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
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with no
Invariant Sections and no cover texts. A copy of the license is
included in the section entitled "GNU Free Documentation License". -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library – Reference Manual: Hessenberg-Triangular Decomposition of Real Matrices</title>
<meta name="description" content="GNU Scientific Library – Reference Manual: Hessenberg-Triangular Decomposition of Real Matrices">
<meta name="keywords" content="GNU Scientific Library – Reference Manual: Hessenberg-Triangular Decomposition of Real Matrices">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-Index.html#Function-Index" rel="index" title="Function Index">
<link href="Linear-Algebra.html#Linear-Algebra" rel="up" title="Linear Algebra">
<link href="Bidiagonalization.html#Bidiagonalization" rel="next" title="Bidiagonalization">
<link href="Hessenberg-Decomposition-of-Real-Matrices.html#Hessenberg-Decomposition-of-Real-Matrices" rel="previous" title="Hessenberg Decomposition of Real Matrices">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Hessenberg_002dTriangular-Decomposition-of-Real-Matrices"></a>
<div class="header">
<p>
Next: <a href="Bidiagonalization.html#Bidiagonalization" accesskey="n" rel="next">Bidiagonalization</a>, Previous: <a href="Hessenberg-Decomposition-of-Real-Matrices.html#Hessenberg-Decomposition-of-Real-Matrices" accesskey="p" rel="previous">Hessenberg Decomposition of Real Matrices</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Hessenberg_002dTriangular-Decomposition-of-Real-Matrices-1"></a>
<h3 class="section">14.9 Hessenberg-Triangular Decomposition of Real Matrices</h3>
<a name="index-Hessenberg-triangular-decomposition"></a>
<p>A general real matrix pair (<em>A</em>, <em>B</em>) can be decomposed by
orthogonal similarity transformations into the form
where <em>U</em> and <em>V</em> are orthogonal, <em>H</em> is an upper
Hessenberg matrix, and <em>R</em> is upper triangular. The
Hessenberg-Triangular reduction is the first step in the generalized
Schur decomposition for the generalized eigenvalue problem.
</p>
<dl>
<dt><a name="index-gsl_005flinalg_005fhesstri_005fdecomp"></a>Function: <em>int</em> <strong>gsl_linalg_hesstri_decomp</strong> <em>(gsl_matrix * <var>A</var>, gsl_matrix * <var>B</var>, gsl_matrix * <var>U</var>, gsl_matrix * <var>V</var>, gsl_vector * <var>work</var>)</em></dt>
<dd><p>This function computes the Hessenberg-Triangular decomposition of the
matrix pair (<var>A</var>, <var>B</var>). On output, <em>H</em> is stored in <var>A</var>,
and <em>R</em> is stored in <var>B</var>. If <var>U</var> and <var>V</var> are provided
(they may be null), the similarity transformations are stored in them.
Additional workspace of length <em>N</em> is needed in <var>work</var>.
</p></dd></dl>
</body>
</html>
|
