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
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>Maxima 5.47.0 Manual: Functions and Variables for hompack</title>
<meta name="description" content="Maxima 5.47.0 Manual: Functions and Variables for hompack">
<meta name="keywords" content="Maxima 5.47.0 Manual: Functions and Variables for hompack">
<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="maxima_toc.html#Top" rel="start" title="Top">
<link href="maxima_423.html#Function-and-Variable-Index" rel="index" title="Function and Variable Index">
<link href="maxima_toc.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="maxima_291.html#hompack_002dpkg" rel="up" title="hompack-pkg">
<link href="maxima_294.html#impdiff_002dpkg" rel="next" title="impdiff-pkg">
<link href="maxima_292.html#Introduction-to-hompack" rel="previous" title="Introduction to hompack">
<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}
body {color: black; background: white; margin-left: 8%; margin-right: 13%;
font-family: "FreeSans", sans-serif}
h1 {font-size: 150%; font-family: "FreeSans", sans-serif}
h2 {font-size: 125%; font-family: "FreeSans", sans-serif}
h3 {font-size: 100%; font-family: "FreeSans", sans-serif}
a[href] {color: rgb(0,0,255); text-decoration: none;}
a[href]:hover {background: rgb(220,220,220);}
div.textbox {border: solid; border-width: thin; padding-top: 1em;
padding-bottom: 1em; padding-left: 2em; padding-right: 2em}
div.titlebox {border: none; padding-top: 1em; padding-bottom: 1em;
padding-left: 2em; padding-right: 2em; background: rgb(200,255,255);
font-family: sans-serif}
div.synopsisbox {
border: none; padding-top: 1em; padding-bottom: 1em; padding-left: 2em;
padding-right: 2em; background: rgb(255,220,255);}
pre.example {border: 1px solid rgb(180,180,180); padding-top: 1em;
padding-bottom: 1em; padding-left: 1em; padding-right: 1em;
background-color: rgb(238,238,255)}
div.spacerbox {border: none; padding-top: 2em; padding-bottom: 2em}
div.image {margin: 0; padding: 1em; text-align: center}
div.categorybox {border: 1px solid gray; padding-top: 1em; padding-bottom: 1em;
padding-left: 1em; padding-right: 1em; background: rgb(247,242,220)}
img {max-width:80%; max-height: 80%; display: block; margin-left: auto; margin-right: auto}
-->
</style>
<link rel="icon" href="figures/favicon.ico">
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6>"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Functions-and-Variables-for-hompack"></a>
<div class="header">
<p>
Previous: <a href="maxima_292.html#Introduction-to-hompack" accesskey="p" rel="previous">Introduction to hompack</a>, Up: <a href="maxima_291.html#hompack_002dpkg" accesskey="u" rel="up">hompack-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-hompack-1"></a>
<h3 class="section">65.2 Functions and Variables for hompack</h3>
<a name="hompack_005fpolsys"></a><a name="Item_003a-hompack_002fdeffn_002fhompack_005fpolsys"></a><dl>
<dt><a name="index-hompack_005fpolsys"></a>Function: <strong>hompack_polsys</strong> <em>(<var>eqnlist</var>, <var>varlist</var> [, <var>iflg1</var>, <var>epsbig</var>, <var>epssml</var>, <var>numrr</var>])</em></dt>
<dd><p>Finds the roots of the system of polynomials in the variables
<var>varlist</var> in the system of equations in <var>eqnlist</var>. The number
of equations must match number of variables. Each equation must be a
polynomial with variables in <var>varlist</var>. The coefficients must be
real numbers.
</p>
<p>The optional keyword arguments provide some control over the
algorithm.
</p>
<dl compact="compact">
<dt><code>epsbig</code></dt>
<dd><p>is the local error tolerance allowed by the
path tracker, defaulting to 1e-4.
</p></dd>
<dt><code>epssml</code></dt>
<dd><p>is the accuracy
desired for the final solution, defaulting to 1d-14.
</p></dd>
<dt><code>numrr</code></dt>
<dd><p>is the number of multiples of 1000 steps that will be tried
before abandoning a path, defaulting to 10.
</p></dd>
<dt><code>iflg1</code></dt>
<dd><p>defaulting to 0, controls the algorithm as follows:
</p><dl compact="compact">
<dt><code>0</code></dt>
<dd><p>If the problem is to be solved without calling <code>polsys</code>’ scaling
routine, <code>sclgnp</code>, and without using the projective
transformation.
</p></dd>
<dt><code>1</code></dt>
<dd><p>If scaling but no projective transformation is to be used.
</p></dd>
<dt><code>10</code></dt>
<dd><p>If no scaling but projective transformation is to be used.
</p></dd>
<dt><code>11</code></dt>
<dd><p>If both scaling and projective transformation are to be used.
</p></dd>
</dl>
</dd>
</dl>
<p><code>hompack_polsys</code> returns a list. The elements of the list are:
</p><dl compact="compact">
<dt><code>retcode</code></dt>
<dd><p>Indicates whether the solution is valid or not.
</p><dl compact="compact">
<dt><code>0</code></dt>
<dd><p>Solution found without problems
</p></dd>
<dt><code>1</code></dt>
<dd><p>Solution succeeded but <code>iflg2</code> indicates some issues with a
root. (That is, <code>iflg2</code> is not all ones.)
</p></dd>
<dt><code>-1</code></dt>
<dd><p><code>NN</code>, the declared dimension of the number of terms in the
polynomials, is too small. (This should not happen.)
</p></dd>
<dt><code>-2</code></dt>
<dd><p><code>MMAXT</code>, the declared dimension for the internal coefficient and
degree arrays, is too small. (This should not happen.)
</p></dd>
<dt><code>-3</code></dt>
<dd><p><code>TTOTDG</code>, the total degree of the equations, is too small.
(This should not happen.)
</p></dd>
<dt><code>-4</code></dt>
<dd><p><code>LENWK</code>, the length of the internal real work array, is too
small. (This should not happen.)
</p></dd>
<dt><code>-5</code></dt>
<dd><p><code>LENIWK</code>, the length of the internal integer work array, is too
small. (This should not happen.)
</p></dd>
<dt><code>-6</code></dt>
<dd><p><var>iflg1</var> is not 0 or 1, or 10 or 11. (This should not happen; an
error should be thrown before <code>polsys</code> is called.)
</p></dd>
</dl>
</dd>
<dt><code>roots</code></dt>
<dd><p>The roots of the system of equations. This is in the same format as
<code>solve</code> would return.
</p></dd>
<dt><code>iflg2</code></dt>
<dd><p>A list containing information on how the path for the m’th root terminated:
</p><dl compact="compact">
<dt><code>1</code></dt>
<dd><p>Normal return
</p></dd>
<dt><code>2</code></dt>
<dd><p>Specified error tolerance cannot be met. Increase <var>epsbig</var> and
<var>epssml</var> and rerun.
</p></dd>
<dt><code>3</code></dt>
<dd><p>Maximum number of steps exceeded. To track the path further, increase
<var>numrr</var> and rerun the path. However, the path may be diverging, if the
lambda value is near 1 and the roots values are large.
</p></dd>
<dt><code>4</code></dt>
<dd><p>Jacobian matrix does not have full rank. The algorithm has failed
(the zero curve of the homotopy map cannot be followed any further).
</p></dd>
<dt><code>5</code></dt>
<dd><p>The tracking algorithm has lost the zero curve of the homotopy map and
is not making progress. The error tolerances <var>epsbig</var> and
<var>epssml</var> were too lenient. The problem should be restarted with
smaller error tolerances.
</p></dd>
<dt><code>6</code></dt>
<dd><p>The normal flow newton iteration in <code>stepnf</code> or <code>rootnf</code>
failed to converge. The error tolerance <var>epsbig</var> may be too
stringent.
</p></dd>
<dt><code>7</code></dt>
<dd><p>Illegal input parameters, a fatal error.
</p></dd>
</dl>
</dd>
<dt><code>lambda</code></dt>
<dd><p>A list of the final lambda value for the m-th root, where lambda is the
continuation parameter.
</p></dd>
<dt><code>arclen</code></dt>
<dd><p>A list of the arc length of the m-th root.
</p></dd>
<dt><code>nfe</code></dt>
<dd><p>A list of the number of jacobian matrix evaluations required to track the m-th
root.
</p></dd>
</dl>
<p>Here are some examples of using <code>hompack_polsys</code>.
</p><div class="example">
<pre class="example">(%i1) load(hompack)$
(%i2) hompack_polsys([x1^2-1, x2^2-2],[x1,x2]);
(%o2) [0,
[[x1 = (-1.354505666901954e-16*%i)-0.9999999999999999,
x2 = 3.52147935979316e-16*%i-1.414213562373095],
[x1 = 1.0-5.536432658639868e-18*%i,
x2 = (-4.213674137126362e-17*%i)-1.414213562373095],
[x1 = (-9.475939894034927e-17*%i)-1.0,
x2 = 2.669654624736742e-16*%i+1.414213562373095],
[x1 = 9.921253413273088e-18*%i+1.0,
x2 = 1.414213562373095-5.305667769855424e-17*%i]],[1,1,1,1],
[1.0,1.0,0.9999999999999996,1.0],
[4.612623769341193,4.612623010859902,4.612623872939383,
4.612623114484402],[40,40,40,40]]
</pre></div>
<p>The analytical solution can be obtained with solve:
</p><div class="example">
<pre class="example">(%i1) solve([x1^2-1, x2^2-2],[x1,x2]);
(%o1) [[x1 = 1,x2 = -sqrt(2)],[x1 = 1,x2 = sqrt(2)],[x1 = -1,x2 = -sqrt(2)],
[x1 = -1,x2 = sqrt(2)]]
</pre></div>
<p>We see that <code>hompack_polsys</code> returned the correct answer except
that the roots are in a different order and there is a small imaginary
part.
</p>
<p>Another example, with corresponding solution from solve:
</p><div class="example">
<pre class="example">(%i1) hompack_polsys([x1^2 + 2*x2^2 + x1*x2 - 5, 2*x1^2 + x2^2 + x2-4],[x1,x2]);
(%o1) [0,
[[x1 = 1.201557301700783-1.004786320788336e-15*%i,
x2 = (-4.376615092392437e-16*%i)-1.667270363480143],
[x1 = 1.871959754090949e-16*%i-1.428529189565313,
x2 = (-6.301586314393093e-17*%i)-0.9106199083334113],
[x1 = 0.5920619420732697-1.942890293094024e-16*%i,
x2 = 6.938893903907228e-17*%i+1.383859154368197],
[x1 = 7.363503717463654e-17*%i+0.08945540033671608,
x2 = 1.557667481081721-4.109128293931921e-17*%i]],[1,1,1,1],
[1.000000000000001,1.0,1.0,1.0],
[6.205795654034752,7.722213259390295,7.228287079174351,
5.611474283583368],[35,41,48,40]]
(%i2) solve([x1^2+2*x2^2+x1*x2 - 5, 2*x1^2+x2^2+x2-4],[x1,x2]);
(%o2) [[x1 = 0.08945540336850383,x2 = 1.557667386609071],
[x1 = 0.5920619554695062,x2 = 1.383859286083807],
[x1 = 1.201557352500749,x2 = -1.66727025803531],
[x1 = -1.428529150636283,x2 = -0.9106198942815954]]
</pre></div>
<p>Note that <code>hompack_polsys</code> can sometimes be very slow. Perhaps
<code>solve</code> can be used. Or perhaps <code>eliminate</code> can be used to
convert the system of polynomials into one polynomial for which
<code>allroots</code> can find all the roots.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-hompack">Package hompack</a>
·</div>
</dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="maxima_292.html#Introduction-to-hompack" accesskey="p" rel="previous">Introduction to hompack</a>, Up: <a href="maxima_291.html#hompack_002dpkg" accesskey="u" rel="up">hompack-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
