1
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html401/loose.dtd">
3
<!-- Created on September, 20 2006 by texi2html 1.76 -->
5
Written by: Lionel Cons <Lionel.Cons@cern.ch> (original author)
6
Karl Berry <karl@freefriends.org>
7
Olaf Bachmann <obachman@mathematik.uni-kl.de>
9
Maintained by: Many creative people <dev@texi2html.cvshome.org>
10
Send bugs and suggestions to <users@texi2html.cvshome.org>
14
<title>Maxima Manual: 70. zeilberger</title>
16
<meta name="description" content="Maxima Manual: 70. zeilberger">
17
<meta name="keywords" content="Maxima Manual: 70. zeilberger">
18
<meta name="resource-type" content="document">
19
<meta name="distribution" content="global">
20
<meta name="Generator" content="texi2html 1.76">
21
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
22
<style type="text/css">
24
a.summary-letter {text-decoration: none}
25
pre.display {font-family: serif}
26
pre.format {font-family: serif}
27
pre.menu-comment {font-family: serif}
28
pre.menu-preformatted {font-family: serif}
29
pre.smalldisplay {font-family: serif; font-size: smaller}
30
pre.smallexample {font-size: smaller}
31
pre.smallformat {font-family: serif; font-size: smaller}
32
pre.smalllisp {font-size: smaller}
33
span.sansserif {font-family:sans-serif; font-weight:normal;}
34
ul.toc {list-style: none}
47
font-family: sans-serif
53
font-family: sans-serif
59
font-family: sans-serif
62
h2,h3,h4,h5,h6 { margin-left: +4%; }
82
background: rgb(200,255,255);
83
font-family: sans-serif
93
background: rgb(255,220,255);
94
/*background: rgb(200,255,255); */
95
/* font-family: fixed */
105
background: rgb(247,242,180); /* kind of sandy */
106
/* background: rgb(200,255,255); */ /* sky blue */
107
font-family: "Lucida Console", monospace
126
<link rel="icon" href="http://maxima.sourceforge.net/favicon.ico"/>
129
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
131
<a name="zeilberger"></a>
132
<a name="SEC255"></a>
133
<table cellpadding="1" cellspacing="1" border="0">
134
<tr><td valign="middle" align="left">[<a href="maxima_69.html#SEC254" title="Previous section in reading order"> < </a>]</td>
135
<td valign="middle" align="left">[<a href="#SEC256" title="Next section in reading order"> > </a>]</td>
136
<td valign="middle" align="left"> </td>
137
<td valign="middle" align="left">[<a href="maxima_69.html#SEC252" title="Beginning of this chapter or previous chapter"> << </a>]</td>
138
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Up section"> Up </a>]</td>
139
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
140
<td valign="middle" align="left"> </td>
141
<td valign="middle" align="left"> </td>
142
<td valign="middle" align="left"> </td>
143
<td valign="middle" align="left"> </td>
144
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
145
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
146
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
147
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
149
<h1 class="chapter"> 70. zeilberger </h1>
151
<table class="menu" border="0" cellspacing="0">
152
<tr><td align="left" valign="top"><a href="#SEC256">70.1 Introduction to zeilberger</a></td><td> </td><td align="left" valign="top">
154
<tr><td align="left" valign="top"><a href="#SEC260">70.2 Definitions for zeilberger</a></td><td> </td><td align="left" valign="top">
159
<a name="Introduction-to-zeilberger"></a>
160
<a name="SEC256"></a>
161
<table cellpadding="1" cellspacing="1" border="0">
162
<tr><td valign="middle" align="left">[<a href="#SEC255" title="Previous section in reading order"> < </a>]</td>
163
<td valign="middle" align="left">[<a href="#SEC257" title="Next section in reading order"> > </a>]</td>
164
<td valign="middle" align="left"> </td>
165
<td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
166
<td valign="middle" align="left">[<a href="#SEC255" title="Up section"> Up </a>]</td>
167
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
168
<td valign="middle" align="left"> </td>
169
<td valign="middle" align="left"> </td>
170
<td valign="middle" align="left"> </td>
171
<td valign="middle" align="left"> </td>
172
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
173
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
174
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
175
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
177
<h2 class="section"> 70.1 Introduction to zeilberger </h2>
179
<p><code>zeilberger</code> is a implementation of Zeilberger's algorithm
180
for definite hypergeometric summation, and also
181
Gosper's algorithm for indefinite hypergeometric
184
<p><code>zeilberger</code> makes use of the "filtering" optimization method developed by Axel Riese.
186
<p><code>zeilberger</code> was developed by Fabrizio Caruso.
188
<p><code>load (zeilberger)</code> loads this package.
191
<a name="SEC257"></a>
192
<table cellpadding="1" cellspacing="1" border="0">
193
<tr><td valign="middle" align="left">[<a href="#SEC256" title="Previous section in reading order"> < </a>]</td>
194
<td valign="middle" align="left">[<a href="#SEC258" title="Next section in reading order"> > </a>]</td>
195
<td valign="middle" align="left"> </td>
196
<td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
197
<td valign="middle" align="left">[<a href="#SEC256" title="Up section"> Up </a>]</td>
198
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
199
<td valign="middle" align="left"> </td>
200
<td valign="middle" align="left"> </td>
201
<td valign="middle" align="left"> </td>
202
<td valign="middle" align="left"> </td>
203
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
204
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
205
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
206
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
208
<h4 class="subsubsection"> 70.1.0.1 The indefinite summation problem </h4>
210
<p><code>zeilberger</code> implements Gosper's algorithm
211
for indefinite hypergeometric summation.
212
Given a hypergeometric term <em>F_k</em> in <em>k</em> we want to find its hypergeometric
213
anti-difference, that is, a hypergeometric term <em>f_k</em> such that <em>F_k = f_(k+1) - f_k</em>.
216
<a name="SEC258"></a>
217
<table cellpadding="1" cellspacing="1" border="0">
218
<tr><td valign="middle" align="left">[<a href="#SEC257" title="Previous section in reading order"> < </a>]</td>
219
<td valign="middle" align="left">[<a href="#SEC259" title="Next section in reading order"> > </a>]</td>
220
<td valign="middle" align="left"> </td>
221
<td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
222
<td valign="middle" align="left">[<a href="#SEC256" title="Up section"> Up </a>]</td>
223
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
224
<td valign="middle" align="left"> </td>
225
<td valign="middle" align="left"> </td>
226
<td valign="middle" align="left"> </td>
227
<td valign="middle" align="left"> </td>
228
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
229
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
230
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
231
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
233
<h4 class="subsubsection"> 70.1.0.2 The definite summation problem </h4>
235
<p><code>zeilberger</code> implements Zeilberger's algorithm
236
for definite hypergeometric summation.
237
Given a proper hypergeometric term (in <em>n</em> and <em>k</em>) <em>F_(n,k)</em> and a
238
positive integer <em>d</em> we want to find a <em>d</em>-th order linear
239
recurrence with polynomial coefficients (in <em>n</em>) for <em>F_(n,k)</em>
240
and a rational function <em>R</em> in <em>n</em> and <em>k</em> such that
242
<p><em>a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_K(R(n,k) F_(n,k))</em>
244
<p>where <em>Delta_k</em> is the <em>k</em>-forward difference operator, i.e.,
245
<em>Delta_k(t_k) := t_(k+1) - t_k</em>.
248
<a name="SEC259"></a>
249
<table cellpadding="1" cellspacing="1" border="0">
250
<tr><td valign="middle" align="left">[<a href="#SEC258" title="Previous section in reading order"> < </a>]</td>
251
<td valign="middle" align="left">[<a href="#SEC260" title="Next section in reading order"> > </a>]</td>
252
<td valign="middle" align="left"> </td>
253
<td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
254
<td valign="middle" align="left">[<a href="#SEC256" title="Up section"> Up </a>]</td>
255
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
256
<td valign="middle" align="left"> </td>
257
<td valign="middle" align="left"> </td>
258
<td valign="middle" align="left"> </td>
259
<td valign="middle" align="left"> </td>
260
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
261
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
262
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
263
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
265
<h3 class="subsection"> 70.1.1 Verbosity levels </h3>
267
<p>There are also verbose versions of the commands
268
which are called by adding one of the following prefixes:
270
<dl compact="compact">
271
<dt> <code>Summary</code></dt>
272
<dd><p>Just a summary at the end is shown
274
<dt> <code>Verbose</code></dt>
275
<dd><p>Some information in the intermidiate steps
277
<dt> <code>VeryVerbose</code></dt>
278
<dd><p>More information
280
<dt> <code>Extra</code></dt>
281
<dd><p>Even more information including information on
282
the linear system in Zeilberger's algorithm
287
<code>GosperVerbose</code>, <code>parGosperVeryVerbose</code>,
288
<code>ZeilbergerExtra</code>, <code>AntiDifferenceSummary</code>.
292
<a name="Definitions-for-zeilberger"></a>
293
<a name="SEC260"></a>
294
<table cellpadding="1" cellspacing="1" border="0">
295
<tr><td valign="middle" align="left">[<a href="#SEC259" title="Previous section in reading order"> < </a>]</td>
296
<td valign="middle" align="left">[<a href="#SEC261" title="Next section in reading order"> > </a>]</td>
297
<td valign="middle" align="left"> </td>
298
<td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
299
<td valign="middle" align="left">[<a href="#SEC255" title="Up section"> Up </a>]</td>
300
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
301
<td valign="middle" align="left"> </td>
302
<td valign="middle" align="left"> </td>
303
<td valign="middle" align="left"> </td>
304
<td valign="middle" align="left"> </td>
305
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
306
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
307
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
308
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
310
<h2 class="section"> 70.2 Definitions for zeilberger </h2>
313
<dt><u>Function:</u> <b>AntiDifference</b><i> (<var>F_k</var>, <var>k</var>)</i>
314
<a name="IDX1980"></a>
316
<dd><p>Returns the hypergeometric anti-difference
317
of <var>F_k</var>, if it exists.
318
Otherwise <code>AntiDifference</code> returns <code>no_hyp_antidifference</code>.
322
<dt><u>Function:</u> <b>Gosper</b><i> (<var>F_k</var>, <var>k</var>)</i>
323
<a name="IDX1981"></a>
325
<dd><p>Returns the rational certificate <var>R(k)</var> for <var>F_k</var>, that is,
326
a rational function such that
328
<p><em>F_k = R(k+1) F_(k+1) - R(k) F_k</em>
331
Otherwise, <code>Gosper</code> returns <code>no_hyp_sol</code>.
335
<dt><u>Function:</u> <b>GosperSum</b><i> (<var>F_k</var>, <var>k</var>, <var>a</var>, <var>b</var>) </i>
336
<a name="IDX1982"></a>
338
<dd><p>Returns the summmation of <var>F_k</var> from <em><var>k</var> = <var>a</var></em> to <em><var>k</var> = <var>b</var></em>
339
if <var>F_k</var> has a hypergeometric anti-difference.
340
Otherwise, <code>GosperSum</code> returns <code>nongosper_summable</code>.
344
<table><tr><td> </td><td><pre class="example">(%i1) load (zeilberger);
345
(%o1) /usr/share/maxima/share/contrib/Zeilberger/zeilberger.mac
346
(%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
348
Dependent equations eliminated: (1)
352
(%o2) - ------------------ - -
355
(%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
359
(%o3) -------------- + -
362
(%i4) GosperSum (x^k, k, 1, n);
367
(%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
370
(%o5) - ------------------------- - ----------
371
a (- n + a - 1)! (n + 1)! a (a - 1)!
372
(%i6) GosperSum (k*k!, k, 1, n);
374
Dependent equations eliminated: (1)
376
(%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
377
(n + 1) (n + 2) (n + 1)!
378
(%o7) ------------------------ - 1
380
(%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
381
(%o8) nonGosper_summable
382
</pre></td></tr></table></dd></dl>
385
<dt><u>Function:</u> <b>parGosper</b><i> (<var>F_{n,k}</var>, <var>k</var>, <var>n</var>, <var>d</var>)</i>
386
<a name="IDX1983"></a>
388
<dd><p>Attempts to find a a <var>d</var>-th order recurrence for <var>F_{n,k}</var>.
390
<p>The algorithm yields a sequence
391
<em>[s_1, s_2, ..., s_m]</em> of solutions.
392
Each solution has the form
394
<p><em>[R(n, k), [a_0, a_1, ..., a_d]]</em>
396
<p><code>parGosper</code> returns <code>[]</code> if it fails to find a recurrence.
400
<dt><u>Function:</u> <b>Zeilberger</b><i> (<var>F_{n,k}</var>, <var>k</var>, <var>n</var>)</i>
401
<a name="IDX1984"></a>
403
<dd><p>Attempts to compute the indefinite hypergeometric summation of <var>F_{n,k}</var>.
405
<p><code>Zeilberger</code> first invokes <code>Gosper</code>, and if that fails to find a solution, then invokes
406
<code>parGosper</code> with order 1, 2, 3, ..., up to <code>MAX_ORD</code>.
407
If Zeilberger finds a solution before reaching <code>MAX_ORD</code>,
408
it stops and returns the solution.
410
<p>The algorithms yields a sequence
411
<em>[s_1, s_2, ..., s_m]</em> of solutions.
412
Each solution has the form
414
<p><em>[R(n,k), [a_0, a_1, ..., a_d]]</em>
416
<p><code>Zeilberger</code> returns <code>[]</code> if it fails to find a solution.
418
<p><code>Zeilberger</code> invokes <code>Gosper</code> only if <code>gosper_in_zeilberger</code> is <code>true</code>.
422
<a name="SEC261"></a>
423
<table cellpadding="1" cellspacing="1" border="0">
424
<tr><td valign="middle" align="left">[<a href="#SEC260" title="Previous section in reading order"> < </a>]</td>
425
<td valign="middle" align="left">[<a href="#SEC262" title="Next section in reading order"> > </a>]</td>
426
<td valign="middle" align="left"> </td>
427
<td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
428
<td valign="middle" align="left">[<a href="#SEC255" title="Up section"> Up </a>]</td>
429
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
430
<td valign="middle" align="left"> </td>
431
<td valign="middle" align="left"> </td>
432
<td valign="middle" align="left"> </td>
433
<td valign="middle" align="left"> </td>
434
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
435
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
436
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
437
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
439
<h2 class="section"> 70.3 General global variables </h2>
442
<dt><u>Global variable:</u> <b>MAX_ORD</b>
443
<a name="IDX1985"></a>
445
<dd><p>Default value: 5
447
<p><code>MAX_ORD</code> is the maximum recurrence order attempted by <code>Zeilberger</code>.
451
<dt><u>Global variable:</u> <b>simplified_output</b>
452
<a name="IDX1986"></a>
454
<dd><p>Default value: <code>false</code>
456
<p>When <code>simplified_output</code> is <code>true</code>,
457
functions in the <code>zeilberger</code> package attempt
458
further simplification of the solution.
462
<dt><u>Global variable:</u> <b>linear_solver</b>
463
<a name="IDX1987"></a>
465
<dd><p>Default value: <code>linsolve</code>
467
<p><code>linear_solver</code> names the solver which is used to solve the system
468
of equations in Zeilberger's algorithm.
472
<dt><u>Global variable:</u> <b>warnings</b>
473
<a name="IDX1988"></a>
475
<dd><p>Default value: <code>true</code>
477
<p>When <code>warnings</code> is <code>true</code>,
478
functions in the <code>zeilberger</code> package print
479
warning messages during execution.
483
<dt><u>Global variable:</u> <b>gosper_in_zeilberger</b>
484
<a name="IDX1989"></a>
486
<dd><p>Default value: <code>true</code>
488
<p>When <code>gosper_in_zeilberger</code> is <code>true</code>,
489
the <code>Zeilberger</code> function calls <code>Gosper</code> before calling <code>parGosper</code>.
490
Otherwise, <code>Zeilberger</code> goes immediately to <code>parGosper</code>.
494
<dt><u>Global variable:</u> <b>trivial_solutions</b>
495
<a name="IDX1990"></a>
497
<dd><p>Default value: <code>true</code>
499
<p>When <code>trivial_solutions</code> is <code>true</code>,
500
<code>Zeilberger</code> returns solutions
501
which have certificate equal to zero, or all coefficients equal to zero.
505
<a name="SEC262"></a>
506
<table cellpadding="1" cellspacing="1" border="0">
507
<tr><td valign="middle" align="left">[<a href="#SEC261" title="Previous section in reading order"> < </a>]</td>
508
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next section in reading order"> > </a>]</td>
509
<td valign="middle" align="left"> </td>
510
<td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
511
<td valign="middle" align="left">[<a href="#SEC255" title="Up section"> Up </a>]</td>
512
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
513
<td valign="middle" align="left"> </td>
514
<td valign="middle" align="left"> </td>
515
<td valign="middle" align="left"> </td>
516
<td valign="middle" align="left"> </td>
517
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
518
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
519
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
520
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
522
<h2 class="section"> 70.4 Variables related to the modular test </h2>
525
<dt><u>Global variable:</u> <b>mod_test</b>
526
<a name="IDX1991"></a>
528
<dd><p>Default value: <code>false</code>
530
<p>When <code>mod_test</code> is <code>true</code>,
531
<code>parGosper</code> executes a
532
modular test for discarding systems with no solutions.
536
<dt><u>Global variable:</u> <b>modular_linear_solver</b>
537
<a name="IDX1992"></a>
539
<dd><p>Default value: <code>linsolve</code>
541
<p><code>modular_linear_solver</code> names the linear solver used by the modular test in <code>parGosper</code>.
545
<dt><u>Global variable:</u> <b>ev_point</b>
546
<a name="IDX1993"></a>
548
<dd><p>Default value: <code>big_primes[10]</code>
550
<p><code>ev_point</code> is the value at which the variable <var>n</var> is evaluated
551
when executing the modular test in <code>parGosper</code>.
555
<dt><u>Global variable:</u> <b>mod_big_prime</b>
556
<a name="IDX1994"></a>
558
<dd><p>Default value: <code>big_primes[1]</code>
560
<p><code>mod_big_prime</code> is the modulus used by the modular test in <code>parGosper</code>.
564
<dt><u>Global variable:</u> <b>mod_threshold</b>
565
<a name="IDX1995"></a>
567
<dd><p>Default value: 4
569
<p><code>mod_threshold</code> is the
570
greatest order for which the modular test in <code>parGosper</code> is attempted.
575
<table cellpadding="1" cellspacing="1" border="0">
576
<tr><td valign="middle" align="left">[<a href="#SEC255" title="Beginning of this chapter or previous chapter"> << </a>]</td>
577
<td valign="middle" align="left">[<a href="maxima_71.html#SEC263" title="Next chapter"> >> </a>]</td>
578
<td valign="middle" align="left"> </td>
579
<td valign="middle" align="left"> </td>
580
<td valign="middle" align="left"> </td>
581
<td valign="middle" align="left"> </td>
582
<td valign="middle" align="left"> </td>
583
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
584
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
585
<td valign="middle" align="left">[<a href="maxima_72.html#SEC264" title="Index">Index</a>]</td>
586
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
590
This document was generated by <em>Robert Dodier</em> on <em>September, 20 2006</em> using <a href="http://texi2html.cvshome.org/"><em>texi2html 1.76</em></a>.
added
removed
doc/info/maxima_70.html
