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
|
<html lang="en">
<head>
<title>cpiv_bard - optim_doc</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="optim_doc">
<meta name="generator" content="makeinfo 4.13">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Helper-functions.html#Helper-functions" title="Helper functions">
<link rel="next" href="gjp.html#gjp" title="gjp">
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
<!--
Additional documentation for the optim package for Octave.
Copyright (C) <Olaf Till <i7tiol@t-online.de>>
You can redistribute this documentation and/or modify it under the terms
of the GNU General Public License as published by the Free Software
Foundation; either version 3 of the License, or (at your option) any
later version.
This documentation is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
You should have received a copy of the GNU General Public License along
with this documentation; if not, see <http://www.gnu.org/licenses/>.-->
<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">
<a name="cpiv_bard"></a>
<a name="cpiv_005fbard"></a>
<p>
Next: <a rel="next" accesskey="n" href="gjp.html#gjp">gjp</a>,
Up: <a rel="up" accesskey="u" href="Helper-functions.html#Helper-functions">Helper functions</a>
<hr>
</div>
<h3 class="section">5.1 A complementary pivoting algorithm</h3>
<p><a name="index-cpiv_005fbard-88"></a>
<h4 class="subheading">Helptext:</h4>
<p><a name="XREFcpiv_005fbard"></a>
<pre class="verbatim">[lb, idx, ridx, mv] = cpiv_bard (v, m[, incl])
v: column vector; m: matrix; incl (optional): index. length (v)
must equal rows (m). Finds column vectors w and l with w == v + m *
l, w >= 0, l >= 0, l.' * w == 0. Chooses idx, w, and l so that
l(~idx) == 0, l(idx) == -inv (m(idx, idx)) * v(idx), w(idx) roughly
== 0, and w(~idx) == v(~idx) + m(idx, ~idx).' * l(idx). idx indexes
at least everything indexed by incl, but l(incl) may be < 0. lb:
l(idx) (column vector); idx: logical index, defined above; ridx:
~idx & w roughly == 0; mv: [m, v] after performing a Gauss-Jordan
'sweep' (with gjp.m) on each diagonal element indexed by idx.
Except the handling of incl (which enables handling of equality
constraints in the calling code), this is called solving the
'complementary pivot problem' (Cottle, R. W. and Dantzig, G. B.,
'Complementary pivot theory of mathematical programming', Linear
Algebra and Appl. 1, 102--125. References for the current
algorithm: Bard, Y.: Nonlinear Parameter Estimation, p. 147--149,
Academic Press, New York and London 1974; Bard, Y., 'An eclectic
approach to nonlinear programming', Proc. ANU Sem. Optimization,
Canberra, Austral. Nat. Univ.).
</pre>
<!-- -->
</body></html>
|
