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<H1> 18. Matrix Manipulation </H1>
<!--docid::SEC147::-->
<P>

There are a number of functions available for checking to see if the
elements of a matrix meet some condition, and for rearranging the
elements of a matrix.  For example, Octave can easily tell you if all
the elements of a matrix are finite, or are less than some specified
value.  Octave can also rotate the elements, extract the upper- or
lower-triangular parts, or sort the columns of a matrix.
</P>
<P>

<TABLE BORDER="0" CELLSPACING="0">
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_19.html#SEC148">18.1 Finding Elements and Checking Conditions</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_19.html#SEC149">18.2 Rearranging Matrices</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_19.html#SEC150">18.3 Special Utility Matrices</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="octave_19.html#SEC151">18.4 Famous Matrices</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
</TABLE>
<P>

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<H2> 18.1 Finding Elements and Checking Conditions </H2>
<!--docid::SEC148::-->
<P>

The functions <CODE>any</CODE> and <CODE>all</CODE> are useful for determining
whether any or all of the elements of a matrix satisfy some condition.
The <CODE>find</CODE> function is also useful in determining which elements of
a matrix meet a specified condition.
</P>
<P>

<A NAME="doc-any"></A>
<A NAME="IDX539"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>any</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>For a vector argument, return 1 if any element of the vector is
nonzero.
<P>

For a matrix argument, return a row vector of ones and
zeros with each element indicating whether any of the elements of the
corresponding column of the matrix are nonzero.  For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>any (eye (2, 4))
     => [ 1, 1, 0, 0 ]
</pre></td></tr></table><P>

If the optional argument <VAR>dim</VAR> is supplied, work along dimension
<VAR>dim</VAR>.  For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>any (eye (2, 4), 2)
     => [ 1; 1 ]
</pre></td></tr></table></DL>
<P>

<A NAME="doc-all"></A>
<A NAME="IDX540"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>all</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>The function <CODE>all</CODE> behaves like the function <CODE>any</CODE>, except
that it returns true only if all the elements of a vector, or all the
elements along dimension <VAR>dim</VAR> of a matrix, are nonzero.
</DL>
<P>

Since the comparison operators (see section <A HREF="octave_11.html#SEC78">10.4 Comparison Operators</A>) return matrices
of ones and zeros, it is easy to test a matrix for many things, not just
whether the elements are nonzero.  For example, 
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>all (all (rand (5) &lt; 0.9))
     => 0
</pre></td></tr></table><P>

tests a random 5 by 5 matrix to see if all of its elements are less
than 0.9.
</P>
<P>

Note that in conditional contexts (like the test clause of <CODE>if</CODE> and
<CODE>while</CODE> statements) Octave treats the test as if you had typed
<CODE>all (all (condition))</CODE>.
</P>
<P>

<A NAME="doc-xor"></A>
<A NAME="IDX541"></A>
</P>
<DL>
<DT><U>Mapping Function:</U>  <B>xor</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD>Return the `exclusive or' of the entries of <VAR>x</VAR> and <VAR>y</VAR>.
For boolean expressions <VAR>x</VAR> and <VAR>y</VAR>,
<CODE>xor (<VAR>x</VAR>, <VAR>y</VAR>)</CODE> is true if and only if <VAR>x</VAR> or <VAR>y</VAR>
is true, but not if both <VAR>x</VAR> and <VAR>y</VAR> are true.
</DL>
<P>

<A NAME="doc-is_duplicate_entry"></A>
<A NAME="IDX542"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>is_duplicate_entry</B> <I>(<VAR>x</VAR>)</I>
<DD>Return non-zero if any entries in <VAR>x</VAR> are duplicates of one
another.
</DL>
<P>

<A NAME="doc-diff"></A>
<A NAME="IDX543"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>diff</B> <I>(<VAR>x</VAR>, <VAR>k</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector of length <VAR>n</VAR>, <CODE>diff (<VAR>x</VAR>)</CODE> is the
vector of first differences
<VAR>x</VAR>(2) - <VAR>x</VAR>(1), <small>...</small>, <VAR>x</VAR>(n) - <VAR>x</VAR>(n-1).
<P>

If <VAR>x</VAR> is a matrix, <CODE>diff (<VAR>x</VAR>)</CODE> is the matrix of column
differences along the first non-singleton dimension.
</P>
<P>

The second argument is optional.  If supplied, <CODE>diff (<VAR>x</VAR>,
<VAR>k</VAR>)</CODE>, where <VAR>k</VAR> is a nonnegative integer, returns the
<VAR>k</VAR>-th differences. It is possible that <VAR>k</VAR> is larger than
then first non-singleton dimension of the matrix. In this case,
<CODE>diff</CODE> continues to take the differences along the next
non-singleton dimension.
</P>
<P>

The dimension along which to take the difference can be explicitly
stated with the optional variable <VAR>dim</VAR>. In this case the 
<VAR>k</VAR>-th order differences are calculated along this dimension.
In the case where <VAR>k</VAR> exceeds <CODE>size (<VAR>x</VAR>, <VAR>dim</VAR>)</CODE>
then an empty matrix is returned.
</P>
</DL>
<P>

<A NAME="doc-isinf"></A>
<A NAME="IDX544"></A>
</P>
<DL>
<DT><U>Mapping Function:</U>  <B>isinf</B> <I>(<VAR>x</VAR>)</I>
<DD>Return 1 for elements of <VAR>x</VAR> that are infinite and zero
otherwise. For example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>isinf ([13, Inf, NA, NaN])
     => [ 0, 1, 0, 0 ]
</pre></td></tr></table></DL>
<P>

<A NAME="doc-isnan"></A>
<A NAME="IDX545"></A>
</P>
<DL>
<DT><U>Mapping Function:</U>  <B>isnan</B> <I>(<VAR>x</VAR>)</I>
<DD>Return 1 for elements of <VAR>x</VAR> that are NaN values and zero
otherwise. For example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>isnan ([13, Inf, NA, NaN])
     => [ 0, 0, 0, 1 ]
</pre></td></tr></table></DL>
<P>

<A NAME="doc-finite"></A>
<A NAME="IDX546"></A>
</P>
<DL>
<DT><U>Mapping Function:</U>  <B>finite</B> <I>(<VAR>x</VAR>)</I>
<DD>Return 1 for elements of <VAR>x</VAR> that are finite values and zero
otherwise. For example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>finite ([13, Inf, NA, NaN])
     => [ 1, 0, 0, 0 ]
</pre></td></tr></table></DL>
<P>

<A NAME="doc-find"></A>
<A NAME="IDX547"></A>
</P>
<DL>
<DT><U>Loadable Function:</U>  <B>find</B> <I>(<VAR>x</VAR>)</I>
<DD>Return a vector of indices of nonzero elements of a matrix.  To obtain a
single index for each matrix element, Octave pretends that the columns
of a matrix form one long vector (like Fortran arrays are stored).  For
example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>find (eye (2))
     => [ 1; 4 ]
</pre></td></tr></table><P>

If two outputs are requested, <CODE>find</CODE> returns the row and column
indices of nonzero elements of a matrix.  For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>[i, j] = find (2 * eye (2))
     => i = [ 1; 2 ]
     => j = [ 1; 2 ]
</pre></td></tr></table><P>

If three outputs are requested, <CODE>find</CODE> also returns a vector
containing the nonzero values.  For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>[i, j, v] = find (3 * eye (2))
     => i = [ 1; 2 ]
     => j = [ 1; 2 ]
     => v = [ 3; 3 ]
</pre></td></tr></table></DL>
<P>

        
<A NAME="doc-common_size"></A>
<A NAME="IDX548"></A>
</P>
<DL>
<DT><U>Function File:</U> [<VAR>err</VAR>, <VAR>y1</VAR>, ...] = <B>common_size</B> <I>(<VAR>x1</VAR>, ...)</I>
<DD>Determine if all input arguments are either scalar or of common
size.  If so, <VAR>err</VAR> is zero, and <VAR>yi</VAR> is a matrix of the
common size with all entries equal to <VAR>xi</VAR> if this is a scalar or
<VAR>xi</VAR> otherwise.  If the inputs cannot be brought to a common size,
errorcode is 1, and <VAR>yi</VAR> is <VAR>xi</VAR>.  For example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>[errorcode, a, b] = common_size ([1 2; 3 4], 5)
=> errorcode = 0
=> a = [ 1, 2; 3, 4 ]
=> b = [ 5, 5; 5, 5 ]
</pre></td></tr></table><P>

This is useful for implementing functions where arguments can either
be scalars or of common size.
</P>
</DL>
<P>

<A NAME="Rearranging Matrices"></A>
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<H2> 18.2 Rearranging Matrices </H2>
<!--docid::SEC149::-->
<P>

<A NAME="doc-fliplr"></A>
<A NAME="IDX549"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>fliplr</B> <I>(<VAR>x</VAR>)</I>
<DD>Return a copy of <VAR>x</VAR> with the order of the columns reversed.  For
example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>fliplr ([1, 2; 3, 4])
=>  2  1
         4  3
</pre></td></tr></table><P>

Note that <CODE>fliplr</CODE> only workw with 2-D arrays.  To flip N-d arrays
use <CODE>flipdim</CODE> instead.
</P>
</DL>
<P>

<A NAME="doc-flipud"></A>
<A NAME="IDX550"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>flipud</B> <I>(<VAR>x</VAR>)</I>
<DD>Return a copy of <VAR>x</VAR> with the order of the rows reversed.  For
example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>flipud ([1, 2; 3, 4])
=>  3  4
         1  2
</pre></td></tr></table><P>

Due to the difficulty of defining which axis about which to flip the 
matrix <CODE>flipud</CODE> only work with 2-d arrays.  To flip N-d arrays
use <CODE>flipdim</CODE> instead.
</P>
</DL>
<P>

<A NAME="doc-flipdim"></A>
<A NAME="IDX551"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>flipdim</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD>Return a copy of <VAR>x</VAR> flipped about the dimension <VAR>dim</VAR>.
For example
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>flipdim ([1, 2; 3, 4], 2)
=>  2  1
         4  3
</pre></td></tr></table></DL>
<P>

<A NAME="doc-rot90"></A>
<A NAME="IDX552"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>rot90</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>Return a copy of <VAR>x</VAR> with the elements rotated counterclockwise in
90-degree increments.  The second argument is optional, and specifies
how many 90-degree rotations are to be applied (the default value is 1).
Negative values of <VAR>n</VAR> rotate the matrix in a clockwise direction.
For example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rot90 ([1, 2; 3, 4], -1)
=>  3  1
         4  2
</pre></td></tr></table><P>

rotates the given matrix clockwise by 90 degrees.  The following are all
equivalent statements:
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rot90 ([1, 2; 3, 4], -1)
==
rot90 ([1, 2; 3, 4], 3)
==
rot90 ([1, 2; 3, 4], 7)
</pre></td></tr></table><P>

Due to the difficulty of defining an axis about which to rotate the 
matrix <CODE>rot90</CODE> only work with 2-D arrays.  To rotate N-d arrays
use <CODE>rotdim</CODE> instead.
</P>
</DL>
<P>

<A NAME="doc-rotdim"></A>
<A NAME="IDX553"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>rotdim</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>, <VAR>plane</VAR>)</I>
<DD>Return a copy of <VAR>x</VAR> with the elements rotated counterclockwise in
90-degree increments.  The second argument is optional, and specifies
how many 90-degree rotations are to be applied (the default value is 1).
The third argument is also optional and defines the plane of the
rotation. As such <VAR>plane</VAR> is a two element vector containing two
different valid dimensions of the matrix. If <VAR>plane</VAR> is not given
Then the first two non-singleton dimensions are used.
<P>

Negative values of <VAR>n</VAR> rotate the matrix in a clockwise direction.
For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rotdim ([1, 2; 3, 4], -1, [1, 2])
=>  3  1
         4  2
</pre></td></tr></table><P>

rotates the given matrix clockwise by 90 degrees.  The following are all
equivalent statements:
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rot90 ([1, 2; 3, 4], -1, [1, 2])
==
rot90 ([1, 2; 3, 4], 3, [1, 2])
==
rot90 ([1, 2; 3, 4], 7, [1, 2])
</pre></td></tr></table></DL>
<P>

<A NAME="doc-cat"></A>
<A NAME="IDX554"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>cat</B> <I>(<VAR>dim</VAR>, <VAR>array1</VAR>, <VAR>array2</VAR>, <small>...</small>, <VAR>arrayN</VAR>)</I>
<DD>Return the concatenation of N-d array objects, <VAR>array1</VAR>,
<VAR>array2</VAR>, <small>...</small>, <VAR>arrayN</VAR> along dimension <VAR>dim</VAR>.
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>A = ones (2, 2);
B = zeros (2, 2);
cat (2, A, B)
=> ans =

     1 1 0 0
     1 1 0 0
</pre></td></tr></table><P>

Alternatively, we can concatenate <VAR>A</VAR> and <VAR>B</VAR> along the
second dimension the following way:
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>[A, B].
</pre></td></tr></table><P>

<VAR>dim</VAR> can be larger than the dimensions of the N-d array objects
and the result will thus have <VAR>dim</VAR> dimensions as the
following example shows:
<TABLE><tr><td>&nbsp;</td><td class=example><pre>cat (4, ones(2, 2), zeros (2, 2))
=> ans =

   ans(:,:,1,1) =

     1 1
     1 1

   ans(:,:,1,2) =
     0 0
     0 0
</pre></td></tr></table><P>

</P>
</DL>
<P>

<A NAME="doc-horzcat"></A>
<A NAME="IDX555"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>horzcat</B> <I>(<VAR>array1</VAR>, <VAR>array2</VAR>, <small>...</small>, <VAR>arrayN</VAR>)</I>
<DD>Return the horizontal concatenation of N-d array objects, <VAR>array1</VAR>,
<VAR>array2</VAR>, <small>...</small>, <VAR>arrayN</VAR> along dimension 2.
</DL>
<P>

<A NAME="doc-vertcat"></A>
<A NAME="IDX556"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>vertcat</B> <I>(<VAR>array1</VAR>, <VAR>array2</VAR>, <small>...</small>, <VAR>arrayN</VAR>)</I>
<DD>Return the vertical concatenation of N-d array objects, <VAR>array1</VAR>,
<VAR>array2</VAR>, <small>...</small>, <VAR>arrayN</VAR> along dimension 1.
</DL>
<P>

<A NAME="doc-permute"></A>
<A NAME="IDX557"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>permute</B> <I>(<VAR>a</VAR>, <VAR>perm</VAR>)</I>
<DD>Return the generalized transpose for an N-d array object <VAR>a</VAR>.
The permutation vector <VAR>perm</VAR> must contain the elements
<CODE>1:ndims(a)</CODE> (in any order, but each element must appear just once).
<P>

</P>
</DL>
<P>

<A NAME="doc-ipermute"></A>
<A NAME="IDX558"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>ipermute</B> <I>(<VAR>a</VAR>, <VAR>iperm</VAR>)</I>
<DD>The inverse of the <CODE>permute</CODE> function.  The expression
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>ipermute (permute (a, perm), perm)
</pre></td></tr></table>returns the original array <VAR>a</VAR>.
<P>

</P>
</DL>
<P>

<A NAME="doc-reshape"></A>
<A NAME="IDX559"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>reshape</B> <I>(<VAR>a</VAR>, <VAR>m</VAR>, <VAR>n</VAR>, <small>...</small>)</I>
<DD><A NAME="IDX560"></A>
<DT><U>Function File:</U>  <B>reshape</B> <I>(<VAR>a</VAR>, <VAR>siz</VAR>)</I>
<DD>Return a matrix with the given dimensions whose elements are taken
from the matrix <VAR>a</VAR>.  The elements of the matrix are access in
column-major order (like Fortran arrays are stored).
<P>

For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>reshape ([1, 2, 3, 4], 2, 2)
     =>  1  3
         2  4
</pre></td></tr></table><P>

Note that the total number of elements in the original
matrix must match the total number of elements in the new matrix.
</P>
<P>

A single dimension of the return matrix can be unknown and is flagged
by an empty argument.
</P>
</DL>
<P>

<A NAME="doc-circshift"></A>
<A NAME="IDX561"></A>
</P>
<DL>
<DT><U>Function File:</U> <VAR>y</VAR> <B>=</B> <I>circshift (<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD>Circularly shifts the values of the array <VAR>x</VAR>. <VAR>n</VAR> must be
a vector of integers no longer than the number of dimensions in 
<VAR>x</VAR>. The values of <VAR>n</VAR> can be either positive or negative,
which determines the direction in which the values or <VAR>x</VAR> are
shifted. If an element of <VAR>n</VAR> is zero, then the corresponding
dimension of <VAR>x</VAR> will not be shifted. For example
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>x = [1, 2, 3; 4, 5, 6, 7, 8, 9];
circshift (x, 1)
=>  7, 8, 9
    1, 2, 3
    4, 5, 6
circshift (x, -2)
=>  7, 8, 9
    1, 2, 3
    4, 5, 6
circshift (x, [0,1])
=>  3, 1, 2
    6, 4, 5
    9, 7, 8
</pre></td></tr></table></DL>
<P>

<A NAME="doc-shiftdim"></A>
<A NAME="IDX562"></A>
</P>
<DL>
<DT><U>Function File:</U> <VAR>y</VAR> <B>=</B> <I>shiftdim (<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX563"></A>
<DT><U>Function File:</U> [<VAR>y</VAR>, <VAR>ns</VAR>] <B>=</B> <I>shiftdim (<VAR>x</VAR>)</I>
<DD>Shifts the dimension of <VAR>x</VAR> by <VAR>n</VAR>, where <VAR>n</VAR> must be
an integer scalar. When <VAR>n</VAR> is negative, the dimensions of
<VAR>x</VAR> are shifted to the left, with the leading dimensions
circulated to the end. If <VAR>n</VAR> is positive, then the dimensions
of <VAR>x</VAR> are shifted to the right, with the <VAR>n</VAR> singleton
dimensions added.
<P>

Called with a single argument, <CODE>shiftdim</CODE>, removes the leading
singleton dimensions, returning the number of dimensions removed
in the second output argument <VAR>ns</VAR>.
</P>
<P>

For example 
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>x = ones (1, 2, 3);
size (shiftdim (x, -1))
=> [2, 3, 1]
size (shiftdim (x, 1))
=> [1, 1, 2, 3]
[b, ns] = shiftdim (x);
=> b =  [1, 1, 1; 1, 1, 1]
=> ns = 1
</pre></td></tr></table></DL>
<P>

<A NAME="doc-shift"></A>
<A NAME="IDX564"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>shift</B> <I>(<VAR>x</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX565"></A>
<DT><U>Function File:</U>  <B>shift</B> <I>(<VAR>x</VAR>, <VAR>b</VAR>, <VAR>dim</VAR>)</I>
<DD>If <VAR>x</VAR> is a vector, perform a circular shift of length <VAR>b</VAR> of
the elements of <VAR>x</VAR>.
<P>

If <VAR>x</VAR> is a matrix, do the same for each column of <VAR>x</VAR>.
If the optional <VAR>dim</VAR> argument is given, operate along this 
dimension
</P>
</DL>
<P>

<A NAME="doc-sort"></A>
<A NAME="IDX566"></A>
</P>
<DL>
<DT><U>Loadable Function:</U> [<VAR>s</VAR>, <VAR>i</VAR>] = <B>sort</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX567"></A>
<DT><U>Loadable Function:</U> [<VAR>s</VAR>, <VAR>i</VAR>] = <B>sort</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>)</I>
<DD><A NAME="IDX568"></A>
<DT><U>Loadable Function:</U> [<VAR>s</VAR>, <VAR>i</VAR>] = <B>sort</B> <I>(<VAR>x</VAR>, <VAR>mode</VAR>)</I>
<DD><A NAME="IDX569"></A>
<DT><U>Loadable Function:</U> [<VAR>s</VAR>, <VAR>i</VAR>] = <B>sort</B> <I>(<VAR>x</VAR>, <VAR>dim</VAR>, <VAR>mode</VAR>)</I>
<DD>Return a copy of <VAR>x</VAR> with the elements elements arranged in
increasing order.  For matrices, <CODE>sort</CODE> orders the elements in each
column.
<P>

For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>sort ([1, 2; 2, 3; 3, 1])
     =>  1  1
         2  2
         3  3
</pre></td></tr></table><P>

The <CODE>sort</CODE> function may also be used to produce a matrix
containing the original row indices of the elements in the sorted
matrix.  For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>[s, i] = sort ([1, 2; 2, 3; 3, 1])
     => s = 1  1
            2  2
            3  3
     => i = 1  3
            2  1
            3  2
</pre></td></tr></table><P>

If the optional argument <VAR>dim</VAR> is given, then the matrix is sorted
along the dimension defined by <VAR>dim</VAR>. The optional argument <CODE>mode</CODE>
defines the order in which the values will be sorted. Valid values of
<CODE>mode</CODE> are `ascend' or `descend'.
</P>
<P>

For equal elements, the indices are such that the equal elements are listed
in the order that appeared in the original list.
</P>
<P>

The <CODE>sort</CODE> function may also be used to sort strings and cell arrays
of strings, it which case the dictionary order of the strings is used.
</P>
<P>

The algorithm used in <CODE>sort</CODE> is optimized for the sorting of partially
ordered lists.
</P>
</DL>
<P>

Since the <CODE>sort</CODE> function does not allow sort keys to be specified,
it can't be used to order the rows of a matrix according to the values
of the elements in various columns<A NAME="DOCF5" HREF="octave_fot.html#FOOT5">(5)</A>
in a single call.  Using the second output, however, it is possible to
sort all rows based on the values in a given column.  Here's an example
that sorts the rows of a matrix based on the values in the second
column.
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>a = [1, 2; 2, 3; 3, 1];
[s, i] = sort (a (:, 2));
a (i, :)
     =>  3  1
         1  2
         2  3
</pre></td></tr></table><P>

<A NAME="doc-tril"></A>
<A NAME="IDX570"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>tril</B> <I>(<VAR>a</VAR>, <VAR>k</VAR>)</I>
<DD><A NAME="IDX571"></A>
<DT><U>Function File:</U>  <B>triu</B> <I>(<VAR>a</VAR>, <VAR>k</VAR>)</I>
<DD>Return a new matrix formed by extracting extract the lower (<CODE>tril</CODE>)
or upper (<CODE>triu</CODE>) triangular part of the matrix <VAR>a</VAR>, and
setting all other elements to zero.  The second argument is optional,
and specifies how many diagonals above or below the main diagonal should
also be set to zero.
<P>

The default value of <VAR>k</VAR> is zero, so that <CODE>triu</CODE> and
<CODE>tril</CODE> normally include the main diagonal as part of the result
matrix.
</P>
<P>

If the value of <VAR>k</VAR> is negative, additional elements above (for
<CODE>tril</CODE>) or below (for <CODE>triu</CODE>) the main diagonal are also
selected.
</P>
<P>

The absolute value of <VAR>k</VAR> must not be greater than the number of
sub- or super-diagonals.
</P>
<P>

For example,
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>tril (ones (3), -1)
=>  0  0  0
         1  0  0
         1  1  0
</pre></td></tr></table><P>

and
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>tril (ones (3), 1)
=>  1  1  0
         1  1  1
         1  1  1
</pre></td></tr></table></DL>
<P>

<A NAME="doc-vec"></A>
<A NAME="IDX572"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>vec</B> <I>(<VAR>x</VAR>)</I>
<DD>Return the vector obtained by stacking the columns of the matrix <VAR>x</VAR>
one above the other.
</DL>
<P>

<A NAME="doc-vech"></A>
<A NAME="IDX573"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>vech</B> <I>(<VAR>x</VAR>)</I>
<DD>Return the vector obtained by eliminating all supradiagonal elements of
the square matrix <VAR>x</VAR> and stacking the result one column above the
other.
</DL>
<P>

<A NAME="doc-prepad"></A>
<A NAME="IDX574"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>prepad</B> <I>(<VAR>x</VAR>, <VAR>l</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX575"></A>
<DT><U>Function File:</U>  <B>postpad</B> <I>(<VAR>x</VAR>, <VAR>l</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX576"></A>
<DT><U>Function File:</U>  <B>postpad</B> <I>(<VAR>x</VAR>, <VAR>l</VAR>, <VAR>c</VAR>, <VAR>dim</VAR>)</I>
<DD><P>

Prepends (appends) the scalar value <VAR>c</VAR> to the vector <VAR>x</VAR>
until it is of length <VAR>l</VAR>.  If the third argument is not
supplied, a value of 0 is used.
</P>
<P>

If <CODE>length (<VAR>x</VAR>) &gt; <VAR>l</VAR></CODE>, elements from the beginning (end) of
<VAR>x</VAR> are removed until a vector of length <VAR>l</VAR> is obtained.
</P>
<P>

If <VAR>x</VAR> is a matrix, elements are prepended or removed from each row.
</P>
<P>

If the optional <VAR>dim</VAR> argument is given, then operate along this
dimension.
</P>
</DL>
<P>

<A NAME="Special Utility Matrices"></A>
<HR SIZE="6">
<A NAME="SEC150"></A>
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</TR></TABLE>
<H2> 18.3 Special Utility Matrices </H2>
<!--docid::SEC150::-->
<P>

<A NAME="doc-eye"></A>
<A NAME="IDX577"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>eye</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX578"></A>
<DT><U>Built-in Function:</U>  <B>eye</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX579"></A>
<DT><U>Built-in Function:</U>  <B>eye</B> <I>(<small>...</small>, <VAR>class</VAR>)</I>
<DD>Return an identity matrix.  If invoked with a single scalar argument,
<CODE>eye</CODE> returns a square matrix with the dimension specified.  If you
supply two scalar arguments, <CODE>eye</CODE> takes them to be the number of
rows and columns.  If given a vector with two elements, <CODE>eye</CODE> uses
the values of the elements as the number of rows and columns,
respectively.  For example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>eye (3)
     =>  1  0  0
         0  1  0
         0  0  1
</pre></td></tr></table><P>

The following expressions all produce the same result:
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>eye (2)
==
eye (2, 2)
==
eye (size ([1, 2; 3, 4])
</pre></td></tr></table><P>

The optional argument <VAR>class</VAR>, allows <CODE>eye</CODE> to return an array of
the specified type, like
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>val = zeros (n,m, &quot;uint8&quot;)
</pre></td></tr></table><P>

For compatibility with MATLAB, calling <CODE>eye</CODE> with no arguments
is equivalent to calling it with an argument of 1.
</P>
</DL>
<P>

<A NAME="doc-ones"></A>
<A NAME="IDX580"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>ones</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX581"></A>
<DT><U>Built-in Function:</U>  <B>ones</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX582"></A>
<DT><U>Built-in Function:</U>  <B>ones</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>, <VAR>k</VAR>, <small>...</small>)</I>
<DD><A NAME="IDX583"></A>
<DT><U>Built-in Function:</U>  <B>ones</B> <I>(<small>...</small>, <VAR>class</VAR>)</I>
<DD>Return a matrix or N-dimensional array whose elements are all 1.
The arguments are handled the same as the arguments for <CODE>eye</CODE>.
<P>

If you need to create a matrix whose values are all the same, you should
use an expression like
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>val_matrix = val * ones (n, m)
</pre></td></tr></table><P>

The optional argument <VAR>class</VAR>, allows <CODE>ones</CODE> to return an array of
the specified type, like
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>val = ones (n,m, &quot;uint8&quot;)
</pre></td></tr></table></DL>
<P>

<A NAME="doc-zeros"></A>
<A NAME="IDX584"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>zeros</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX585"></A>
<DT><U>Built-in Function:</U>  <B>zeros</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX586"></A>
<DT><U>Built-in Function:</U>  <B>zeros</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>, <VAR>k</VAR>, <small>...</small>)</I>
<DD><A NAME="IDX587"></A>
<DT><U>Built-in Function:</U>  <B>zeros</B> <I>(<small>...</small>, <VAR>class</VAR>)</I>
<DD>Return a matrix or N-dimensional array whose elements are all 0.
The arguments are handled the same as the arguments for <CODE>eye</CODE>.
<P>

The optional argument <VAR>class</VAR>, allows <CODE>zeros</CODE> to return an array of
the specified type, like
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>val = zeros (n,m, &quot;uint8&quot;)
</pre></td></tr></table></DL>
<P>

<A NAME="doc-repmat"></A>
<A NAME="IDX588"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>repmat</B> <I>(<VAR>A</VAR>, <VAR>m</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX589"></A>
<DT><U>Function File:</U>  <B>repmat</B> <I>(<VAR>A</VAR>, [<VAR>m</VAR> <VAR>n</VAR>])</I>
<DD>Form a block matrix of size <VAR>m</VAR> by <VAR>n</VAR>, with a copy of matrix
<VAR>A</VAR> as each element.  If <VAR>n</VAR> is not specified, form an 
<VAR>m</VAR> by <VAR>m</VAR> block matrix.
</DL>
<P>

<A NAME="doc-rand"></A>
<A NAME="IDX590"></A>
</P>
<DL>
<DT><U>Loadable Function:</U>  <B>rand</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX591"></A>
<DT><U>Loadable Function:</U>  <B>rand</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX592"></A>
<DT><U>Loadable Function:</U>  <B>rand</B> <I>(<CODE>&quot;seed&quot;</CODE>, <VAR>x</VAR>)</I>
<DD>Return a matrix with random elements uniformly distributed on the
interval (0, 1).  The arguments are handled the same as the arguments
for <CODE>eye</CODE>.  In
addition, you can set the seed for the random number generator using the
form
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rand (&quot;seed&quot;, <VAR>x</VAR>)
</pre></td></tr></table><P>

where <VAR>x</VAR> is a scalar value.  If called as
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rand (&quot;seed&quot;)
</pre></td></tr></table><P>

<CODE>rand</CODE> returns the current value of the seed.
</P>
</DL>
<P>

<A NAME="doc-randn"></A>
<A NAME="IDX593"></A>
</P>
<DL>
<DT><U>Loadable Function:</U>  <B>randn</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX594"></A>
<DT><U>Loadable Function:</U>  <B>randn</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX595"></A>
<DT><U>Loadable Function:</U>  <B>randn</B> <I>(<CODE>&quot;seed&quot;</CODE>, <VAR>x</VAR>)</I>
<DD>Return a matrix with normally distributed random elements.  The
arguments are handled the same as the arguments for <CODE>eye</CODE>.  In
addition, you can set the seed for the random number generator using the
form
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>randn (&quot;seed&quot;, <VAR>x</VAR>)
</pre></td></tr></table><P>

where <VAR>x</VAR> is a scalar value.  If called as
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>randn (&quot;seed&quot;)
</pre></td></tr></table><P>

<CODE>randn</CODE> returns the current value of the seed.
</P>
</DL>
<P>

The <CODE>rand</CODE> and <CODE>randn</CODE> functions use separate generators.
This ensures that
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rand (&quot;seed&quot;, 13);
randn (&quot;seed&quot;, 13);
u = rand (100, 1);
n = randn (100, 1);
</pre></td></tr></table><P>

and
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>rand (&quot;seed&quot;, 13);
randn (&quot;seed&quot;, 13);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
  u(i) = rand ();
  n(i) = randn ();
end
</pre></td></tr></table><P>

produce equivalent results.
</P>
<P>

Normally, <CODE>rand</CODE> and <CODE>randn</CODE> obtain their initial
seeds from the system clock, so that the sequence of random numbers is
not the same each time you run Octave.  If you really do need for to
reproduce a sequence of numbers exactly, you can set the seed to a
specific value.
</P>
<P>

If it is invoked without arguments, <CODE>rand</CODE> and <CODE>randn</CODE> return a
single element of a random sequence.
</P>
<P>

The <CODE>rand</CODE> and <CODE>randn</CODE> functions use Fortran code from
RANLIB, a library of fortran routines for random number generation,
compiled by Barry W. Brown and James Lovato of the Department of
Biomathematics at The University of Texas, M.D. Anderson Cancer Center,
Houston, TX 77030.
</P>
<P>

<A NAME="doc-randperm"></A>
<A NAME="IDX596"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>randperm</B> <I>(<VAR>n</VAR>)</I>
<DD>Return a row vector containing a random permutation of the
integers from 1 to <VAR>n</VAR>.
</DL>
<P>

<A NAME="doc-diag"></A>
<A NAME="IDX597"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>diag</B> <I>(<VAR>v</VAR>, <VAR>k</VAR>)</I>
<DD>Return a diagonal matrix with vector <VAR>v</VAR> on diagonal <VAR>k</VAR>.  The
second argument is optional.  If it is positive, the vector is placed on
the <VAR>k</VAR>-th super-diagonal.  If it is negative, it is placed on the
<VAR>-k</VAR>-th sub-diagonal.  The default value of <VAR>k</VAR> is 0, and the
vector is placed on the main diagonal.  For example,
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>diag ([1, 2, 3], 1)
     =>  0  1  0  0
         0  0  2  0
         0  0  0  3
         0  0  0  0
</pre></td></tr></table></DL>
<P>

The functions <CODE>linspace</CODE> and <CODE>logspace</CODE> make it very easy to
create vectors with evenly or logarithmically spaced elements.
See section <A HREF="octave_5.html#SEC53">4.2 Ranges</A>.
</P>
<P>

<A NAME="doc-linspace"></A>
<A NAME="IDX598"></A>
</P>
<DL>
<DT><U>Built-in Function:</U>  <B>linspace</B> <I>(<VAR>base</VAR>, <VAR>limit</VAR>, <VAR>n</VAR>)</I>
<DD>Return a row vector with <VAR>n</VAR> linearly spaced elements between
<VAR>base</VAR> and <VAR>limit</VAR>.  The number of elements, <VAR>n</VAR>, must be
greater than 1.  The <VAR>base</VAR> and <VAR>limit</VAR> are always included in
the range.  If <VAR>base</VAR> is greater than <VAR>limit</VAR>, the elements are
stored in decreasing order.  If the number of points is not specified, a
value of 100 is used.
<P>

The <CODE>linspace</CODE> function always returns a row vector.
</P>
</DL>
<P>

<A NAME="doc-logspace"></A>
<A NAME="IDX599"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>logspace</B> <I>(<VAR>base</VAR>, <VAR>limit</VAR>, <VAR>n</VAR>)</I>
<DD>Similar to <CODE>linspace</CODE> except that the values are logarithmically
spaced from
10^base to 10^limit.
<P>

If <VAR>limit</VAR> is equal to
pi,
the points are between
10^base and pi,
<EM>not</EM>
10^base and 10^pi,
in order to  be compatible with the corresponding MATLAB function.
</P>
</DL>
<P>

<A NAME="doc-warn_neg_dim_as_zero"></A>
<A NAME="IDX600"></A>
</P>
<DL>
<DT><U>Built-in Variable:</U> <B>warn_neg_dim_as_zero</B>
<DD>If the value of <CODE>warn_neg_dim_as_zero</CODE> is nonzero, print a warning
for expressions like
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>eye (-1)
</pre></td></tr></table><P>

The default value is 0.
</P>
</DL>
<P>

<A NAME="doc-warn_imag_to_real"></A>
<A NAME="IDX601"></A>
</P>
<DL>
<DT><U>Built-in Variable:</U> <B>warn_imag_to_real</B>
<DD>If the value of <CODE>warn_imag_to_real</CODE> is nonzero, a warning is
printed for implicit conversions of complex numbers to real numbers.
The default value is 0.
</DL>
<P>

<A NAME="Famous Matrices"></A>
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<H2> 18.4 Famous Matrices </H2>
<!--docid::SEC151::-->
<P>

The following functions return famous matrix forms.
</P>
<P>

<A NAME="doc-hankel"></A>
<A NAME="IDX602"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>hankel</B> <I>(<VAR>c</VAR>, <VAR>r</VAR>)</I>
<DD>Return the Hankel matrix constructed given the first column <VAR>c</VAR>, and
(optionally) the last row <VAR>r</VAR>.  If the last element of <VAR>c</VAR> is
not the same as the first element of <VAR>r</VAR>, the last element of
<VAR>c</VAR> is used.  If the second argument is omitted, it is assumed to
be a vector of zeros with the same size as <VAR>c</VAR>.
<P>

A Hankel matrix formed from an m-vector <VAR>c</VAR>, and an n-vector
<VAR>r</VAR>, has the elements
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>H(i,j) = c(i+j-1),  i+j-1 &lt;= m;
H(i,j) = r(i+j-m),  otherwise
</pre></td></tr></table></DL>
<P>

<A NAME="doc-hilb"></A>
<A NAME="IDX603"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>hilb</B> <I>(<VAR>n</VAR>)</I>
<DD>Return the Hilbert matrix of order <VAR>n</VAR>.  The
i, j
element of a Hilbert matrix is defined as
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>H (i, j) = 1 / (i + j - 1)
</pre></td></tr></table></DL>
<P>

<A NAME="doc-invhilb"></A>
<A NAME="IDX604"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>invhilb</B> <I>(<VAR>n</VAR>)</I>
<DD>Return the inverse of a Hilbert matrix of order <VAR>n</VAR>.  This can be 
computed computed exactly using
<TABLE><tr><td>&nbsp;</td><td class=example><pre>
            (i+j)         /n+i-1\  /n+j-1\   /i+j-2\ 2
 A(i,j) = -1      (i+j-1)(       )(       ) (       )
                          \ n-j /  \ n-i /   \ i-2 /

        = p(i) p(j) / (i+j-1)

</pre></td></tr></table>where
<TABLE><tr><td>&nbsp;</td><td class=example><pre>             k  /k+n-1\   /n\
    p(k) = -1  (       ) (   )
                \ k-1 /   \k/
</pre></td></tr></table><P>

The validity of this formula can easily be checked by expanding 
the binomial coefficients in both formulas as factorials.  It can 
be derived more directly via the theory of Cauchy matrices: 
see J. W. Demmel, Applied Numerical Linear Algebra, page 92.
</P>
<P>

Compare this with the numerical calculation of <CODE>inverse (hilb (n))</CODE>,
which suffers from the ill-conditioning of the Hilbert matrix, and the
finite precision of your computer's floating point arithmetic.
</P>
<P>

</P>
</DL>
<P>

<A NAME="doc-sylvester_matrix"></A>
<A NAME="IDX605"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>sylvester_matrix</B> <I>(<VAR>k</VAR>)</I>
<DD>Return the Sylvester matrix of order
n = 2^k.
</DL>
<P>

<A NAME="doc-toeplitz"></A>
<A NAME="IDX606"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>toeplitz</B> <I>(<VAR>c</VAR>, <VAR>r</VAR>)</I>
<DD>Return the Toeplitz matrix constructed given the first column <VAR>c</VAR>,
and (optionally) the first row <VAR>r</VAR>.  If the first element of <VAR>c</VAR>
is not the same as the first element of <VAR>r</VAR>, the first element of
<VAR>c</VAR> is used.  If the second argument is omitted, the first row is
taken to be the same as the first column.
<P>

A square Toeplitz matrix has the form:
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>c(0)  r(1)   r(2)  ...  r(n)
c(1)  c(0)   r(1)  ... r(n-1)
c(2)  c(1)   c(0)  ... r(n-2)
 .     ,      ,   .      .
 .     ,      ,     .    .
 .     ,      ,       .  .
c(n) c(n-1) c(n-2) ...  c(0)
</pre></td></tr></table></DL>
<P>

<A NAME="doc-vander"></A>
<A NAME="IDX607"></A>
</P>
<DL>
<DT><U>Function File:</U>  <B>vander</B> <I>(<VAR>c</VAR>)</I>
<DD>Return the Vandermonde matrix whose next to last column is <VAR>c</VAR>.
<P>

A Vandermonde matrix has the form:
</P>
<P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>c(1)^(n-1) ... c(1)^2  c(1)  1
c(2)^(n-1) ... c(2)^2  c(2)  1
    .     .      .      .    .
    .       .    .      .    .
    .         .  .      .    .
c(n)^(n-1) ... c(n)^2  c(n)  1
</pre></td></tr></table></DL>
<P>

<A NAME="Arithmetic"></A>
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