <a href="#zeros_standalone">zeros</a>&lt;mat&gt;(k,k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A = ones(k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A = <a href="#ones_standalone">ones</a>&lt;mat&gt;(k,k)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      C = complex(A,B)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      cx_mat C = <a href="#Mat">cx_mat</a>(A,B)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A .* B
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A % B
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      <a href="#operators">element-wise multiplication</a>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A ./ B
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A / B
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      <a href="#operators">element-wise division</a>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A \ B
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      <a href="#solve">solve</a>(A,B)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      conceptually similar to <a href="#inv">inv</a>(A)*B, but more efficient
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A = A + 1;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A++
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A = A - 1;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A--
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A = [ 1 2; 3 4; ]
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
A&nbsp;<font size=-1>&lt;&lt;</font> 1 <font size=-1>&lt;&lt;</font> 2 <font size=-1>&lt;&lt;</font> endr<br>
&nbsp;&nbsp;&nbsp;<font size=-1>&lt;&lt;</font> 3 <font size=-1>&lt;&lt;</font> 4 <font size=-1>&lt;&lt;</font> endr;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      <a href="#element_initialisation">element initialisation</a>,
      with special element <i>endr</i> indicating <i>end of row</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      X = A(:)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      X = <a href="#vectorise">vectorise</a>(A)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      X = [&nbsp;A&nbsp;&nbsp;B&nbsp;]
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      X = <a href="#join">join_horiz</a>(A,B)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      X = [&nbsp;A;&nbsp;B&nbsp;]
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      X = <a href="#join">join_vert</a>(A,B)
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      cout <font size=-1>&lt;&lt;</font> A <font size=-1>&lt;&lt;</font> endl;
      <br><font size=-1>or</font>
      <br>A<a href="#print">.print</a>("A =");
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      save&nbsp;&#8209;ascii&nbsp;'A.dat'&nbsp;A
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#save_load_mat">.save</a>("A.dat",&nbsp;raw_ascii);
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      Matlab/Octave matrices saved as ascii are readable by Armadillo (and vice-versa)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      load&nbsp;&#8209;ascii&nbsp;'A.dat'
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      A<a href="#save_load_mat">.load</a>("A.dat",&nbsp;raw_ascii);
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      A&nbsp;=&nbsp;randn(2,3);
      <br>B&nbsp;=&nbsp;randn(4,5);
      <br>F&nbsp;=&nbsp;{&nbsp;A;&nbsp;B&nbsp;}
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      mat&nbsp;A&nbsp;=&nbsp;randn(2,3);
      <br>mat&nbsp;B&nbsp;=&nbsp;randn(4,5);
      <br><a href="#field">field</a>&lt;mat&gt;&nbsp;F(2,1);
      <br>F(0,0)&nbsp;=&nbsp;A;
      <br>F(1,0)&nbsp;=&nbsp;B;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: center;">
      <a href="#field">fields</a> store arbitrary objects, such as matrices
    </td>
  </tr>
</tbody>
</table>
</ul>
<br>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="example_prog"></a>
<b>example program</b>
<br>
<ul>
<pre>
#include &lt;iostream&gt;
#include &lt;armadillo&gt;

using namespace std;
using namespace arma;

int main()
  {
  mat A = randu&lt;mat&gt;(4,5);
  mat B = randu&lt;mat&gt;(4,5);
  
  cout &lt;&lt; A*B.t() &lt;&lt; endl;
  
  return 0;
  }
</pre>
<li>
If you save the above program as <i>example.cpp</i>,
under Linux and Mac OS X it can be compiled using:
<br>
<code>g++ example.cpp -o example -O2 -larmadillo</code>
</li>
<br>
<li>
As Armadillo is a template library, we strongly recommend to enable optimisation when compiling programs
(eg. when compiling with GCC or clang, use the -O2 or -O3 options)
</li>
<br>
<li>
See also the example program that comes with the Armadillo archive
</li>
</ul>
<br>

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="config_hpp"></a>
<b>config.hpp</b>
<br>
<ul>
<li>
Armadillo can be configured via editing the file <i>include/armadillo_bits/config.hpp</i>.
Specific functionality can be enabled or disabled by uncommenting or commenting out a particular <i>#define</i>, listed below.
<br>
<br>
<table style="text-align: left;" border="0" cellpadding="2" cellspacing="2">
<tbody>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_WRAPPER</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable going through the run-time Armadillo wrapper library (<i>libarmadillo.so</i>) when calling LAPACK, BLAS, ARPACK, SuperLU and HDF5 functions.
You will need to directly link with BLAS, LAPACK, etc (eg. <i>-lblas -llapack </i>)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_LAPACK</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Enable use of LAPACK, or a high-speed replacement for LAPACK (eg. Intel MKL, AMD ACML or the Accelerate framework).
Armadillo requires LAPACK for functions such as <a href="#svd">svd()</a>, <a href="#inv">inv()</a>, <a href="#eig_sym">eig_sym()</a>, <a href="#solve">solve()</a>, etc.
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_LAPACK</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable use of LAPACK. Overrides <i>ARMA_USE_LAPACK</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_BLAS</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Enable use of BLAS, or a high-speed replacement for BLAS (eg. OpenBLAS, Intel MKL, AMD ACML or the Accelerate framework).
BLAS is used for <a href="#operators">matrix multiplication</a>.
Without BLAS, Armadillo will use a built-in matrix multiplication routine, which might be slower for large matrices.
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_BLAS</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable use of BLAS. Overrides <i>ARMA_USE_BLAS</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_NEWARP</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Enable use of the built-in reimplementation of ARPACK (Armadillo 7.x).
This is used for the eigen decomposition of real (non-complex) sparse matrices, ie. <a href="#eigs_gen">eigs_gen()</a>, <a href="#eigs_sym">eigs_sym()</a> and <a href="#svds">svds().</a>
Requires ARMA_USE_LAPACK to be enabled.
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_NEWARP</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable use of the built-in reimplementation of ARPACK. Overrides <i>ARMA_USE_NEWARP</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_ARPACK</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Enable use of ARPACK, or a high-speed replacement for ARPACK.
Armadillo requires ARPACK for the eigen decomposition of complex sparse matrices, ie. <a href="#eigs_gen">eigs_gen()</a>, <a href="#eigs_sym">eigs_sym()</a> and <a href="#svds">svds()</a>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_ARPACK</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable use of ARPACK. Overrides <i>ARMA_USE_ARPACK</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_SUPERLU</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Enable use of SuperLU, which is used by <a href="#spsolve">spsolve()</a> for finding the solutions of sparse systems;
you will need to link with the superlu library, for example <code><i>-lsuperlu</i></code>
<br><b>Caveat:</b> Armadillo&nbsp;6.x requires SuperLU&nbsp;4.3, while Armadillo&nbsp;7.x requires SuperLU&nbsp;5.2
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_SUPERLU</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable use of SuperLU. Overrides <i>ARMA_USE_SUPERLU</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_HDF5</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Enable the ability to <a href="#save_load_mat">save and load</a> matrices stored in the HDF5 format;
the <i>hdf5.h</i> header file must be available on your system and you will need to link with the hdf5 library (eg. -lhdf5)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_HDF5</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable the use of the HDF5 library. Overrides <i>ARMA_USE_HDF5</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<a name="config_hpp_arma_use_cxx11"></a>
<code>ARMA_USE_CXX11</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Use C++11 features, such as <a href="#element_initialisation">initialiser lists</a>;
automatically enabled when using a compiler in C++11 mode, for example, g++&nbsp;-std=c++11
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_USE_CXX11</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable use of C++11 features. Overrides <i>ARMA_USE_CXX11</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_BLAS_CAPITALS</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Use capitalised (uppercase) BLAS and LAPACK function names (eg. DGEMM vs dgemm)
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_BLAS_UNDERSCORE</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Append an underscore to BLAS and LAPACK function names (eg. dgemm_ vs dgemm). Enabled by default.
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_BLAS_LONG</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Use "long" instead of "int" when calling BLAS and LAPACK functions
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_BLAS_LONG_LONG</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Use "long&nbsp;long" instead of "int" when calling BLAS and LAPACK functions
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_TBB_ALLOC</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Use Intel TBB <i>scalable_malloc()</i> and <i>scalable_free()</i> instead of standard <i>malloc()</i> and <i>free()</i> for managing matrix memory
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_USE_MKL_ALLOC</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Use Intel MKL <i>mkl_malloc()</i> and <i>mkl_free()</i> instead of standard <i>malloc()</i> and <i>free()</i> for managing matrix memory
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<a name="config_hpp_arma_64bit_word"></a>
<code>ARMA_64BIT_WORD</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Use 64 bit integers.
Automatically enabled when using a C++11 compiler on 64-bit platforms.
Useful if you require matrices/vectors capable of holding more than 4 billion elements.
Your machine and compiler must have support for 64 bit integers (eg. via "long" or "long&nbsp;long").
This can also be enabled by adding <i>#define&nbsp;ARMA_64BIT_WORD</i> before each instance of <i>#include&nbsp;&lt;armadillo&gt;</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<a name="config_hpp_arma_no_debug"></a>
<code>ARMA_NO_DEBUG</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Disable all run-time checks, such as <a href="#element_access">bounds checking</a>.
This will result in faster code, but you first need to make sure that your code runs correctly!
We strongly recommend to have the run-time checks enabled during development,
as this greatly aids in finding mistakes in your code, and hence speeds up development.
We recommend that run-time checks be disabled <b>only</b> for the shipped version of your program
(ie. final release build).
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_EXTRA_DEBUG</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Print out the trace of internal functions used for evaluating expressions.
Not recommended for normal use.
This is mainly useful for debugging the library.
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_MAT_PREALLOC</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
The number of pre-allocated elements used by matrices and vectors.
Must be always enabled and set to an integer that is at least&nbsp;1.
By default set to 16.
If you mainly use lots of very small vectors (eg.&nbsp;&le;&nbsp;4&nbsp;elements), change the number to the size of your vectors.
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DEFAULT_OSTREAM</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
The default stream used for printing <a href="#logging">error messages</a> and by <a href="#print">.print()</a>.
Must be always enabled.
By default defined to <i>std::cout</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_PRINT_ERRORS</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Print errors and warnings encountered during program execution
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
<code>ARMA_DONT_PRINT_ERRORS</code>
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
Do not print errors or warnings. Overrides <i>ARMA_PRINT_ERRORS</i>
    </td>
  </tr>
  <tr>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
    <td style="vertical-align: top;">
      &nbsp;
    </td>
  </tr>
</tbody>
</table>
</li>

<br>
<li>
See also:
<ul>
<li><a href="#logging">logging of warnings and errors</a></li>
<li><a href="#element_access">element access</a></li>
<li><a href="#element_initialisation">element initialisation</a></li>
<li><a href="#uword">uword/sword</a></li>
</ul>
</li>
<br>
</ul>

<!--
<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="catching_exceptions"></a>
<br>
<b>how to catch std::runtime_error exceptions</b>
<br>
<br>
<ul>
<li>
If a function such as <a href="#inv">inv()</a> fails to find a solution,
an error message is printed and a <i>std::runtime_error</i> exception is thrown.
If the exception is not caught, the program typically terminates.
Below is an example of how to catch exceptions:
<ul>
<pre>
#include &lt;iostream&gt;
#include &lt;armadillo&gt;

using namespace std;
using namespace arma;

int main(int argc, char** argv)
  {
  // create a non-invertible matrix
  mat A = zeros&lt;mat&gt;(5,5);
  
  mat B;
  
  try
    {
    B = inv(A);
    }
  catch (std::runtime_error&amp; x)
    {
    cout &lt;&lt; "caught an exception" &lt;&lt; endl;
    }
  
  return 0;
  }
</pre>
</ul>
<li>
See also:
<ul>
<li><a href="#logging">logging of warnings and errors</a></li>
<li><a href="http://cplusplus.com/doc/tutorial/exceptions/">tutorial on exceptions</a></li>
<li><a href="http://cplusplus.com/reference/std/stdexcept/runtime_error/">std::runtime_error</a></li>
</ul>
</li>
<br>
</ul>
<br>
-->

<div class="pagebreak"></div><div class="noprint"><hr class="greyline"><br></div>
<a name="api_additions"></a>
<a name="api_changes"></a>
<b>History of API Additions, Changes and Deprecations</b>
<br>
<ul>
<li>API Version Policy:
<br>
<br>
<ul>
<li>
Armadillo's version number is <i>A.B.C</i>, where <i>A</i> is a major version, <i>B</i> is a minor version, and <i>C</i> is the patch level (indicating bug fixes)
</li>
<br>
<li>
Within each major version (eg. 4.x), minor versions are backwards compatible with earlier minor versions.
For example, code written for version 4.000 will work with version 4.100, 4.120, 4.200, etc.
However, as each minor version may have more features (ie. API extensions) than earlier versions,
code specifically written for version 4.100 doesn't necessarily work with 4.000
</li>
<br>
<li>
We don't like changes to existing APIs and strongly prefer not to break any user software.
However, to allow evolution, we reserve the right to alter the APIs in future <i>major</i> versions of Armadillo
while remaining backwards compatible in as many cases as possible
(eg. version 5.x may have slightly different APIs than 4.x).
In a rare instance the user API may need to be tweaked if a bug fix absolutely requires it
</li>
<br>
<li>
This policy is applicable to the APIs described in this documentation; it is not applicable to internal functions
(ie. the underlying internal implementation details may change across consecutive minor versions)
</li>
</ul>
</li>

<!--
<br>
<li>
<a name="deprecated"></a>
List of deprecated functionality; this functionality will be <b>removed</b> in version 5.0:
<ul>
<li>
...
</li>
</ul>
</li>
-->

<br>
<br>
<li>
List of additions and changes for each version:
<br>
<br>
<ul>

<a name="version_7600"></a>
<li>Version 7.600:
<ul>
<li>more accurate <a href="#eigs_sym">eigs_sym()</a> and <a href="#eigs_gen">eigs_gen()</a></li>
<li>expanded <a href="#misc_fns">floor()</a>, <a href="#misc_fns">ceil()</a>, <a href="#misc_fns">round()</a>, <a href="#misc_fns">trunc()</a>, <a href="#misc_fns">sign()</a> to handle sparse matrices</li>
<li>added <a href="#arg">arg()</a>, <a href="#trig_fns">atan2()</a>, <a href="#trig_fns">hypot()</a></li>
</ul>
</li>
<br>
<a name="version_7500"></a>
<li>Version 7.500:
<ul>
<li>expanded <a href="#qz">qz()</a> to optionally specify ordering of the Schur form</li>
<li>expanded <a href="#each_slice">.each_slice()</a> to support matrix multiplication</li>
</ul>
</li>
<br>
<a name="version_7400"></a>
<li>Version 7.400:
<ul>
<li>added <a href="#expmat_sym">expmat_sym()</a>, <a href="#logmat_sympd">logmat_sympd()</a>, <a href="#sqrtmat_sympd">sqrtmat_sympd()</a></li>
<li>added <a href="#replace">.replace()</a></li>
</ul>
</li>
<br>
<a name="version_7300"></a>
<li>Version 7.300:
<ul>
<li>added <a href="#index_min_and_index_max_standalone">index_min()</a> and <a href="#index_min_and_index_max_standalone">index_max()</a> standalone functions</li>
<li>expanded <a href="#submat">.subvec()</a> to accept <a href="#size">size()</a> arguments</li>
<li>more robust handling of non-square matrices by <a href="#lu">lu()</a></li>
</ul>
</li>
<br>
<a name="version_7200"></a>
<li>Version 7.200:
<ul>
<li>added <a href="#index_min_and_index_max_member">.index_min()</a> and <a href="#index_min_and_index_max_member">.index_max()</a> member functions</li>
<li>expanded <a href="#ind2sub">ind2sub()</a> to handle vectors of indices</li>
<li>expanded <a href="#sub2ind">sub2ind()</a> to handle matrix of subscripts</li>
<li>expanded <a href="#expmat">expmat()</a>, <a href="#logmat">logmat()</a> and <a href="#sqrtmat">sqrtmat()</a> to optionally return a bool indicating success</li>
<li>faster handling of compound expressions by <a href="#vectorise">vectorise()</a></li>
</ul>
</li>
<br>
<a name="version_7100"></a>
<li>Version 7.100:
<ul>
<li>added <a href="#misc_fns">erf()</a>, <a href="#misc_fns">erfc()</a>, <a href="#misc_fns">lgamma()</a></li>
<li>added <a href="#subcube">.head_slices()</a> and <a href="#subcube">.tail_slices()</a> to subcube views</li>
<li><a href="#spsolve">spsolve()</a> now requires SuperLU 5.2</li>
<li><a href="#eigs_sym">eigs_sym()</a>, <a href="#eigs_gen">eigs_gen()</a> and <a href="#svds">svds()</a> now use a built-in reimplementation of ARPACK for real (non-complex) matrices; contributed by Yixuan Qiu</li>
</ul>
</li>
<br>
<a name="version_6700"></a>
<li>Version 6.700:
<ul>
<li>added <a href="#trapz">trapz()</a> for numerical integration</li>
<li>added <a href="#logmat">logmat()</a> for calculating the matrix logarithm</li>
<li>added <a href="#regspace">regspace()</a> for generating vectors with regularly spaced elements</li>
<li>added <a href="#logspace">logspace()</a> for generating vectors with logarithmically spaced elements</li>
<li>added <a href="#approx_equal">approx_equal()</a> for determining approximate equality</li>
<li>expanded <a href="#save_load_mat">.save()</a> and <a href="#save_load_mat">.load()</a> with <i>hdf5_binary_trans</i> file type, to save/load data with columns transposed to rows 
</ul>
</li>
<br>
<a name="version_6600"></a>
<li>Version 6.600:
<ul>
<li>expanded <a href="#sum">sum()</a>, <a href="#stats_fns">mean()</a>, <a href="#min_and_max">min()</a>, <a href="#min_and_max">max()</a> to handle cubes</li>
<li>expanded <a href="#Cube">Cube</a> class to handle arbitrarily sized empty cubes (eg. 0x5x2)</li>
<li>added <a href="#shift">shift()</a> for circular shifts of elements</li>
<li>added <a href="#sqrtmat">sqrtmat()</a> for finding the square root of a matrix</li>
</ul>
</li>
<br>
<a name="version_6500"></a>
<li>Version 6.500:
<ul>
<li>added <a href="#conv2">conv2()</a> for 2D convolution</li>
<li>added stand-alone <a href="#kmeans">kmeans()</a> function for clustering data</li>
<li>added <a href="#misc_fns">trunc()</a></li>
<li>extended <a href="#conv">conv()</a> to optionally provide central convolution</li>
<li>faster handling of multiply-and-accumulate by <a href="#accu">accu()</a> when using Intel MKL, ATLAS or OpenBLAS</li>
</ul>
</li>
<br>
<a name="version_6400"></a>
<li>Version 6.400:
<ul>
<li>expanded <a href="#each_colrow">each_col()</a>, <a href="#each_colrow">each_row()</a> and <a href="#each_slice">each_slice()</a> to handle C++11 lambda functions</li>
<li>added <a href="#ind2sub">ind2sub()</a> and <a href="#sub2ind">sub2ind()</a></li>
</ul>
</li>
<br>
<a name="version_6300"></a>
<li>Version 6.300:
<ul>
<li>expanded <a href="#solve">solve()</a> to find approximate solutions for rank-deficient systems</li>
<li>faster handling of <a href="#submat">non-contiguous submatrix views</a> in compound expressions</li>
<li>added <a href="#for_each">.for_each()</a> to <i>Mat</i>, <i>Row</i>, <i>Col</i>, <i>Cube</i> and <i>field</i> classes</li>
<li>added <a href="#rcond">rcond()</a> for estimating the reciprocal condition number</li>
</ul>
</li>
<br>
<a name="version_6200"></a>
<li>Version 6.200:
<ul>
<li>expanded <a href="#diagmat">diagmat()</a> to handle non-square matrices and arbitrary diagonals</li>
<li>expanded <a href="#trace">trace()</a> to handle non-square matrices</li>
</ul>
</li>
<br>
<a name="version_6100"></a>
<li>Version 6.100:
<ul>
<li>faster <a href="#norm">norm()</a> and <a href="#normalise">normalise()</a> when using Intel MKL, ATLAS or OpenBLAS</li>
<li>added Schur decomposition: <a href="#schur">schur()</a></li>
<li>stricter handling of matrix objects by <a href="#hist">hist()</a> and <a href="#histc">histc()</a></li>
<li><a href="#adv_constructors_mat">advanced constructors</a> for using auxiliary memory by Mat, Col, Row and Cube now have the default of <i>strict = false</i></li>
<li><a href="#Cube">Cube</a> class now delays allocation of <a href="#subcube">.slice()</a> related structures until needed</li>
<li>expanded <a href="#join_slices">join_slices()</a> to handle joining cubes with matrices</li>
</ul>
</li>
<br>
<a name="version_5600"></a>
<li>Version 5.600:
<ul>
<li>added <a href="#each_slice">.each_slice()</a> for repeated matrix operations on each slice of a cube</li>
<li>expanded <a href="#each_colrow">.each_col()</a> and <a href="#each_colrow">.each_row()</a> to handle out-of-place operations</li>
</ul>
</li>
<br>
<a name="version_5500"></a>
<li>Version 5.500:
<ul>
<li>expanded object constructors and generators to handle <a href="#size">size()</a> based specification of dimensions</li>
</ul>
</li>
<br>
<a name="version_5400"></a>
<li>Version 5.400:
<ul>
<li>added <a href="#find_unique">find_unique()</a> for finding indices of unique values</li>
<li>added <a href="#diff">diff()</a> for calculating differences between consecutive elements</li>
<li>added <a href="#cumprod">cumprod()</a> for calculating cumulative product</li>
<li>added <a href="#null">null()</a> for finding the orthonormal basis of null space</li>
<li>expanded <a href="#interp1">interp1()</a> to handle repeated locations</li>
<li>expanded <a href="#unique">unique()</a> to handle complex numbers</li>
<li>faster <a href="#flip">flipud()</a></li>
<li>faster row-wise <a href="#cumsum">cumsum()</a></li>
</ul>
</li>
<br>
<a name="version_5300"></a>
<li>Version 5.300:
<ul>
<li>added generalised Schur decomposition: <a href="#qz">qz()</a></li>
<li>added <a href="#has_inf">.has_inf()</a> and <a href="#has_nan">.has_nan()</a></li>
<li>expanded <a href="#interp1">interp1()</a> to handle out-of-domain locations</li>
<li>expanded <a href="#SpMat">sparse matrix</a> class with <a href="#set_imag">.set_imag()</a> and <a href="#set_imag">.set_real()</a></li>
<li>expanded <a href="#imag_real">imag()</a>, <a href="#imag_real">real()</a> and <a href="#conj">conj()</a> to handle sparse matrices</li>
<li>expanded <a href="#diagmat">diagmat()</a>, <a href="#reshape">reshape()</a> and <a href="#resize">resize()</a> to handle sparse matrices</li>
<li>faster sparse <a href="#sum">sum()</a></li>
<li>faster row-wise <a href="#sum">sum()</a>, <a href="#stats_fns">mean()</a>, <a href="#min_and_max">min()</a>, <a href="#min_and_max">max()</a></li>
<li>updated <a href="#constants">physical constants</a> to NIST 2014 CODATA values</li>
</ul>
</li>
<br>
<a name="version_5200"></a>
<li>Version 5.200:
<ul>
<li>added <a href="#orth">orth()</a> for finding the orthonormal basis of the range space of a matrix</li>
<li>expanded <a href="#element_initialisation">element initialisation</a> to handle nested initialiser lists (C++11)</li>
</ul>
</li>
<br>
<a name="version_5100"></a>
<li>Version 5.100:
<ul>
<li>added <a href="#interp1">interp1()</a> for 1D interpolation</li>
<li>added <a href="#is_sorted">.is_sorted()</a> for checking whether a vector or matrix has sorted elements</li>
<li>updated <a href="#constants">physical constants</a> to NIST 2010 CODATA values</li>
</ul>
</li>
<br>
<a name="version_5000"></a>
<li>Version 5.000:
<ul>
<li>added <a href="#spsolve">spsolve()</a> for solving sparse systems of linear equations</li>
<li>added <a href="#svds">svds()</a> for singular value decomposition of sparse matrices</li>
<li>added <a href="#nonzeros">nonzeros()</a> for extracting non-zero values from matrices</li>
<li>added handling of <a href="#diag">diagonal views</a> by sparse matrices
<li>expanded <a href="#repmat">repmat()</a> to handle sparse matrices
<li>expanded <a href="#join">join_rows()</a> and <a href="#join">join_cols()</a> to handle sparse matrices
<li><a href="#sort_index">sort_index()</a> and <a href="#sort_index">stable_sort_index()</a> have been placed in the delayed operations framework for increased efficiency
<li>use of <a href="#uword">64 bit integers</a> is automatically enabled when using a C++11 compiler</li>
</ul>
</li>
<br>
<a name="version_4650"></a>
<li>Version 4.650:
<ul>
<li>added <a href="#randg">randg()</a> for generating random values from gamma distributions (C++11 only)</li>
<li>added <i>.head_rows()</i> and <i>.tail_rows()</i> to <a href="#submat">submatrix views</a></li>
<li>added <i>.head_cols()</i> and <i>.tail_cols()</i> to <a href="#submat">submatrix views</a></li>
<li>expanded <a href="#eigs_sym">eigs_sym()</a> to optionally calculate eigenvalues with smallest/largest algebraic values</li>
</ul>
</li>
<br>
<a name="version_4600"></a>
<li>Version 4.600:
<ul>
<li>added <i>.head()</i> and <i>.tail()</i> to <a href="#submat">submatrix views</a></li>
<li>faster matrix transposes within compound expressions</li>
<li>faster in-place matrix multiplication</li>
<li>faster <a href="#accu">accu()</a> and <a href="#norm">norm()</a> when compiling with <i>-O3 -ffast-math -march=native</i> (gcc and clang)
</ul>
</li>
<br>
<a name="version_4550"></a>
<li>Version 4.550:
<ul>
<li>added matrix exponential function: <a href="#expmat">expmat()</a></li>
<li>faster <i>.log_p()</i> and <i>.avg_log_p()</i> functions in the <a href="#gmm_diag">gmm_diag</a> class when compiling with OpenMP enabled</li>
<li>faster handling of in-place addition/subtraction of expressions with an outer product
</ul>
</li>
<br>
<a name="version_4500"></a>
<li>Version 4.500:
<ul>
<li>faster handling of complex vectors by <a href="#norm">norm()</a></li>
<li>expanded <a href="#chol">chol()</a> to optionally specify output matrix as upper or lower triangular</li>
<li>better handling of non-finite values when <a href="#save_load_mat">saving</a> matrices as text files</li>
</ul>
</li>
<br>
<a name="version_4450"></a>
<li>Version 4.450:
<ul>
<li>faster handling of matrix transposes within compound expressions</li>
<li>expanded <a href="#symmat">symmatu()/symmatl()</a> to optionally disable taking the complex conjugate of elements</li>
<li>expanded <a href="#sort_index">sort_index()</a> to handle complex vectors</li>
<li>expanded the <a href="#gmm_diag">gmm_diag</a> class with functions to generate random samples</li>
</ul>
</li>
<br>
<a name="version_4400"></a>
<li>Version 4.400:
<ul>
<li>faster handling of subvectors by <a href="#dot">dot()</a></li>
<li>faster handling of aliasing by <a href="#submat">submatrix</a> views</li>
<li>added <a href="#clamp">clamp()</a> for clamping values to be between lower and upper limits</li>
<li>added <a href="#gmm_diag">gmm_diag</a> class for statistical modelling of data using Gaussian Mixture Models</li>
<li>expanded <a href="#batch_constructors_sp_mat">batch insertion constructors</a> for sparse matrices to add values at repeated locations</li>
</ul>
</li>
<br>
<a name="version_4320"></a>
<li>Version 4.320:
<ul>
<li>expanded <a href="#eigs_sym">eigs_sym()</a> and <a href="#eigs_gen">eigs_gen()</a> to use an optional tolerance parameter</li>
<li>expanded <a href="#eig_sym">eig_sym()</a> to automatically fall back to standard decomposition method if divide-and-conquer fails</li>
<li>cmake-based installer enables use of C++11 random number generator when using gcc 4.8.3+ in C++11 mode</li>
</ul>
</li>
<br>
<a name="version_4300"></a>
<li>Version 4.300:
<ul>
<li>added <a href="#find_finite">find_finite()</a> and <a href="#find_nonfinite">find_nonfinite()</a></li>
<li>expressions <i>X=<a href="#inv">inv</a>(A)*B*C</i> and <i>X=A<a href="#i_member">.i()</a>*B*C</i> are automatically converted to <i>X=<a href="#solve">solve</a>(A,B*C)</i></li>
</ul>
</li>
<br>
<a name="version_4200"></a>
<li>Version 4.200:
<ul>
<li>faster transpose of sparse matrices</li>
<li>more efficient handling of aliasing during matrix multiplication</li>
<li>faster <a href="#inv">inverse</a> of matrices marked as diagonal</li>
</ul>
</li>
<br>
<a name="version_4100"></a>
<li>Version 4.100:
<ul>
<li>added <a href="#normalise">normalise()</a> for normalising vectors to unit <i>p</i>-norm</li>
<li>extended the <a href="#field">field class</a> to handle 3D layout</li>
<li>extended <a href="#eigs_sym">eigs_sym()</a> and <a href="#eigs_gen">eigs_gen()</a> to obtain eigenvalues of various forms (eg. largest or smallest magnitude)</li>
<li>automatic SIMD vectorisation of elementary expressions (eg. matrix addition) when using Clang 3.4+ with -O3 optimisation</li>
<li>faster handling of sparse submatrix views</li>
</ul>
</li>
<br>
<a name="version_4000"></a>
<li>Version 4.000:
<ul>
<li>added eigen decompositions of sparse matrices: <a href="#eigs_sym">eigs_sym()</a> and <a href="#eigs_gen">eigs_gen()</a></li>
<li>added eigen decomposition for pair of matrices: <a href="#eig_pair">eig_pair()</a></li>
<li>added simpler forms of <a href="#eig_gen">eig_gen()</a></li>
<li>added condition number of matrices: <a href="#cond">cond()</a></li>
<li>expanded <a href="#find">find()</a> to handle cubes</li>
<li>expanded <a href="#subcube">subcube views</a> to access elements specified in a vector</li>
<li>template argument for <a href="#running_stat_vec">running_stat_vec</a> expanded to accept vector types</li>
<li>more robust fast <a href="#inv">inverse</a> of 4x4 matrices</li>
<li>faster divide-and-conquer decompositions are now used by default for
<a href="#eig_sym">eig_sym()</a>,
<a href="#pinv">pinv()</a>,
<a href="#princomp">princomp()</a>,
<a href="#rank">rank()</a>,
<a href="#svd">svd()</a>,
<a href="#svd_econ">svd_econ()</a>
</li>
<li>
the form <i>inv(sympd(X))</i> no longer assumes that <i>X</i> is positive definite; use <a href="#inv_sympd">inv_sympd()</a> instead 
</li>
</ul>
</li>
<br>
<a name="version_3930"></a>
<li>Version 3.930:
<ul>
<li>added <a href="#size">size()</a> based specifications of <a href="#submat">submatrix view</a> sizes</li>
<li>added element-wise variants of <a href="#min_and_max">min()</a> and <a href="#min_and_max">max()</a></li>
<li>added divide-and-conquer variant of <a href="#svd_econ">svd_econ()</a></li>
<li>added divide-and-conquer variant of <a href="#pinv">pinv()</a></li>
<li>added <a href="#randi">randi()</a> for generating matrices with random integer values</li>
<li>added <a href="#inplace_trans">inplace_trans()</a> for memory efficient in-place transposes</li>
<li>added more intuitive specification of sort direction in <a href="#sort">sort()</a> and <a href="#sort_index">sort_index()</a></li>
<li>added more intuitive specification of method in <a href="#det">det()</a>, <a href="#i_member">.i()</a>, <a href="#inv">inv()</a> and <a href="#solve">solve()</a></li>
<li>more precise timer for the <a href="#wall_clock">wall_clock</a> class when using C++11</li>
</ul>
</li>
<br>
<a name="version_3920"></a>
<li>Version 3.920:
<ul>
<li>faster <a href="#zeros_member">.zeros()</a></li>
<li>faster <a href="#misc_fns">round()</a>, <a href="#misc_fns">exp2()</a> and <a href="#misc_fns">log2()</a> when using C++11</li>
<li>added signum function: <a href="#misc_fns">sign()</a></li>
<li>added move constructors when using C++11</li>
<li>added 2D fast Fourier transform: <a href="#fft2">fft2()</a></li>
<li>added <a href="#subcube">.tube()</a> for easier extraction of vectors and subcubes from cubes</li>
<li>added specification of a fill type during construction of <a href="#Mat">Mat</a>, <a href="#Col">Col</a>, <a href="#Row">Row</a> and <a href="#Cube">Cube</a> classes,
eg. <i>mat&nbsp;X(4,&nbsp;5,&nbsp;fill::zeros)</i></li>
</ul>
</li>
<br>
<a name="version_3910"></a>
<li>Version 3.910:
<ul>
<li>faster multiplication of a matrix with a transpose of itself, ie. <i>X*X.t()</i> and <i>X.t()*X</i></li>
<li>added <a href="#vectorise">vectorise()</a> for reshaping matrices into vectors</li>
<li>added <a href="#all">all()</a> and <a href="#any">any()</a> for indicating presence of elements satisfying a relational condition</li>
</ul>
</li>
<br>
<a name="version_3900"></a>
<li>Version 3.900:
<ul>
<li>added automatic SSE2 vectorisation of elementary expressions (eg. matrix addition) when using GCC 4.7+ with -O3 optimisation</li>
<li>faster <a href="#stats_fns">median()</a></li>
<li>faster handling of compound expressions with transposes of <a href="#submat">submatrix</a> rows</li>
<li>faster handling of compound expressions with transposes of complex vectors</li>
<li>added support for <a href="#save_load_mat">saving &amp; loading</a> of <a href="#Cube">cubes</a> in HDF5 format</li>
</ul>
</li>
<br>
<a name="version_3820"></a>
<li>Version 3.820:
<ul>
<li>faster <a href="#as_scalar">as_scalar()</a> for compound expressions</li>
<li>faster transpose of small vectors</li>
<li>faster matrix-vector product for small vectors</li>
<li>faster multiplication of small <a href="#adv_constructors_mat_fixed">fixed size matrices</a></li>
</ul>
</li>
<br>
<a name="version_3810"></a>
<li>Version 3.810:
<ul>
<li>added fast Fourier transform: <a href="#fft">fft()</a></li>
<li>added handling of <a href="#imbue">.imbue()</a> and <a href="#transform">.transform()</a> by submatrices and subcubes</li>
<li>added <a href="#batch_constructors_sp_mat">batch insertion constructors</a> for sparse matrices
</ul>
</li>
<br>
<a name="version_3800"></a>
<li>Version 3.800:
<ul>
<li>added <a href="#imbue">.imbue()</a> for filling a matrix/cube with values provided by a functor or lambda expression</li>
<li>added <a href="#swap">.swap()</a> for swapping contents with another matrix</li>
<li>added <a href="#transform">.transform()</a> for transforming a matrix/cube using a functor or lambda expression</li>
<li>added <a href="#misc_fns">round()</a> for rounding matrix elements towards nearest integer</li>
<li>faster <a href="#find">find()</a></li>
<li>changed license to the <a href="http://www.mozilla.org/MPL/2.0/">Mozilla Public License 2.0</a>;
see also the associated <a href="http://www.mozilla.org/MPL/2.0/FAQ.html">frequently asked questions</a> about the license</li>
</ul>
<br>
</li>
<a name="version_36"></a>
<li>Version 3.6:
<ul>
<li>faster handling of compound expressions with submatrices and subcubes</li>
<li>faster <a href="#trace">trace()</a></li>
<li>added support for loading matrices as text files with <i>NaN</i> and <i>Inf</i> elements</li>
<li>added <a href="#sort_index">stable_sort_index()</a>, which preserves the relative order of elements with equivalent values</li>
<li>added handling of <a href="#SpMat">sparse matrices</a> by <a href="#stats_fns">mean()</a>, <a href="#stats_fns">var()</a>, <a href="#norm">norm()</a>, <a href="#abs">abs()</a>, <a href="#misc_fns">square()</a>, <a href="#misc_fns">sqrt()</a></li>
<li>added saving and loading of sparse matrices in <i>arma_binary</i> format</li>
</ul>
<br>
</li>
<a name="version_34"></a>
<li>Version 3.4:
<ul>
<li>added economical QR decomposition: <a href="#qr_econ">qr_econ()</a></li>
<li>added <a href="#each_colrow">.each_col() &amp; .each_row()</a> for vector operations repeated on each column or row of a matrix</li>
<li>added preliminary support for <a href="#SpMat">sparse matrices</a></li>
<li>added ability to <a href="#save_load_mat">save and load</a> matrices in HDF5 format</li>
<li>faster <a href="#svd">singular value decomposition</a> via optional use of divide-and-conquer algorithm</li>
<li>faster <a href="#randu_randn_member">.randn()</a></li>
<li>faster <a href="#dot">dot() and cdot()</a> for complex numbers</li>
</ul>
<br>
</li>
<a name="version_32"></a>
<li>Version 3.2:
<ul>
<li>added <a href="#unique">unique()</a>, for finding unique elements of a matrix</li>
<li>added <a href="#eval_member">.eval()</a>, for forcing the evaluation of delayed expressions</li>
<li>faster <a href="#eig_sym">eigen decomposition</a> via optional use of divide-and-conquer algorithm</li>
<li>faster <a href="#t_st_members">transpose</a> of vectors and compound expressions</li>
<li>faster handling of <a href="#diag">diagonal views</a></li>
<li>faster handling of tiny <a href="#adv_constructors_col_fixed">fixed size</a> vectors (&le; 4 elements)</li>
</ul>
<br>
</li>
<a name="version_30"></a>
<li>Version 3.0:
<ul>
<li>added shorthand for inverse: <a href="#i_member">.i()</a></li>
<li>added <a href="#constants">datum</a> class</li>
<li>added <a href="#hist">hist()</a> and <a href="#histc">histc()</a></li>
<li>added non-contiguous <a href="#submat">submatrix views</a></li>
<li>faster handling of <a href="#submat">submatrix views</a> with a single row or column</li>
<li>faster element access in <a href="#adv_constructors_mat_fixed">fixed size matrices</a></li>
<li>faster <a href="#repmat">repmat()</a></li>
<li>expressions <i>X=<a href="#inv">inv</a>(A)*B</i> and <i>X=A<a href="#i_member">.i()</a>*B</i> are automatically converted to <i>X=<a href="#solve">solve</a>(A,B)</i>
<li>better detection of vector expressions by <a href="#sum">sum()</a>, <a href="#cumsum">cumsum()</a>, <a href="#prod">prod()</a>, <a href="#min_and_max">min()</a>, <a href="#min_and_max">max()</a>, <a href="#stats_fns">mean()</a>, <a href="#stats_fns">median()</a>, <a href="#stats_fns">stddev()</a>, <a href="#stats_fns">var()</a>
<li>faster generation of random numbers
(eg. <a href="#randu_randn_standalone">randu()</a> and <a href="#randu_randn_standalone">randn()</a>),
via an algorithm that produces slightly different numbers than in 2.x
</li>
<li>
support for tying writable auxiliary (external) memory to fixed size matrices has been removed;
instead, you can use standard matrices with <a href="#adv_constructors_mat">writable auxiliary memory</a>,
or initialise fixed size matrices by <a href="#adv_constructors_mat">copying the memory</a>;
using auxiliary memory with standard matrices is unaffected
</li>
<li>
<i>.print_trans()</i> and <i>.raw_print_trans()</i> have been removed;
instead, you can chain <i><a href="#t_st_members">.t()</a></i> and <i><a href="#print">.print()</a></i> to achieve a similar result: <i>X.t().print()</i>
</li>
</ul>
<br>
</li>
<a name="version_24"></a>
<li>Version 2.4:
<ul>
<li>added shorter forms of transposes: <a href="#t_st_members">.t()</a> and <a href="#t_st_members">.st()</a></li>
<li>added <a href="#resize_member">.resize()</a> and <a href="#resize">resize()</a></li>
<li>added optional use of 64 bit indices (allowing matrices to have more than 4 billion elements), enabled via ARMA_64BIT_WORD in <i>include/armadillo_bits/config.hpp</i></li>
<li>added experimental support for C++11 initialiser lists, enabled via ARMA_USE_CXX11 in <i>include/armadillo_bits/config.hpp</i></li>
<li>refactored code to eliminate warnings when using the Clang C++ compiler</li>
<li><a href="#Mat">umat</a>, <a href="#Col">uvec</a>, <a href="#min_and_max_member">.min()</a> and <a href="#min_and_max_member">.max()</a>
have been changed to use the <a href="#uword"><i>uword</i></a> type instead of the <i>u32</i> type;
by default the <i>uword</i> and <i>u32</i> types are equivalent (ie. unsigned integer type with a minimum width 32 bits);
however, when the use of 64 bit indices is enabled via ARMA_64BIT_WORD in <i>include/armadillo_bits/config.hpp</i>,
the <i>uword</i> type then has a minimum width of 64 bits
</ul>
</li>
<br>
<li>Version 2.2:
<ul>
<li>added <a href="#svd_econ">svd_econ()</a></li>
<li>added <a href="#toeplitz">circ_toeplitz()</a></li>
<li>added <a href="#is_vec">.is_colvec()</a> and <a href="#is_vec">.is_rowvec()</a></li>
</ul>
<br>
<li>Version 2.0:
<ul>
<li><a href="#det">det()</a>, <a href="#inv">inv()</a> and <a href="#solve">solve()</a> can be forced to use more precise algorithms for tiny matrices (&le;&nbsp;4x4)</li>
<li>added <a href="#syl">syl()</a>, for solving Sylvester's equation</li>
<li>added <a href="#trans">strans()</a>, for transposing a complex matrix without taking the complex conjugate</li>
<li>added <a href="#symmat">symmatu()</a> and <a href="#symmat">symmatl()</a></li>
<li>added submatrices of <a href="#submat">submatrices</a></li>
<li>faster <a href="#inv">inverse</a> of symmetric positive definite matrices</li>
<li>faster element access for <a href="#adv_constructors_mat_fixed">fixed size</a> matrices</li>
<li>faster multiplication of tiny matrices (eg. 4x4)</li>
<li>faster compound expressions containing <a href="#submat">submatrices</a></li>
<li>added handling of arbitrarily sized empty matrices (eg. 5x0)</li>
<li>added .count() member function in <a href="#running_stat">running_stat</a> and <a href="#running_stat_vec">running_stat_vec</a></li>
<li>added <a href="#save_load_mat">loading &amp; saving</a> of matrices as CSV text files</li>
<li><a href="#trans">trans()</a> now takes the complex conjugate when transposing a complex matrix</li>
<li>forms of
<a href="#chol">chol()</a>, <a href="#eig_sym">eig_sym()</a>, <a href="#eig_gen">eig_gen()</a>,
<a href="#inv">inv()</a>, <a href="#lu">lu()</a>, <a href="#pinv">pinv()</a>, <a href="#princomp">princomp()</a>,
<a href="#qr">qr()</a>, <a href="#solve">solve()</a>, <a href="#svd">svd()</a>, <a href="#syl">syl()</a>
that do not return a bool indicating success now throw <i>std::runtime_error</i> exceptions when failures are detected</li>
<li>princomp_cov() has been removed; <a href="#eig_sym">eig_sym()</a> in conjunction with <a href="#cov">cov()</a> can be used instead</li>
<li><a href="#is_vec">.is_vec()</a> now outputs <i>true</i> for empty vectors (eg. 0x1)</li>
<li>set_log_stream() &amp; get_log_stream() have been replaced by <a href="#logging">set_stream_err1()</a> &amp; <a href="#logging">get_stream_err1()</a></li>
</ul>
<br>
<li>Version 1.2:
<ul>
<li>added <a href="#min_and_max_member">.min() &amp; .max()</a> member functions of Mat and Cube</li>
<li>added <a href="#misc_fns">floor()</a> and <a href="#misc_fns">ceil()</a></li>
<li>added representation of &ldquo;not a number&rdquo;: math::nan()</li>
<li>added representation of infinity: math::inf()</li>
<li>added standalone <a href="#is_finite">is_finite()</a></li>
<li><a href="#in_range">.in_range()</a> expanded to use <b>span()</b> arguments</li>
<li><a href="#adv_constructors_mat">fixed size</a> matrices and vectors can use auxiliary (external) memory</li>
<li><a href="#submat">submatrices</a> and <a href="#subfield">subfields</a> can be accessed via <i><b>X(</b>&nbsp;<b>span(</b>a,b<b>)</b>,&nbsp;<b>span(</b>c,d<b>)</b>&nbsp;<b>)</b></i></li>
<li><a href="#subcube">subcubes</a> can be accessed via <i><b>X(</b>&nbsp;<b>span(</b>a,b<b>)</b>,&nbsp;<b>span(</b>c,d<b>)</b>,&nbsp;<b>span(</b>e,f<b>)</b>&nbsp;<b>)</b></i></li>
<li>the two argument version of <i><b>span</b></i> can be replaced by
<i><b>span::all</b></i> or <i><b>span()</b></i>, to indicate an entire range
</li>
<li>for cubes, the two argument version of <i><b>span</b></i> can be replaced by
a single argument version, <i><b>span(</b>a<b>)</b></i>, to indicate a single column, row or slice
</li>
<li>arbitrary "flat" subcubes can be interpreted as matrices; for example:
<ul>
<pre>
cube Q = randu&lt;cube&gt;(5,3,4);
mat  A = Q(&nbsp;span(1),&nbsp;span(1,2),&nbsp;span::all&nbsp;);
// A has a size of 2x4

vec v = ones&lt;vec&gt;(4);
Q(&nbsp;span(1),&nbsp;span(1),&nbsp;span::all&nbsp;)&nbsp;=&nbsp;v;
</pre>
</ul>
</li>
<li>added interpretation of matrices as triangular through <a href="#trimat">trimatu() / trimatl()</a></li>
<li>added explicit handling of triangular matrices by <a href="#solve">solve()</a> and <a href="#inv">inv()</a></li>
<li>extended syntax for <a href="#submat">submatrices</a>, including access to elements whose indices are specified in a vector</li>
<li>added ability to change the stream used for <a href="#logging">logging</a> of errors and warnings</li>
<li>added ability to <a href="#save_load_mat">save/load matrices</a> in raw binary format</li>
<li>added cumulative sum function: <a href="#cumsum">cumsum()</a></li>
</ul>
</li>
<br>
<li>
Changed in 1.0 (compared to earlier 0.x development versions):
<ul>
<li>
the 3 argument version of <a href="#lu">lu()</a>,
eg. lu(L,U,X),
provides L and U which should be the same as produced by Octave 3.2
(this was not the case in versions prior to 0.9.90)
</li>
<br>
<li>
rand() has been replaced by <a href="#randu_randn_standalone">randu()</a>;
this has been done to avoid confusion with <a href="http://cplusplus.com/reference/clibrary/cstdlib/rand/">std::rand()</a>,
which generates random numbers in a different interval
</li>
<br>
<li>
In versions earlier than 0.9.0,
some multiplication operations directly converted result matrices with a size of 1x1 into scalars.
This is no longer the case.
If you know the result of an expression will be a 1x1 matrix and wish to treat it as a pure scalar,
use the <a href="#as_scalar">as_scalar()</a> wrapping function
</li>
<br>
<li>
Almost all functions have been placed in the delayed operations framework (for speed purposes).
This may affect code which assumed that the output of some functions was a pure matrix.
The solution is easy, as explained below.
<br>
<br>
In general, Armadillo queues operations before executing them.
As such, the direct output of an operation or function cannot be assumed to be a directly accessible matrix.
The queued operations are executed when the output needs to be stored in a matrix,
eg. <i>mat&nbsp;B&nbsp;=&nbsp;trans(A)</i> or <i>mat&nbsp;B(trans(A))</i>.
If you need to force the execution of the delayed operations,
place the operation or function inside the corresponding Mat constructor.
For example, if your code assumed that the output of some functions was a pure matrix,
eg. <i>chol(m).diag()</i>, change the code to <i>mat(chol(m)).diag()</i>.
Similarly, if you need to pass the result of an operation such as <i>A+B</i> to one of your own functions,
use <i>my_function(&nbsp;mat(A+B)&nbsp;)</i>.
</li>
</ul>
</li>
</ul>
</li>
<br>
</ul>

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