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<h3 class="section" id="Functions-of-a-Matrix-1"><span>18.4 Functions of a Matrix<a class="copiable-link" href="#Functions-of-a-Matrix-1"> ¶</a></span></h3>
<a class="index-entry-id" id="index-matrix_002c-functions-of"></a>
<a class="anchor" id="XREFexpm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-expm"><span><code class="def-type"><var class="var">r</var> =</code> <strong class="def-name">expm</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-expm"> ¶</a></span></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-preformatted">expm (A) = I + A + A^2/2! + A^3/3! + ...
</pre></div>
<p>However, the Taylor series is <em class="emph">not</em> the way to compute the matrix
exponential; see Moler and Van Loan, <cite class="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">
<div class="group"><pre class="example-preformatted"> -1
D (A) N (A)
</pre></div></div>
<p>whose Taylor series matches the first
<code class="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 class="code">Dq(A)</code>
is ill-conditioned.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFlogm">logm</a>, <a class="ref" href="#XREFsqrtm">sqrtm</a>.
</p></dd></dl>
<a class="anchor" id="XREFlogm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-logm"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">logm</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-logm"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-logm-1"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">logm</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">opt_iters</var>)</code><a class="copiable-link" href="#index-logm-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-logm-2"><span><code class="def-type">[<var class="var">s</var>, <var class="var">iters</var>] =</code> <strong class="def-name">logm</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-logm-2"> ¶</a></span></dt>
<dd><p>Compute the matrix logarithm of the square matrix <var class="var">A</var>.
</p>
<p>The implementation utilizes a Padé approximant and the identity
</p>
<div class="example">
<pre class="example-preformatted">logm (<var class="var">A</var>) = 2^k * logm (<var class="var">A</var>^(1 / 2^k))
</pre></div>
<p>The optional input <var class="var">opt_iters</var> is the maximum number of square roots
to compute and defaults to 100.
</p>
<p>The optional output <var class="var">iters</var> is the number of square roots actually
computed.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFexpm">expm</a>, <a class="ref" href="#XREFsqrtm">sqrtm</a>.
</p></dd></dl>
<a class="anchor" id="XREFsqrtm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-sqrtm"><span><code class="def-type"><var class="var">s</var> =</code> <strong class="def-name">sqrtm</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-sqrtm"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-sqrtm-1"><span><code class="def-type">[<var class="var">s</var>, <var class="var">error_estimate</var>] =</code> <strong class="def-name">sqrtm</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-sqrtm-1"> ¶</a></span></dt>
<dd><p>Compute the matrix square root of the square matrix <var class="var">A</var>.
</p>
<p>Ref: N.J. Higham. <cite class="cite">A New sqrtm for <small class="sc">MATLAB</small></cite>. Numerical
Analysis Report No. 336, Manchester Centre for Computational
Mathematics, Manchester, England, January 1999.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFexpm">expm</a>, <a class="ref" href="#XREFlogm">logm</a>.
</p></dd></dl>
<a class="anchor" id="XREFkron"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-kron"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">kron</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>)</code><a class="copiable-link" href="#index-kron"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-kron-1"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">kron</strong> <code class="def-code-arguments">(<var class="var">A1</var>, <var class="var">A2</var>, …)</code><a class="copiable-link" href="#index-kron-1"> ¶</a></span></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-preformatted">c = [ a(i,j)*b ]
</pre></div>
<p>For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">kron (1:4, ones (3, 1))
⇒ 1 2 3 4
1 2 3 4
1 2 3 4
</pre></div></div>
<p>If there are more than two input arguments <var class="var">A1</var>, <var class="var">A2</var>, …,
<var class="var">An</var> the Kronecker product is computed as
</p>
<div class="example">
<pre class="example-preformatted">kron (kron (<var class="var">A1</var>, <var class="var">A2</var>), ..., <var class="var">An</var>)
</pre></div>
<p>Since the Kronecker product is associative, this is well-defined.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFtensorprod">tensorprod</a>.
</p></dd></dl>
<a class="anchor" id="XREFtensorprod"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-tensorprod"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">tensorprod</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>, <var class="var">dimA</var>, <var class="var">dimB</var>)</code><a class="copiable-link" href="#index-tensorprod"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-tensorprod-1"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">tensorprod</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-tensorprod-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-tensorprod-2"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">tensorprod</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>)</code><a class="copiable-link" href="#index-tensorprod-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-tensorprod-3"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">tensorprod</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>, "all")</code><a class="copiable-link" href="#index-tensorprod-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-tensorprod-4"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">tensorprod</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>, …, "NumDimensionsA", <var class="var">value</var>)</code><a class="copiable-link" href="#index-tensorprod-4"> ¶</a></span></dt>
<dd><p>Compute the tensor product between numeric tensors <var class="var">A</var> and <var class="var">B</var>.
</p>
<p>The dimensions of <var class="var">A</var> and <var class="var">B</var> that are contracted are defined by
<var class="var">dimA</var> and <var class="var">dimB</var>, respectively. <var class="var">dimA</var> and <var class="var">dimB</var> are
scalars or equal length vectors that define the dimensions to match up.
The matched dimensions of <var class="var">A</var> and <var class="var">B</var> must have the same number of
elements.
</p>
<p>When only <var class="var">dim</var> is used, it is equivalent to
<code class="code"><var class="var">dimA</var> = <var class="var">dimB</var> = <var class="var">dim</var></code>.
</p>
<p>When no dimensions are specified, <code class="code"><var class="var">dimA</var> = <var class="var">dimB</var> = []</code>. This
computes the outer product between <var class="var">A</var> and <var class="var">B</var>.
</p>
<p>Using the <code class="code">"all"</code> option results in the inner product between <var class="var">A</var>
and <var class="var">B</var>. This requires <code class="code">size (<var class="var">A</var>) == size (<var class="var">B</var>)</code>.
</p>
<p>Use the property-value pair with the property name
<code class="code">"NumDimensionsA"</code> when <var class="var">A</var> has trailing singleton
dimensions that should be transferred to <var class="var">C</var>. The specified <var class="var">value</var>
should be the total number of dimensions of <var class="var">A</var>.
</p>
<p><small class="sc">MATLAB</small> Compatibility: Octave does not currently support the
<code class="code">"<var class="var">property_name</var>=<var class="var">value</var>"</code> syntax for the
<code class="code">"NumDimensionsA"</code> parameter.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFkron">kron</a>, <a class="ref" href="Utility-Functions.html#XREFdot">dot</a>, <a class="ref" href="Arithmetic-Ops.html#XREFmtimes">mtimes</a>.
</p></dd></dl>
<a class="anchor" id="XREFblkmm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-blkmm"><span><code class="def-type"><var class="var">C</var> =</code> <strong class="def-name">blkmm</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>)</code><a class="copiable-link" href="#index-blkmm"> ¶</a></span></dt>
<dd><p>Compute products of matrix blocks.
</p>
<p>The blocks are given as 2-dimensional subarrays of the arrays <var class="var">A</var>,
<var class="var">B</var>. The size of <var class="var">A</var> must have the form <code class="code">[m,k,…]</code> and
size of <var class="var">B</var> must be <code class="code">[k,n,…]</code>. The result is then of size
<code class="code">[m,n,…]</code> and is computed as follows:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">for i = 1:prod (size (<var class="var">A</var>)(3:end))
<var class="var">C</var>(:,:,i) = <var class="var">A</var>(:,:,i) * <var class="var">B</var>(:,:,i)
endfor
</pre></div></div>
</dd></dl>
<a class="anchor" id="XREFsylvester"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-sylvester"><span><code class="def-type"><var class="var">X</var> =</code> <strong class="def-name">sylvester</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">B</var>, <var class="var">C</var>)</code><a class="copiable-link" href="#index-sylvester"> ¶</a></span></dt>
<dd><p>Solve the Sylvester equation.
</p>
<p>The Sylvester equation is defined as:
</p>
<div class="example">
<pre class="example-preformatted">A X + X B = C
</pre></div>
<p>The solution is computed using standard <small class="sc">LAPACK</small> subroutines.
</p>
<p>For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">sylvester ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
⇒ [ 0.50000, 0.66667; 0.66667, 0.50000 ]
</pre></div></div>
</dd></dl>
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