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<span id="Functions-of-a-Matrix"></span><div class="header">
<p>
Next: <a href="Specialized-Solvers.html" accesskey="n" rel="next">Specialized Solvers</a>, Previous: <a href="Matrix-Factorizations.html" accesskey="p" rel="prev">Matrix Factorizations</a>, Up: <a href="Linear-Algebra.html" accesskey="u" rel="up">Linear Algebra</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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<hr>
<span id="Functions-of-a-Matrix-1"></span><h3 class="section">18.4 Functions of a Matrix</h3>
<span id="index-matrix_002c-functions-of"></span>
<span id="XREFexpm"></span><dl>
<dt id="index-expm">: <em></em> <strong>expm</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Return the exponential of a matrix.
</p>
<p>The matrix exponential is defined as the infinite Taylor series
</p>
<div class="example">
<pre class="example">expm (A) = I + A + A^2/2! + A^3/3! + …
</pre></div>
<p>However, 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
</p>
<div class="example">
<pre class="example"> -1
D (A) N (A)
</pre></div>
<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.
</p>
<p><strong>See also:</strong> <a href="#XREFlogm">logm</a>, <a href="#XREFsqrtm">sqrtm</a>.
</p></dd></dl>
<span id="XREFlogm"></span><dl>
<dt id="index-logm">: <em><var>s</var> =</em> <strong>logm</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-logm-1">: <em><var>s</var> =</em> <strong>logm</strong> <em>(<var>A</var>, <var>opt_iters</var>)</em></dt>
<dt id="index-logm-2">: <em>[<var>s</var>, <var>iters</var>] =</em> <strong>logm</strong> <em>(…)</em></dt>
<dd><p>Compute the matrix logarithm of the square matrix <var>A</var>.
</p>
<p>The implementation utilizes a Padé approximant and the identity
</p>
<div class="example">
<pre class="example">logm (<var>A</var>) = 2^k * logm (<var>A</var>^(1 / 2^k))
</pre></div>
<p>The optional input <var>opt_iters</var> is the maximum number of square roots
to compute and defaults to 100.
</p>
<p>The optional output <var>iters</var> is the number of square roots actually
computed.
</p>
<p><strong>See also:</strong> <a href="#XREFexpm">expm</a>, <a href="#XREFsqrtm">sqrtm</a>.
</p></dd></dl>
<span id="XREFsqrtm"></span><dl>
<dt id="index-sqrtm">: <em><var>s</var> =</em> <strong>sqrtm</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-sqrtm-1">: <em>[<var>s</var>, <var>error_estimate</var>] =</em> <strong>sqrtm</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the matrix square root of the square matrix <var>A</var>.
</p>
<p>Ref: N.J. Higham. <cite>A New sqrtm for <small>MATLAB</small></cite>. Numerical
Analysis Report No. 336, Manchester Centre for Computational
Mathematics, Manchester, England, January 1999.
</p>
<p><strong>See also:</strong> <a href="#XREFexpm">expm</a>, <a href="#XREFlogm">logm</a>.
</p></dd></dl>
<span id="XREFkron"></span><dl>
<dt id="index-kron">: <em></em> <strong>kron</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt id="index-kron-1">: <em></em> <strong>kron</strong> <em>(<var>A1</var>, <var>A2</var>, …)</em></dt>
<dd><p>Form the Kronecker product of two or more matrices.
</p>
<p>This is defined block by block as
</p>
<div class="example">
<pre class="example">x = [ a(i,j)*b ]
</pre></div>
<p>For example:
</p>
<div class="example">
<pre class="example">kron (1:4, ones (3, 1))
⇒ 1 2 3 4
1 2 3 4
1 2 3 4
</pre></div>
<p>If there are more than two input arguments <var>A1</var>, <var>A2</var>, …,
<var>An</var> the Kronecker product is computed as
</p>
<div class="example">
<pre class="example">kron (kron (<var>A1</var>, <var>A2</var>), …, <var>An</var>)
</pre></div>
<p>Since the Kronecker product is associative, this is well-defined.
</p></dd></dl>
<span id="XREFblkmm"></span><dl>
<dt id="index-blkmm">: <em></em> <strong>blkmm</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dd><p>Compute products of matrix blocks.
</p>
<p>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:
</p>
<div class="example">
<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></div>
</dd></dl>
<span id="XREFsylvester"></span><dl>
<dt id="index-sylvester">: <em><var>X</var> =</em> <strong>sylvester</strong> <em>(<var>A</var>, <var>B</var>, <var>C</var>)</em></dt>
<dd><p>Solve the Sylvester equation.
</p>
<p>The Sylvester equation is defined as:
</p>
<div class="example">
<pre class="example">A X + X B = C
</pre></div>
<p>The solution is computed using standard <small>LAPACK</small> subroutines.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">sylvester ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
⇒ [ 0.50000, 0.66667; 0.66667, 0.50000 ]
</pre></div>
</dd></dl>
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