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<h1 class="chapter"> 60. lapack </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC297">60.1 Introduction to lapack</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC298">60.2 Functions and Variables for lapack</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
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<p><a name="Item_003a-Introduction-to-lapack"></a>
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<h2 class="section"> 60.1 Introduction to lapack </h2>

<p><code>lapack</code> is a Common Lisp translation (via the program <code>f2cl</code>) of the Fortran library LAPACK,
as obtained from the SLATEC project.
</p>
<div class=categorybox>
&middot;
<p>@ref{Category: Numerical methods} &middot;
@ref{Category: Share packages} &middot;
@ref{Category: Package lapack}
</div>
</p>

<p><a name="Item_003a-Functions-and-Variables-for-lapack"></a>
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<h2 class="section"> 60.2 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="IDX2482"></a>
</dt>
<dt><u>Function:</u> <b>dgeev</b><i> (<var>A</var>, <var>right_p</var>, <var>left_p</var>)</i>
<a name="IDX2483"></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>
&middot;
<p>@ref{Category: Package lapack}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-dgeqrf"></a>
</p><dl>
<dt><u>Function:</u> <b>dgeqrf</b><i> (<var>A</var>)</i>
<a name="IDX2484"></a>
</dt>
<dd><p>Computes the QR decomposition of the matrix <var>A</var>.
All elements of <var>A</var> must be integer or floating point numbers.
<var>A</var> may or may not have the same number of rows and columns.
</p>
<p>A list of two items is returned.
The first item is the matrix <var>Q</var>, which is a square, orthonormal matrix
which has the same number of rows as <var>A</var>.
The second item is the matrix <var>R</var>, which is the same size as <var>A</var>,
and which has all elements equal to zero below the diagonal.
The product <code><var>Q</var> . <var>R</var></code>, where &quot;.&quot; is the noncommutative multiplication operator,
is equal to <var>A</var> (ignoring floating point round-off errors).
</p>
<pre class="example">(%i1) load (lapack) $
(%i2) fpprintprec : 6 $
(%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]) $
(%i4) [q, r] : dgeqrf (M);
       [ - .0905357  .995893  ]
(%o4) [[                      ], 
       [  .995893    .0905357 ]
                               [ - 11.0454   2.97863   5.15148 ]
                               [                               ]]
                               [     0      - 2.94241  8.50131 ]
(%i5) q . r - M;
         [ - 7.77156E-16   1.77636E-15   - 8.88178E-16 ]
(%o5)    [                                             ]
         [      0.0       - 1.33227E-15   8.88178E-16  ]
(%i6) mat_norm (%, 1);
(%o6)                      3.10862E-15
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lapack}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-dgesv"></a>
</p><dl>
<dt><u>Function:</u> <b>dgesv</b><i> (<var>A</var>, <var>b</var>)</i>
<a name="IDX2485"></a>
</dt>
<dd><p>Computes the solution <var>x</var> of the linear equation <em><var>A</var> <var>x</var> = <var>b</var></em>,
where <var>A</var> is a square matrix, and <var>b</var> is a matrix of the same number of rows
as <var>A</var> and any number of columns.
The return value <var>x</var> is the same size as <var>b</var>.
</p>
<p>The elements of <var>A</var> and <var>b</var> must evaluate to real floating point numbers via <code>float</code>;
thus elements may be any numeric type, symbolic numerical constants, or expressions which evaluate to floats.
The elements of <var>x</var> are always floating point numbers.
All arithmetic is carried out as floating point operations.
</p>
<p><code>dgesv</code> computes the solution via the LU decomposition of <var>A</var>.
</p>
<p>Examples:
</p>
<p><code>dgesv</code> computes the solution of the linear equation <em><var>A</var> <var>x</var> = <var>b</var></em>.
</p>
<pre class="example">(%i1) A : matrix ([1, -2.5], [0.375, 5]);
                               [   1    - 2.5 ]
(%o1)                          [              ]
                               [ 0.375    5   ]
(%i2) b : matrix ([1.75], [-0.625]);
                                  [  1.75   ]
(%o2)                             [         ]
                                  [ - 0.625 ]
(%i3) x : dgesv (A, b);
                            [  1.210526315789474  ]
(%o3)                       [                     ]
                            [ - 0.215789473684211 ]
(%i4) dlange (inf_norm, b - A.x);
(%o4)                                 0.0
</pre>
<p><var>b</var> is a matrix with the same number of rows as <var>A</var> and any number of columns.
<var>x</var> is the same size as <var>b</var>.
</p>
<pre class="example">(%i1) A : matrix ([1, -0.15], [1.82, 2]);
                               [  1    - 0.15 ]
(%o1)                          [              ]
                               [ 1.82    2    ]
(%i2) b : matrix ([3.7, 1, 8], [-2.3, 5, -3.9]);
                              [  3.7   1    8   ]
(%o2)                         [                 ]
                              [ - 2.3  5  - 3.9 ]
(%i3) x : dgesv (A, b);
      [  3.103827540695117  1.20985481742191    6.781786185657722 ]
(%o3) [                                                           ]
      [ -3.974483062032557  1.399032116146062  -8.121425428948527 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4)                       1.1102230246251565E-15
</pre>
<p>The elements of <var>A</var> and <var>b</var> must evaluate to real floating point numbers.
</p>
<pre class="example">(%i1) A : matrix ([5, -%pi], [1b0, 11/17]);
                               [   5    - %pi ]
                               [              ]
(%o1)                          [         11   ]
                               [ 1.0b0   --   ]
                               [         17   ]
(%i2) b : matrix ([%e], [sin(1)]);
                                  [   %e   ]
(%o2)                             [        ]
                                  [ sin(1) ]
(%i3) x : dgesv (A, b);
                             [ 0.690375643155986 ]
(%o3)                        [                   ]
                             [ 0.233510982552952 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4)                        2.220446049250313E-16
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lapack} &middot;
@ref{Category: Linear equations}
</div>
</p>
</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="IDX2486"></a>
</dt>
<dt><u>Function:</u> <b>dgesvd</b><i> (<var>A</var>, <var>left_p</var>, <var>right_p</var>)</i>
<a name="IDX2487"></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>
&middot;
<p>@ref{Category: Package lapack}
</div>
</p>
</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="IDX2488"></a>
</dt>
<dt><u>Function:</u> <b>zlange</b><i> (<var>norm</var>, <var>A</var>)</i>
<a name="IDX2489"></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>
&middot;
<p>@ref{Category: Package lapack}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-dgemm"></a>
</p><dl>
<dt><u>Function:</u> <b>dgemm</b><i> (<var>A</var>, <var>B</var>)</i>
<a name="IDX2490"></a>
</dt>
<dt><u>Function:</u> <b>dgemm</b><i> (<var>A</var>, <var>B</var>, <var>options</var>)</i>
<a name="IDX2491"></a>
</dt>
<dd><p>Compute the product of two matrices and optionally add the product to
a third matrix.
</p>
<p>In the simplest form, <code>dgemm(<var>A</var>, <var>B</var>)</code> computes the
product of the two real matrices, <var>A</var> and <var>B</var>.
</p>
<p>In the second form, <code>dgemm</code> computes the <em><var>alpha</var> *
<var>A</var> * <var>B</var> + <var>beta</var> * <var>C</var></em> where <var>A</var>, <var>B</var>,
<var>C</var> are real matrices of the appropriate sizes and <var>alpha</var> and
<var>beta</var> are real numbers.  Optionally, <var>A</var> and/or <var>B</var> can
be transposed before computing the product.  The extra parameters are
specifed by optional keyword arguments: The keyword arguments are
optional and may be specified in any order.  They all take the form
<code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt> <code>C</code></dt>
<dd><p>The matrix <var>C</var> that should be added.  The default is <code>false</code>,
which means no matrix is added.
</p></dd>
<dt> <code>alpha</code></dt>
<dd><p>The product of <var>A</var> and <var>B</var> is multiplied by this value.  The
default is 1.
</p></dd>
<dt> <code>beta</code></dt>
<dd><p>If a matrix <var>C</var> is given, this value multiplies <var>C</var> before it
is added.  The default value is 0, which implies that <var>C</var> is not
added, even if <var>C</var> is given.  Hence, be sure to specify a non-zero
value for <var>beta</var>.
</p></dd>
<dt> <code>transpose_a</code></dt>
<dd><p>If <code>true</code>, the transpose of <var>A</var> is used instead of <var>A</var>
for the product.  The default is <code>false</code>.
</p></dd>
<dt> <code>transpose_b</code></dt>
<dd><p>If <code>true</code>, the transpose of <var>B</var> is used instead of <var>B</var>
for the product.  The default is <code>false</code>.
</p></dd>
</dl>

<pre class="example">(%i1) load (lapack)$
(%i2) A : matrix([1,2,3],[4,5,6],[7,8,9]);
                                  [ 1  2  3 ]
                                  [         ]
(%o2)                             [ 4  5  6 ]
                                  [         ]
                                  [ 7  8  9 ]
(%i3) B : matrix([-1,-2,-3],[-4,-5,-6],[-7,-8,-9]);
                               [ - 1  - 2  - 3 ]
                               [               ]
(%o3)                          [ - 4  - 5  - 6 ]
                               [               ]
                               [ - 7  - 8  - 9 ]
(%i4) C : matrix([3,2,1],[6,5,4],[9,8,7]);
                                  [ 3  2  1 ]
                                  [         ]
(%o4)                             [ 6  5  4 ]
                                  [         ]
                                  [ 9  8  7 ]
(%i5) dgemm(A,B);
                         [ - 30.0   - 36.0   - 42.0  ]
                         [                           ]
(%o5)                    [ - 66.0   - 81.0   - 96.0  ]
                         [                           ]
                         [ - 102.0  - 126.0  - 150.0 ]
(%i6) A . B;
                            [ - 30   - 36   - 42  ]
                            [                     ]
(%o6)                       [ - 66   - 81   - 96  ]
                            [                     ]
                            [ - 102  - 126  - 150 ]
(%i7) dgemm(A,B,transpose_a=true);
                         [ - 66.0  - 78.0   - 90.0  ]
                         [                          ]
(%o7)                    [ - 78.0  - 93.0   - 108.0 ]
                         [                          ]
                         [ - 90.0  - 108.0  - 126.0 ]
(%i8) transpose(A) . B;
                           [ - 66  - 78   - 90  ]
                           [                    ]
(%o8)                      [ - 78  - 93   - 108 ]
                           [                    ]
                           [ - 90  - 108  - 126 ]
(%i9) dgemm(A,B,c=C,beta=1);
                         [ - 27.0  - 34.0   - 41.0  ]
                         [                          ]
(%o9)                    [ - 60.0  - 76.0   - 92.0  ]
                         [                          ]
                         [ - 93.0  - 118.0  - 143.0 ]
(%i10) A . B + C;
                            [ - 27  - 34   - 41  ]
                            [                    ]
(%o10)                      [ - 60  - 76   - 92  ]
                            [                    ]
                            [ - 93  - 118  - 143 ]
(%i11) dgemm(A,B,c=C,beta=1, alpha=-1);
                            [ 33.0   38.0   43.0  ]
                            [                     ]
(%o11)                      [ 72.0   86.0   100.0 ]
                            [                     ]
                            [ 111.0  134.0  157.0 ]
(%i12) -A . B + C;
                               [ 33   38   43  ]
                               [               ]
(%o12)                         [ 72   86   100 ]
                               [               ]
                               [ 111  134  157 ]

</pre><div class=categorybox>
&middot;
<p>@ref{Category: Package lapack}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-zgeev"></a>
</p><dl>
<dt><u>Function:</u> <b>zgeev</b><i> (<var>A</var>)</i>
<a name="IDX2492"></a>
</dt>
<dt><u>Function:</u> <b>zgeev</b><i> (<var>A</var>, <var>right_p</var>, <var>left_p</var>)</i>
<a name="IDX2493"></a>
</dt>
<dd><p>Like <code>dgeev</code>, but the matrix <var>A</var> is complex.
</p>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lapack}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-zheev"></a>
</p><dl>
<dt><u>Function:</u> <b>zheev</b><i> (<var>A</var>)</i>
<a name="IDX2494"></a>
</dt>
<dt><u>Function:</u> <b>zheev</b><i> (<var>A</var>, <var>eigvec_p</var>)</i>
<a name="IDX2495"></a>
</dt>
<dd><p>Like <code>zheev</code>, but the matrix <var>A</var> is assumed to be a square
complex Hermitian matrix. If <var>eigvec_p</var> is <code>true</code>, then the
eigenvectors of the matrix are also computed.
</p>
<p>No check is made that the matrix <var>A</var> is, in fact, Hermitian.
</p>
<p>A list of two items is returned, as in <code>dgeev</code>: a list of
eigenvalues, and <code>false</code> or the matrix of the eigenvectors.
</p>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lapack}
</div>
</p>
</dd></dl>


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