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<h1 class="chapter"> 58. lapack </h1>
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<tr><td align="left" valign="top"><a href="#SEC250">58.1 Introduction to lapack</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC251">58.2 Functions and Variables for lapack</a></td><td> </td><td align="left" valign="top">
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<p><a name="Item_003a-Introduction-to-lapack"></a>
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<h2 class="section"> 58.1 Introduction to lapack </h2>
<p><code>lapack</code> is a Common Lisp translation (via the program <code>f2c</code>) of the Fortran library LAPACK,
as obtained from the SLATEC project.
</p>
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<p>Categories: <a href="maxima_95.html#Category_003a-Numerical-methods">Numerical methods</a>
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<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
·
<a href="maxima_95.html#Category_003a-Package-lapack">Package lapack</a>
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<h2 class="section"> 58.2 Functions and Variables for lapack </h2>
<p><a name="Item_003a-dgeev"></a>
</p><dl>
<dt><u>Function:</u> <b>dgeev</b><i> (<var>A</var>)</i>
<a name="IDX2292"></a>
</dt>
<dt><u>Function:</u> <b>dgeev</b><i> (<var>A</var>, <var>right_p</var>, <var>left_p</var>)</i>
<a name="IDX2293"></a>
</dt>
<dd><p>Computes the eigenvalues and, optionally, the eigenvectors of a matrix <var>A</var>.
All elements of <var>A</var> must be integer or floating point numbers.
<var>A</var> must be square (same number of rows and columns).
<var>A</var> might or might not be symmetric.
</p>
<p><code>dgeev(<var>A</var>)</code> computes only the eigenvalues of <var>A</var>.
<code>dgeev(<var>A</var>, <var>right_p</var>, <var>left_p</var>)</code> computes the eigenvalues of <var>A</var>
and the right eigenvectors when <em><var>right_p</var> = <code>true</code></em>
and the left eigenvectors when <em><var>left_p</var> = <code>true</code></em>.
</p>
<p>A list of three items is returned.
The first item is a list of the eigenvalues.
The second item is <code>false</code> or the matrix of right eigenvectors.
The third item is <code>false</code> or the matrix of left eigenvectors.
</p>
<p>The right eigenvector <em>v(j)</em> (the <em>j</em>-th column of the right eigenvector matrix) satisfies
</p>
<p><em>A . v(j) = lambda(j) . v(j)</em>
</p>
<p>where <em>lambda(j)</em> is the corresponding eigenvalue.
The left eigenvector <em>u(j)</em> (the <em>j</em>-th column of the left eigenvector matrix) satisfies
</p>
<p><em>u(j)**H . A = lambda(j) . u(j)**H</em>
</p>
<p>where <em>u(j)**H</em> denotes the conjugate transpose of <em>u(j)</em>.
The Maxima function <code>ctranspose</code> computes the conjugate transpose.
</p>
<p>The computed eigenvectors are normalized to have Euclidean norm
equal to 1, and largest component has imaginary part equal to zero.
</p>
<p>Example:
</p>
<pre class="example">(%i1) load (lapack)$
(%i2) fpprintprec : 6;
(%o2) 6
(%i3) M : matrix ([9.5, 1.75], [3.25, 10.45]);
[ 9.5 1.75 ]
(%o3) [ ]
[ 3.25 10.45 ]
(%i4) dgeev (M);
(%o4) [[7.54331, 12.4067], false, false]
(%i5) [L, v, u] : dgeev (M, true, true);
[ - .666642 - .515792 ]
(%o5) [[7.54331, 12.4067], [ ],
[ .745378 - .856714 ]
[ - .856714 - .745378 ]
[ ]]
[ .515792 - .666642 ]
(%i6) D : apply (diag_matrix, L);
[ 7.54331 0 ]
(%o6) [ ]
[ 0 12.4067 ]
(%i7) M . v - v . D;
[ 0.0 - 8.88178E-16 ]
(%o7) [ ]
[ - 8.88178E-16 0.0 ]
(%i8) transpose (u) . M - D . transpose (u);
[ 0.0 - 4.44089E-16 ]
(%o8) [ ]
[ 0.0 0.0 ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-lapack">Package lapack</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-dgesvd"></a>
</p><dl>
<dt><u>Function:</u> <b>dgesvd</b><i> (<var>A</var>)</i>
<a name="IDX2294"></a>
</dt>
<dt><u>Function:</u> <b>dgesvd</b><i> (<var>A</var>, <var>left_p</var>, <var>right_p</var>)</i>
<a name="IDX2295"></a>
</dt>
<dd><p>Computes the singular value decomposition (SVD) of a matrix <var>A</var>,
comprising the singular values and, optionally, the left and right singular vectors.
All elements of <var>A</var> must be integer or floating point numbers.
<var>A</var> might or might not be square (same number of rows and columns).
</p>
<p>Let <em>m</em> be the number of rows, and <em>n</em> the number of columns of <var>A</var>.
The singular value decomposition of <var>A</var> comprises three matrices,
<var>U</var>, <var>Sigma</var>, and <var>V^T</var>,
such that
</p>
<p><em><var>A</var> = <var>U</var> . <var>Sigma</var> . <var>V</var>^T</em>
</p>
<p>where <var>U</var> is an <em>m</em>-by-<em>m</em> unitary matrix,
<var>Sigma</var> is an <em>m</em>-by-<em>n</em> diagonal matrix,
and <var>V^T</var> is an <em>n</em>-by-<em>n</em> unitary matrix.
</p>
<p>Let <em>sigma[i]</em> be a diagonal element of <em>Sigma</em>,
that is, <em><var>Sigma</var>[i, i] = <var>sigma</var>[i]</em>.
The elements <em>sigma[i]</em> are the so-called singular values of <var>A</var>;
these are real and nonnegative, and returned in descending order.
The first <em>min(m, n)</em> columns of <var>U</var> and <var>V</var> are
the left and right singular vectors of <var>A</var>.
Note that <code>dgesvd</code> returns the transpose of <var>V</var>, not <var>V</var> itself.
</p>
<p><code>dgesvd(<var>A</var>)</code> computes only the singular values of <var>A</var>.
<code>dgesvd(<var>A</var>, <var>left_p</var>, <var>right_p</var>)</code> computes the singular values of <var>A</var>
and the left singular vectors when <em><var>left_p</var> = <code>true</code></em>
and the right singular vectors when <em><var>right_p</var> = <code>true</code></em>.
</p>
<p>A list of three items is returned.
The first item is a list of the singular values.
The second item is <code>false</code> or the matrix of left singular vectors.
The third item is <code>false</code> or the matrix of right singular vectors.
</p>
<p>Example:
</p>
<pre class="example">(%i1) load (lapack)$
(%i2) fpprintprec : 6;
(%o2) 6
(%i3) M: matrix([1, 2, 3], [3.5, 0.5, 8], [-1, 2, -3], [4, 9, 7]);
[ 1 2 3 ]
[ ]
[ 3.5 0.5 8 ]
(%o3) [ ]
[ - 1 2 - 3 ]
[ ]
[ 4 9 7 ]
(%i4) dgesvd (M);
(%o4) [[14.4744, 6.38637, .452547], false, false]
(%i5) [sigma, U, VT] : dgesvd (M, true, true);
(%o5) [[14.4744, 6.38637, .452547],
[ - .256731 .00816168 .959029 - .119523 ]
[ ]
[ - .526456 .672116 - .206236 - .478091 ]
[ ],
[ .107997 - .532278 - .0708315 - 0.83666 ]
[ ]
[ - .803287 - .514659 - .180867 .239046 ]
[ - .374486 - .538209 - .755044 ]
[ ]
[ .130623 - .836799 0.5317 ]]
[ ]
[ - .917986 .100488 .383672 ]
(%i6) m : length (U);
(%o6) 4
(%i7) n : length (VT);
(%o7) 3
(%i8) Sigma:
genmatrix(lambda ([i, j], if i=j then sigma[i] else 0),
m, n);
[ 14.4744 0 0 ]
[ ]
[ 0 6.38637 0 ]
(%o8) [ ]
[ 0 0 .452547 ]
[ ]
[ 0 0 0 ]
(%i9) U . Sigma . VT - M;
[ 1.11022E-15 0.0 1.77636E-15 ]
[ ]
[ 1.33227E-15 1.66533E-15 0.0 ]
(%o9) [ ]
[ - 4.44089E-16 - 8.88178E-16 4.44089E-16 ]
[ ]
[ 8.88178E-16 1.77636E-15 8.88178E-16 ]
(%i10) transpose (U) . U;
[ 1.0 5.55112E-17 2.498E-16 2.77556E-17 ]
[ ]
[ 5.55112E-17 1.0 5.55112E-17 4.16334E-17 ]
(%o10) [ ]
[ 2.498E-16 5.55112E-17 1.0 - 2.08167E-16 ]
[ ]
[ 2.77556E-17 4.16334E-17 - 2.08167E-16 1.0 ]
(%i11) VT . transpose (VT);
[ 1.0 0.0 - 5.55112E-17 ]
[ ]
(%o11) [ 0.0 1.0 5.55112E-17 ]
[ ]
[ - 5.55112E-17 5.55112E-17 1.0 ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-lapack">Package lapack</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-dlange"></a>
</p><dl>
<dt><u>Function:</u> <b>dlange</b><i> (<var>norm</var>, <var>A</var>)</i>
<a name="IDX2296"></a>
</dt>
<dt><u>Function:</u> <b>zlange</b><i> (<var>norm</var>, <var>A</var>)</i>
<a name="IDX2297"></a>
</dt>
<dd><p>Computes a norm or norm-like function of the matrix <var>A</var>.
</p>
<dl compact="compact">
<dt> <code>max</code></dt>
<dd><p>Compute <em>max(abs(A(i, j)))</em> where <em>i</em> and <em>j</em> range over
the rows and columns, respectively, of <var>A</var>.
Note that this function is not a proper matrix norm.
</p></dd>
<dt> <code>one_norm</code></dt>
<dd><p>Compute the <em>L[1]</em> norm of <var>A</var>,
that is, the maximum of the sum of the absolute value of elements in each column.
</p></dd>
<dt> <code>inf_norm</code></dt>
<dd><p>Compute the <em>L[inf]</em> norm of <var>A</var>,
that is, the maximum of the sum of the absolute value of elements in each row.
</p></dd>
<dt> <code>frobenius</code></dt>
<dd><p>Compute the Frobenius norm of <var>A</var>,
that is, the square root of the sum of squares of the matrix elements.
</p></dd>
</dl>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-lapack">Package lapack</a>
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</div>
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</p><hr size="6">
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This document was generated by <em>Robert Dodier</em> on <em>April, 24 2010</em> using <a href="http://texi2html.cvshome.org/"><em>texi2html 1.76</em></a>.
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