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<a name="Functions-and-Variables-for-lapack"></a>
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Previous: <a href="maxima_202.html#Introduction-to-lapack" accesskey="p" rel="previous">Introduction to lapack</a>, Up: <a href="maxima_201.html#lapack_002dpkg" accesskey="u" rel="up">lapack-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-lapack-1"></a>
<h3 class="section">55.2 Functions and Variables for lapack</h3>
<a name="dgeev"></a><a name="Item_003a-lapack_002fdeffn_002fdgeev"></a><dl>
<dt><a name="index-dgeev"></a>Function: <strong>dgeev</strong> <em><br> <tt>dgeev</tt> (<var>A</var>) <br> <tt>dgeev</tt> (<var>A</var>, <var>right_p</var>, <var>left_p</var>)</em></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>
<div class="example">
<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>
</dd></dl>
<a name="dgeqrf"></a><a name="Item_003a-lapack_002fdeffn_002fdgeqrf"></a><dl>
<dt><a name="index-dgeqrf"></a>Function: <strong>dgeqrf</strong> <em>(<var>A</var>)</em></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 "." is the noncommutative multiplication operator,
is equal to <var>A</var> (ignoring floating point round-off errors).
</p>
<div class="example">
<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>
</dd></dl>
<a name="dgesv"></a><a name="Item_003a-lapack_002fdeffn_002fdgesv"></a><dl>
<dt><a name="index-dgesv"></a>Function: <strong>dgesv</strong> <em>(<var>A</var>, <var>b</var>)</em></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>
<div class="example">
<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></div>
<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>
<div class="example">
<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></div>
<p>The elements of <var>A</var> and <var>b</var> must evaluate to real floating point numbers.
</p>
<div class="example">
<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>
</dd></dl>
<a name="dgesvd"></a><a name="Item_003a-lapack_002fdeffn_002fdgesvd"></a><dl>
<dt><a name="index-dgesvd"></a>Function: <strong>dgesvd</strong> <em><br> <tt>dgesvd</tt> (<var>A</var>) <br> <tt>dgesvd</tt> (<var>A</var>, <var>left_p</var>, <var>right_p</var>)</em></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>
<div class="example">
<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>
</dd></dl>
<a name="dlange"></a><a name="zlange"></a><a name="Item_003a-lapack_002fdeffn_002fdlange"></a><dl>
<dt><a name="index-dlange"></a>Function: <strong>dlange</strong> <em>(<var>norm</var>, <var>A</var>)</em></dt>
<dd><a name="Item_003a-lapack_002fdeffn_002fzlange"></a></dd><dt><a name="index-zlange"></a>Function: <strong>zlange</strong> <em>(<var>norm</var>, <var>A</var>)</em></dt>
<dd>
<p>Computes a norm or norm-like function of the matrix <var>A</var>. If
<var>A</var> is a real matrix, use <code>dlange</code>. For a matrix with
complex elements, use <code>zlange</code>.
</p>
<p><code>norm</code> specifies the kind of norm to be computed:
</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>
</dd></dl>
<a name="dgemm"></a><a name="Item_003a-lapack_002fdeffn_002fdgemm"></a><dl>
<dt><a name="index-dgemm"></a>Function: <strong>dgemm</strong> <em><br> <tt>dgemm</tt> (<var>A</var>, <var>B</var>) <br> <tt>dgemm</tt> (<var>A</var>, <var>B</var>, <var>options</var>)</em></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
specified 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>
<div class="example">
<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>
</dd></dl>
<a name="zgeev"></a><a name="Item_003a-lapack_002fdeffn_002fzgeev"></a><dl>
<dt><a name="index-zgeev"></a>Function: <strong>zgeev</strong> <em><br> <tt>zgeev</tt> (<var>A</var>) <br> <tt>zgeev</tt> (<var>A</var>, <var>right_p</var>, <var>left_p</var>)</em></dt>
<dd>
<p>Like <code>dgeev</code>, but the matrix <var>A</var> is complex.
</p>
</dd></dl>
<a name="zheev"></a><a name="Item_003a-lapack_002fdeffn_002fzheev"></a><dl>
<dt><a name="index-zheev"></a>Function: <strong>zheev</strong> <em><br> <tt>zheev</tt> (<var>A</var>) <br> <tt>zheev</tt> (<var>A</var>, <var>eigvec_p</var>)</em></dt>
<dd>
<p>Like <code>dgeev</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>
<p>An example of computing the eigenvalues and then eigenvalues and
eigenvectors of an Hermitian matrix.
</p><div class="example">
<pre class="example">(%i1) load("lapack")$
(%i2) A: matrix(
[9.14 +%i*0.00 , -4.37 -%i*9.22 , -1.98 -%i*1.72 , -8.96 -%i*9.50],
[-4.37 +%i*9.22 , -3.35 +%i*0.00 , 2.25 -%i*9.51 , 2.57 +%i*2.40],
[-1.98 +%i*1.72 , 2.25 +%i*9.51 , -4.82 +%i*0.00 , -3.24 +%i*2.04],
[-8.96 +%i*9.50 , 2.57 -%i*2.40 , -3.24 -%i*2.04 , 8.44 +%i*0.00]);
(%o2)
[ 9.14 (- 9.22 %i) - 4.37 (- 1.72 %i) - 1.98 (- 9.5 %i) - 8.96 ]
[ ]
[ 9.22 %i - 4.37 - 3.35 2.25 - 9.51 %i 2.4 %i + 2.57 ]
[ ]
[ 1.72 %i - 1.98 9.51 %i + 2.25 - 4.82 2.04 %i - 3.24 ]
[ ]
[ 9.5 %i - 8.96 2.57 - 2.4 %i (- 2.04 %i) - 3.24 8.44 ]
(%i3) zheev(A);
(%o3) [[- 16.00474647209473, - 6.764970154793324, 6.665711453507098,
25.51400517338097], false]
(%i4) E:zheev(A,true)$
(%i5) E[1];
(%o5) [- 16.00474647209474, - 6.764970154793325, 6.665711453507101,
25.51400517338096]
(%i6) E[2];
[ 0.2674650533172745 %i + 0.2175453586665017 ]
[ ]
[ 0.002696730886619885 %i + 0.6968836773391712 ]
(%o6) Col 1 = [ ]
[ (- 0.6082406376714117 %i) - 0.01210614292697931 ]
[ ]
[ 0.1593081858095037 ]
[ 0.2644937470667444 %i + 0.4773693349937472 ]
[ ]
[ (- 0.2852389036031621 %i) - 0.1414362742011673 ]
Col 2 = [ ]
[ 0.2654607680986639 %i + 0.4467818117184174 ]
[ ]
[ 0.5750762708542709 ]
[ 0.2810649767305922 %i - 0.1335263928245182 ]
[ ]
[ 0.2866310132869556 %i - 0.4536971347853274 ]
Col 3 = [ ]
[ (- 0.2933684323754295 %i) - 0.4954972425541057 ]
[ ]
[ 0.5325337537576771 ]
[ (- 0.5737316575503476 %i) - 0.3966146799427706 ]
[ ]
[ 0.01826502619021457 %i + 0.3530557704387017 ]
Col 4 = [ ]
[ 0.1673700900085425 %i + 0.01476684746229564 ]
[ ]
[ 0.6002632636961784 ]
</pre></div>
</dd></dl>
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