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<h3 class="section">18.4 Functions of a Matrix</h3>
<!-- expm scripts/linear-algebra/expm.m -->
<p><a name="doc_002dexpm"></a>
<div class="defun">
— Function File: <b>expm</b> (<var>A</var>)<var><a name="index-expm-2101"></a></var><br>
<blockquote><p>Return the exponential of a matrix, defined as the infinite Taylor
series
<pre class="example"> expm (A) = I + A + A^2/2! + A^3/3! + ...
</pre>
<p>The Taylor series is <em>not</em> the way to compute the matrix
exponential; see Moler and Van Loan, <cite>Nineteen Dubious Ways to
Compute the Exponential of a Matrix</cite>, SIAM Review, 1978. This routine
uses Ward's diagonal Padé approximation method with three step
preconditioning (SIAM Journal on Numerical Analysis, 1977). Diagonal
Padé approximations are rational polynomials of matrices
<pre class="example"> -1
D (A) N (A)
</pre>
<p>whose Taylor series matches the first
<code>2q+1</code>
terms of the Taylor series above; direct evaluation of the Taylor series
(with the same preconditioning steps) may be desirable in lieu of the
Padé approximation when
<code>Dq(A)</code>
is ill-conditioned.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dlogm.html#doc_002dlogm">logm</a>, <a href="doc_002dsqrtm.html#doc_002dsqrtm">sqrtm</a>.
</p></blockquote></div>
<!-- logm scripts/linear-algebra/logm.m -->
<p><a name="doc_002dlogm"></a>
<div class="defun">
— Function File: <var>s</var> = <b>logm</b> (<var>A</var>)<var><a name="index-logm-2102"></a></var><br>
— Function File: <var>s</var> = <b>logm</b> (<var>A, opt_iters</var>)<var><a name="index-logm-2103"></a></var><br>
— Function File: [<var>s</var>, <var>iters</var>] = <b>logm</b> (<var><small class="dots">...</small></var>)<var><a name="index-logm-2104"></a></var><br>
<blockquote><p>Compute the matrix logarithm of the square matrix <var>A</var>. The
implementation utilizes a Padé approximant and the identity
<pre class="example"> logm (<var>A</var>) = 2^k * logm (<var>A</var>^(1 / 2^k))
</pre>
<p>The optional argument <var>opt_iters</var> is the maximum number of square roots
to compute and defaults to 100. The optional output <var>iters</var> is the
number of square roots actually computed.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dexpm.html#doc_002dexpm">expm</a>, <a href="doc_002dsqrtm.html#doc_002dsqrtm">sqrtm</a>.
</p></blockquote></div>
<!-- sqrtm src/DLD-FUNCTIONS/sqrtm.cc -->
<p><a name="doc_002dsqrtm"></a>
<div class="defun">
— Loadable Function: <var>s</var> = <b>sqrtm</b> (<var>A</var>)<var><a name="index-sqrtm-2105"></a></var><br>
— Loadable Function: [<var>s</var>, <var>error_estimate</var>] = <b>sqrtm</b> (<var>A</var>)<var><a name="index-sqrtm-2106"></a></var><br>
<blockquote><p>Compute the matrix square root of the square matrix <var>A</var>.
<p>Ref: N.J. Higham. <cite>A New sqrtm for </cite><span class="sc">matlab</span>. Numerical
Analysis Report No. 336, Manchester Centre for Computational
Mathematics, Manchester, England, January 1999.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dexpm.html#doc_002dexpm">expm</a>, <a href="doc_002dlogm.html#doc_002dlogm">logm</a>.
</p></blockquote></div>
<!-- kron src/DLD-FUNCTIONS/kron.cc -->
<p><a name="doc_002dkron"></a>
<div class="defun">
— Loadable Function: <b>kron</b> (<var>A, B</var>)<var><a name="index-kron-2107"></a></var><br>
— Loadable Function: <b>kron</b> (<var>A1, A2, <small class="dots">...</small></var>)<var><a name="index-kron-2108"></a></var><br>
<blockquote><p>Form the Kronecker product of two or more matrices, defined block by
block as
<pre class="example"> x = [a(i, j) b]
</pre>
<p>For example:
<pre class="example"> kron (1:4, ones (3, 1))
⇒ 1 2 3 4
1 2 3 4
1 2 3 4
</pre>
<p>If there are more than two input arguments <var>A1</var>, <var>A2</var>, <small class="dots">...</small>,
<var>An</var> the Kronecker product is computed as
<pre class="example"> kron (kron (<var>A1</var>, <var>A2</var>), ..., <var>An</var>)
</pre>
<p class="noindent">Since the Kronecker product is associative, this is well-defined.
</p></blockquote></div>
<!-- blkmm src/DLD-FUNCTIONS/dot.cc -->
<p><a name="doc_002dblkmm"></a>
<div class="defun">
— Loadable Function: <b>blkmm</b> (<var>A, B</var>)<var><a name="index-blkmm-2109"></a></var><br>
<blockquote><p>Compute products of matrix blocks. The blocks are given as
2-dimensional subarrays of the arrays <var>A</var>, <var>B</var>.
The size of <var>A</var> must have the form <code>[m,k,...]</code> and
size of <var>B</var> must be <code>[k,n,...]</code>. The result is
then of size <code>[m,n,...]</code> and is computed as follows:
<pre class="example"> for i = 1:prod (size (<var>A</var>)(3:end))
<var>C</var>(:,:,i) = <var>A</var>(:,:,i) * <var>B</var>(:,:,i)
endfor
</pre>
</blockquote></div>
<!-- syl src/DLD-FUNCTIONS/syl.cc -->
<p><a name="doc_002dsyl"></a>
<div class="defun">
— Loadable Function: <var>x</var> = <b>syl</b> (<var>A, B, C</var>)<var><a name="index-syl-2110"></a></var><br>
<blockquote><p>Solve the Sylvester equation
<pre class="example"> A X + X B + C = 0
</pre>
<p>using standard <span class="sc">lapack</span> subroutines. For example:
<pre class="example"> syl ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
⇒ [ -0.50000, -0.66667; -0.66667, -0.50000 ]
</pre>
</blockquote></div>
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