1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
|
<html lang="en">
<head>
<title>Functions of a Matrix - Untitled</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="Untitled">
<meta name="generator" content="makeinfo 4.11">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Linear-Algebra.html#Linear-Algebra" title="Linear Algebra">
<link rel="prev" href="Matrix-Factorizations.html#Matrix-Factorizations" title="Matrix Factorizations">
<link rel="next" href="Specialized-Solvers.html#Specialized-Solvers" title="Specialized Solvers">
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
<meta http-equiv="Content-Style-Type" content="text/css">
<style type="text/css"><!--
pre.display { font-family:inherit }
pre.format { font-family:inherit }
pre.smalldisplay { font-family:inherit; font-size:smaller }
pre.smallformat { font-family:inherit; font-size:smaller }
pre.smallexample { font-size:smaller }
pre.smalllisp { font-size:smaller }
span.sc { font-variant:small-caps }
span.roman { font-family:serif; font-weight:normal; }
span.sansserif { font-family:sans-serif; font-weight:normal; }
--></style>
</head>
<body>
<div class="node">
<p>
<a name="Functions-of-a-Matrix"></a>
Next: <a rel="next" accesskey="n" href="Specialized-Solvers.html#Specialized-Solvers">Specialized Solvers</a>,
Previous: <a rel="previous" accesskey="p" href="Matrix-Factorizations.html#Matrix-Factorizations">Matrix Factorizations</a>,
Up: <a rel="up" accesskey="u" href="Linear-Algebra.html#Linear-Algebra">Linear Algebra</a>
<hr>
</div>
<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.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
<!-- A simple blank line produces the correct behavior. -->
<!-- @sp 1 -->
<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>
</body></html>
|
