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<h3 class="section">18.4 Functions of a Matrix</h3>
<!-- ./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-1622"></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
Pade'
approximation method with three step preconditioning (SIAM Journal on
Numerical Analysis, 1977). Diagonal
Pade'
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
Pade'
approximation when
<code>Dq(a)</code>
is ill-conditioned.
</p></blockquote></div>
<!-- ./linear-algebra/logm.m -->
<p><a name="doc_002dlogm"></a>
<div class="defun">
— Function File: <b>logm</b> (<var>a</var>)<var><a name="index-logm-1623"></a></var><br>
<blockquote><p>Compute the matrix logarithm of the square matrix <var>a</var>. Note that
this is currently implemented in terms of an eigenvalue expansion and
needs to be improved to be more robust.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/sqrtm.cc -->
<p><a name="doc_002dsqrtm"></a>
<div class="defun">
— Loadable Function: [<var>result</var>, <var>error_estimate</var>] = <b>sqrtm</b> (<var>a</var>)<var><a name="index-sqrtm-1624"></a></var><br>
<blockquote><p>Compute the matrix square root of the square matrix <var>a</var>.
<p>Ref: Nicholas J. Higham. A new sqrtm for <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>
<!-- ./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-1625"></a></var><br>
<blockquote><p>Form the kronecker product of two 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>
</blockquote></div>
<!-- ./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-1626"></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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