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1<!--Converted with LaTeX2HTML 97.1 (release) (July 13th, 1997)
2 by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
3* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
4* with significant contributions from:
5 Jens Lippman, Marek Rouchal, Martin Wilck and others -->
6<HTML>
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8<TITLE>The Dynamical Systems package</TITLE>
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34<BR>
35<B> Next:</B> <A NAME="tex2html947"
36 HREF="node8.html">The Geometric Theorem Prover</A>
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38 HREF="manual.html">CGBLisp User Guide and</A>
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40 HREF="node6.html">The Division Package</A>
41<BR>
42<BR>
43<!--End of Navigation Panel-->
44<!--Table of Child-Links-->
45<A NAME="CHILD_LINKS"><strong>Subsections</strong></A>
46<UL>
47<LI><A NAME="tex2html948"
48 HREF="node7.html#SECTION00070010000000000000">
49<I>poly<MATH CLASS="INLINE">
50-
51</MATH>scalar<MATH CLASS="INLINE">
52-
53</MATH>composition</I></A>
54<LI><A NAME="tex2html949"
55 HREF="node7.html#SECTION00070020000000000000">
56<I>poly<MATH CLASS="INLINE">
57-
58</MATH>composition</I></A>
59<LI><A NAME="tex2html950"
60 HREF="node7.html#SECTION00070030000000000000">
61<I>poly<MATH CLASS="INLINE">
62-
63</MATH>dynamic<MATH CLASS="INLINE">
64-
65</MATH>power</I></A>
66<LI><A NAME="tex2html951"
67 HREF="node7.html#SECTION00070040000000000000">
68<I>poly<MATH CLASS="INLINE">
69-
70</MATH>scalar<MATH CLASS="INLINE">
71-
72</MATH>evaluate</I></A>
73<LI><A NAME="tex2html952"
74 HREF="node7.html#SECTION00070050000000000000">
75<I>poly<MATH CLASS="INLINE">
76-
77</MATH>evaluate</I></A>
78<LI><A NAME="tex2html953"
79 HREF="node7.html#SECTION00070060000000000000">
80<I>factorial</I></A>
81<LI><A NAME="tex2html954"
82 HREF="node7.html#SECTION00070070000000000000">
83<I>poly<MATH CLASS="INLINE">
84-
85</MATH>scalar<MATH CLASS="INLINE">
86-
87</MATH>diff</I></A>
88<LI><A NAME="tex2html955"
89 HREF="node7.html#SECTION00070080000000000000">
90<I>poly<MATH CLASS="INLINE">
91-
92</MATH>diff</I></A>
93<LI><A NAME="tex2html956"
94 HREF="node7.html#SECTION00070090000000000000">
95<I>standard<MATH CLASS="INLINE">
96-
97</MATH>vector</I></A>
98<LI><A NAME="tex2html957"
99 HREF="node7.html#SECTION000700100000000000000">
100<I>scalar<MATH CLASS="INLINE">
101-
102</MATH>partial</I></A>
103<LI><A NAME="tex2html958"
104 HREF="node7.html#SECTION000700110000000000000">
105<I>partial</I></A>
106<LI><A NAME="tex2html959"
107 HREF="node7.html#SECTION000700120000000000000">
108<I>determinant</I></A>
109<LI><A NAME="tex2html960"
110 HREF="node7.html#SECTION000700130000000000000">
111<I>minor</I></A>
112<LI><A NAME="tex2html961"
113 HREF="node7.html#SECTION000700140000000000000">
114<I>drop<MATH CLASS="INLINE">
115-
116</MATH>row</I></A>
117<LI><A NAME="tex2html962"
118 HREF="node7.html#SECTION000700150000000000000">
119<I>drop<MATH CLASS="INLINE">
120-
121</MATH>column</I></A>
122<LI><A NAME="tex2html963"
123 HREF="node7.html#SECTION000700160000000000000">
124<I>drop<MATH CLASS="INLINE">
125-
126</MATH>elt</I></A>
127<LI><A NAME="tex2html964"
128 HREF="node7.html#SECTION000700170000000000000">
129<I>matrix<MATH CLASS="INLINE">
130-
131</MATH></I></A>
132<LI><A NAME="tex2html965"
133 HREF="node7.html#SECTION000700180000000000000">
134<I>scalar<MATH CLASS="INLINE">
135-
136</MATH>times<MATH CLASS="INLINE">
137-
138</MATH>matrix</I></A>
139<LI><A NAME="tex2html966"
140 HREF="node7.html#SECTION000700190000000000000">
141<I>monom<MATH CLASS="INLINE">
142-
143</MATH>times<MATH CLASS="INLINE">
144-
145</MATH>matrix</I></A>
146<LI><A NAME="tex2html967"
147 HREF="node7.html#SECTION000700200000000000000">
148<I>term<MATH CLASS="INLINE">
149-
150</MATH>times<MATH CLASS="INLINE">
151-
152</MATH>matrix</I></A>
153<LI><A NAME="tex2html968"
154 HREF="node7.html#SECTION000700210000000000000">
155<I>poly<MATH CLASS="INLINE">
156-
157</MATH>list<MATH CLASS="INLINE">
158-
159</MATH></I></A>
160<LI><A NAME="tex2html969"
161 HREF="node7.html#SECTION000700220000000000000">
162<I>scalar<MATH CLASS="INLINE">
163-
164</MATH>times<MATH CLASS="INLINE">
165-
166</MATH>poly<MATH CLASS="INLINE">
167-
168</MATH>list</I></A>
169<LI><A NAME="tex2html970"
170 HREF="node7.html#SECTION000700230000000000000">
171<I>monom<MATH CLASS="INLINE">
172-
173</MATH>times<MATH CLASS="INLINE">
174-
175</MATH>poly<MATH CLASS="INLINE">
176-
177</MATH>list</I></A>
178<LI><A NAME="tex2html971"
179 HREF="node7.html#SECTION000700240000000000000">
180<I>term<MATH CLASS="INLINE">
181-
182</MATH>times<MATH CLASS="INLINE">
183-
184</MATH>poly<MATH CLASS="INLINE">
185-
186</MATH>list</I></A>
187<LI><A NAME="tex2html972"
188 HREF="node7.html#SECTION000700250000000000000">
189<I>characteristic<MATH CLASS="INLINE">
190-
191</MATH>combination</I></A>
192<LI><A NAME="tex2html973"
193 HREF="node7.html#SECTION000700260000000000000">
194<I>characteristic<MATH CLASS="INLINE">
195-
196</MATH>combination<MATH CLASS="INLINE">
197-
198</MATH>poly<MATH CLASS="INLINE">
199-
200</MATH>list</I></A>
201<LI><A NAME="tex2html974"
202 HREF="node7.html#SECTION000700270000000000000">
203<I>characteristic<MATH CLASS="INLINE">
204-
205</MATH>matrix</I></A>
206<LI><A NAME="tex2html975"
207 HREF="node7.html#SECTION000700280000000000000">
208<I>characteristic<MATH CLASS="INLINE">
209-
210</MATH>polynomial</I></A>
211<LI><A NAME="tex2html976"
212 HREF="node7.html#SECTION000700290000000000000">
213<I>identity<MATH CLASS="INLINE">
214-
215</MATH>matrix</I></A>
216<LI><A NAME="tex2html977"
217 HREF="node7.html#SECTION000700300000000000000">
218<I>print<MATH CLASS="INLINE">
219-
220</MATH>matrix</I></A>
221<LI><A NAME="tex2html978"
222 HREF="node7.html#SECTION000700310000000000000">
223<I>jacobi<MATH CLASS="INLINE">
224-
225</MATH>matrix</I></A>
226<LI><A NAME="tex2html979"
227 HREF="node7.html#SECTION000700320000000000000">
228<I>jacobian</I></A>
229</UL>
230<!--End of Table of Child-Links-->
231<HR>
232<H1><A NAME="SECTION00070000000000000000">
233The Dynamical Systems package</A>
234</H1>
235<H4><A NAME="SECTION00070010000000000000">
236<I>poly<MATH CLASS="INLINE">
237-
238</MATH>scalar<MATH CLASS="INLINE">
239-
240</MATH>composition</I></A>
241</H4>
242<P><IMG WIDTH="435" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
243 SRC="img116.gif"
244 ALT="$\textstyle\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
245<BLOCKQUOTE>
246Returns a polynomial obtained by substituting a list of polynomials
247G=(G1,G2,...,GN) into a polynomial F(X1,X2,...,XN). All polynomials
248are assumed to be in the internal form, so variables do not
249explicitly apprear in the calculation. </BLOCKQUOTE><H4><A NAME="SECTION00070020000000000000">
250<I>poly<MATH CLASS="INLINE">
251-
252</MATH>composition</I></A>
253</H4>
254<P><IMG WIDTH="435" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
255 SRC="img116.gif"
256 ALT="$\textstyle\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
257<BLOCKQUOTE>
258Return the composition of a polynomial map F with a polynomial map
259G. Both maps are represented as lists of polynomials, and each
260polynomial is in the internal alist representation. The restriction
261is that the length of the list G must be the number of variables in
262the list F. </BLOCKQUOTE><H4><A NAME="SECTION00070030000000000000">
263<I>poly<MATH CLASS="INLINE">
264-
265</MATH>dynamic<MATH CLASS="INLINE">
266-
267</MATH>power</I></A>
268</H4>
269<P><IMG WIDTH="459" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
270 SRC="img117.gif"
271 ALT="$\textstyle\parbox{\pboxargslen}{\em f n {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
272<BLOCKQUOTE>
273Calculate the composition FoFo...oF (n times), where
274F is a polynomial map represented as a list of polynomials.</BLOCKQUOTE><H4><A NAME="SECTION00070040000000000000">
275<I>poly<MATH CLASS="INLINE">
276-
277</MATH>scalar<MATH CLASS="INLINE">
278-
279</MATH>evaluate</I></A>
280</H4>
281<P><IMG WIDTH="461" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
282 SRC="img118.gif"
283 ALT="$\textstyle\parbox{\pboxargslen}{\em f x {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
284<BLOCKQUOTE>
285Evaluate a polynomial F at a point X. This operation is implemented
286through POLY<MATH CLASS="INLINE">
287-
288</MATH>SCALAR<MATH CLASS="INLINE">
289-
290</MATH>COMPOSITION.</BLOCKQUOTE><H4><A NAME="SECTION00070050000000000000">
291<I>poly<MATH CLASS="INLINE">
292-
293</MATH>evaluate</I></A>
294</H4>
295<P><IMG WIDTH="461" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
296 SRC="img118.gif"
297 ALT="$\textstyle\parbox{\pboxargslen}{\em f x {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
298<BLOCKQUOTE>
299Evaluate a polynomial map F, represented as list of polynomials, at a
300point X. </BLOCKQUOTE><H4><A NAME="SECTION00070060000000000000">
301<I>factorial</I></A>
302</H4>
303<P><IMG WIDTH="556" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
304 SRC="img119.gif"
305 ALT="$\textstyle\parbox{\pboxargslen}{\em n {\sf \&optional} (k n) {\sf \&aux} (result 1) \/}$"> [<EM>FUNCTION</EM>]
306<BLOCKQUOTE>
307Return N!/(N<MATH CLASS="INLINE">
308-
309</MATH>K)!=N(N<MATH CLASS="INLINE">
310-
311</MATH>1)(N<MATH CLASS="INLINE">
312-
313</MATH>K+1).</BLOCKQUOTE><H4><A NAME="SECTION00070070000000000000">
314<I>poly<MATH CLASS="INLINE">
315-
316</MATH>scalar<MATH CLASS="INLINE">
317-
318</MATH>diff</I></A>
319</H4>
320<P><IMG WIDTH="496" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
321 SRC="img120.gif"
322 ALT="$\textstyle\parbox{\pboxargslen}{\em f m \/}$"> [<EM>FUNCTION</EM>]
323<BLOCKQUOTE>
324Return the partial derivative of a polynomial F over multiple
325 variables according to multiindex M.</BLOCKQUOTE><H4><A NAME="SECTION00070080000000000000">
326<I>poly<MATH CLASS="INLINE">
327-
328</MATH>diff</I></A>
329</H4>
330<P><IMG WIDTH="496" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
331 SRC="img120.gif"
332 ALT="$\textstyle\parbox{\pboxargslen}{\em f m \/}$"> [<EM>FUNCTION</EM>]
333<BLOCKQUOTE>
334Return the partial derivative of a polynomial map F, represented as a
335list of polynomials, with respect to several variables, according to
336multi<MATH CLASS="INLINE">
337-
338</MATH>index M. </BLOCKQUOTE><H4><A NAME="SECTION00070090000000000000">
339<I>standard<MATH CLASS="INLINE">
340-
341</MATH>vector</I></A>
342</H4>
343<P><IMG WIDTH="497" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
344 SRC="img121.gif"
345 ALT="$\textstyle\parbox{\pboxargslen}{\em n k {\sf \&optional} (coeff
346 1) {\sf \&aux} (v
347 (make$-$list
348 n
349 :initial$-$element
350 0)) \/}$"> [<EM>FUNCTION</EM>]
351<BLOCKQUOTE>
352Returns vector (0 0 ... 1 ... 0 0) of length N, where 1 appears on
353K<MATH CLASS="INLINE">
354-
355</MATH>th place. </BLOCKQUOTE><H4><A NAME="SECTION000700100000000000000">
356<I>scalar<MATH CLASS="INLINE">
357-
358</MATH>partial</I></A>
359</H4>
360<P><IMG WIDTH="516" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
361 SRC="img122.gif"
362 ALT="$\textstyle\parbox{\pboxargslen}{\em f k {\sf \&optional} (l 1) \/}$"> [<EM>FUNCTION</EM>]
363<BLOCKQUOTE>
364Returns the L<MATH CLASS="INLINE">
365-
366</MATH>th partial derivative of a polynomial F over the
367K<MATH CLASS="INLINE">
368-
369</MATH>th variable. </BLOCKQUOTE><H4><A NAME="SECTION000700110000000000000">
370<I>partial</I></A>
371</H4>
372<P><IMG WIDTH="516" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
373 SRC="img122.gif"
374 ALT="$\textstyle\parbox{\pboxargslen}{\em f k {\sf \&optional} (l 1) \/}$"> [<EM>FUNCTION</EM>]
375<BLOCKQUOTE>
376Returns the L<MATH CLASS="INLINE">
377-
378</MATH>th partial derivative over the K<MATH CLASS="INLINE">
379-
380</MATH>th variable, of a
381polynomial map F, represented as a list of polynomials.</BLOCKQUOTE><H4><A NAME="SECTION000700120000000000000">
382<I>determinant</I></A>
383</H4>
384<P><IMG WIDTH="528" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
385 SRC="img123.gif"
386 ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (order \char93 'lex$\gt$) {\sf \&aux} (result nil) \/}$"> [<EM>FUNCTION</EM>]
387<BLOCKQUOTE>
388Returns the determinant of a polynomial matrix F, which is a list of
389rows of the matrix, each row is a list of polynomials. The algorithm
390is recursive expansion along columns.</BLOCKQUOTE><H4><A NAME="SECTION000700130000000000000">
391<I>minor</I></A>
392</H4>
393<P><IMG WIDTH="572" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
394 SRC="img124.gif"
395 ALT="$\textstyle\parbox{\pboxargslen}{\em f i j {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
396<BLOCKQUOTE>
397Calculate the minor of a polynomial matrix F with respect to entry
398(I,J). </BLOCKQUOTE><H4><A NAME="SECTION000700140000000000000">
399<I>drop<MATH CLASS="INLINE">
400-
401</MATH>row</I></A>
402</H4>
403<P><IMG WIDTH="543" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
404 SRC="img125.gif"
405 ALT="$\textstyle\parbox{\pboxargslen}{\em f i \/}$"> [<EM>FUNCTION</EM>]
406<BLOCKQUOTE>
407Discards the I<MATH CLASS="INLINE">
408-
409</MATH>th row from a polynomial matrix F.</BLOCKQUOTE><H4><A NAME="SECTION000700150000000000000">
410<I>drop<MATH CLASS="INLINE">
411-
412</MATH>column</I></A>
413</H4>
414<P><IMG WIDTH="517" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
415 SRC="img126.gif"
416 ALT="$\textstyle\parbox{\pboxargslen}{\em f j \/}$"> [<EM>FUNCTION</EM>]
417<BLOCKQUOTE>
418Discards the J<MATH CLASS="INLINE">
419-
420</MATH>th column from a polynomial matrix F.</BLOCKQUOTE><H4><A NAME="SECTION000700160000000000000">
421<I>drop<MATH CLASS="INLINE">
422-
423</MATH>elt</I></A>
424</H4>
425<P><IMG WIDTH="550" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
426 SRC="img127.gif"
427 ALT="$\textstyle\parbox{\pboxargslen}{\em lst j \/}$"> [<EM>FUNCTION</EM>]
428<BLOCKQUOTE>
429Discards the J<MATH CLASS="INLINE">
430-
431</MATH>th element from a list LST.</BLOCKQUOTE><H4><A NAME="SECTION000700170000000000000">
432<I>matrix<MATH CLASS="INLINE">
433-
434</MATH></I></A>
435</H4>
436<P><IMG WIDTH="435" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
437 SRC="img116.gif"
438 ALT="$\textstyle\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
439<BLOCKQUOTE>
440Returns difference of two polynomial matrices F and G.</BLOCKQUOTE><H4><A NAME="SECTION000700180000000000000">
441<I>scalar<MATH CLASS="INLINE">
442-
443</MATH>times<MATH CLASS="INLINE">
444-
445</MATH>matrix</I></A>
446</H4>
447<P><IMG WIDTH="466" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
448 SRC="img128.gif"
449 ALT="$\textstyle\parbox{\pboxargslen}{\em s f \/}$"> [<EM>FUNCTION</EM>]
450<BLOCKQUOTE>
451Returns a product of a polynomial S by a polynomial matrix F.</BLOCKQUOTE><H4><A NAME="SECTION000700190000000000000">
452<I>monom<MATH CLASS="INLINE">
453-
454</MATH>times<MATH CLASS="INLINE">
455-
456</MATH>matrix</I></A>
457</H4>
458<P><IMG WIDTH="453" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
459 SRC="img129.gif"
460 ALT="$\textstyle\parbox{\pboxargslen}{\em m f \/}$"> [<EM>FUNCTION</EM>]
461<BLOCKQUOTE>
462Returns a product of a monomial M by a polynomial matrix F.</BLOCKQUOTE><H4><A NAME="SECTION000700200000000000000">
463<I>term<MATH CLASS="INLINE">
464-
465</MATH>times<MATH CLASS="INLINE">
466-
467</MATH>matrix</I></A>
468</H4>
469<P><IMG WIDTH="472" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
470 SRC="img130.gif"
471 ALT="$\textstyle\parbox{\pboxargslen}{\em term f \/}$"> [<EM>FUNCTION</EM>]
472<BLOCKQUOTE>
473Returns a product of a term TERM by a polynomial matrix F.</BLOCKQUOTE><H4><A NAME="SECTION000700210000000000000">
474<I>poly<MATH CLASS="INLINE">
475-
476</MATH>list<MATH CLASS="INLINE">
477-
478</MATH></I></A>
479</H4>
480<P><IMG WIDTH="435" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
481 SRC="img116.gif"
482 ALT="$\textstyle\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
483<BLOCKQUOTE>
484Returns the list of differences of two lists of polynomials
485F and G (polynomial maps).</BLOCKQUOTE><H4><A NAME="SECTION000700220000000000000">
486<I>scalar<MATH CLASS="INLINE">
487-
488</MATH>times<MATH CLASS="INLINE">
489-
490</MATH>poly<MATH CLASS="INLINE">
491-
492</MATH>list</I></A>
493</H4>
494<P><IMG WIDTH="466" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
495 SRC="img128.gif"
496 ALT="$\textstyle\parbox{\pboxargslen}{\em s f \/}$"> [<EM>FUNCTION</EM>]
497<BLOCKQUOTE>
498Returns the list of products of a polynomial S by the
499list of polynomials F.</BLOCKQUOTE><H4><A NAME="SECTION000700230000000000000">
500<I>monom<MATH CLASS="INLINE">
501-
502</MATH>times<MATH CLASS="INLINE">
503-
504</MATH>poly<MATH CLASS="INLINE">
505-
506</MATH>list</I></A>
507</H4>
508<P><IMG WIDTH="453" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
509 SRC="img129.gif"
510 ALT="$\textstyle\parbox{\pboxargslen}{\em m f \/}$"> [<EM>FUNCTION</EM>]
511<BLOCKQUOTE>
512Returns the list of products of a monomial M by the
513list of polynomials F.</BLOCKQUOTE><H4><A NAME="SECTION000700240000000000000">
514<I>term<MATH CLASS="INLINE">
515-
516</MATH>times<MATH CLASS="INLINE">
517-
518</MATH>poly<MATH CLASS="INLINE">
519-
520</MATH>list</I></A>
521</H4>
522<P><IMG WIDTH="472" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
523 SRC="img130.gif"
524 ALT="$\textstyle\parbox{\pboxargslen}{\em term f \/}$"> [<EM>FUNCTION</EM>]
525<BLOCKQUOTE>
526Returns the list of products of a term TERM by the
527list of polynomials F.</BLOCKQUOTE><H4><A NAME="SECTION000700250000000000000">
528<I>characteristic<MATH CLASS="INLINE">
529-
530</MATH>combination</I></A>
531</H4>
532<P><IMG WIDTH="423" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
533 SRC="img131.gif"
534 ALT="$\textstyle\parbox{\pboxargslen}{\em a b {\sf \&optional} (order
535 \char93 'lex$\gt$) {\sf \&aux} (n
536 (length
537 b)) \/}$"> [<EM>FUNCTION</EM>]
538<BLOCKQUOTE>
539Returns A <MATH CLASS="INLINE">
540-
541</MATH> U1 * B1 <MATH CLASS="INLINE">
542-
543</MATH> U2 * B2 <MATH CLASS="INLINE">
544-
545</MATH> ... <MATH CLASS="INLINE">
546-
547</MATH> UM * BM where A is a
548polynomial and B=(B1,B2,...,BM) is a polynomial list, where U1, U2,
549... , UM are new variables. These variables will be added to every
550polynomial A and BI as the last M variables.</BLOCKQUOTE><H4><A NAME="SECTION000700260000000000000">
551<I>characteristic<MATH CLASS="INLINE">
552-
553</MATH>combination<MATH CLASS="INLINE">
554-
555</MATH>poly<MATH CLASS="INLINE">
556-
557</MATH>list</I></A>
558</H4>
559<P><IMG WIDTH="348" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
560 SRC="img132.gif"
561 ALT="$\textstyle\parbox{\pboxargslen}{\em a b {\sf \&optional} (order
562 \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
563<BLOCKQUOTE>
564Returns A <MATH CLASS="INLINE">
565-
566</MATH> U1 * B1 <MATH CLASS="INLINE">
567-
568</MATH> U2 * B2 <MATH CLASS="INLINE">
569-
570</MATH> ... <MATH CLASS="INLINE">
571-
572</MATH> UM * BM where A is a
573polynomial list and B=(B1, B2, ... , BM) is a list of polynomial
574lists, where U1, U2, ... ,UM are new variables. These variables will
575be added to every polynomial A and BI as the last M variables. Se
576also CHARACTERISTIC<MATH CLASS="INLINE">
577-
578</MATH>COMBINATION. </BLOCKQUOTE><H4><A NAME="SECTION000700270000000000000">
579<I>characteristic<MATH CLASS="INLINE">
580-
581</MATH>matrix</I></A>
582</H4>
583<P><IMG WIDTH="463" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
584 SRC="img133.gif"
585 ALT="$\textstyle\parbox{\pboxargslen}{\em a {\sf \&optional} (order
586 \char93 'lex$\gt$) (b
587 (list
588 (identity$-$matrix
589 (length a)
590 (length
591 (caaaar a))))) \/}$"> [<EM>FUNCTION</EM>]
592<BLOCKQUOTE>
593Returns A <MATH CLASS="INLINE">
594-
595</MATH> U1*B1 <MATH CLASS="INLINE">
596-
597</MATH> U2*B2 <MATH CLASS="INLINE">
598-
599</MATH> ... <MATH CLASS="INLINE">
600-
601</MATH> UM * BM where A is a
602polynomial matrix and B=(B1,B2,...,BM) is a list of polynomial
603matrices, where U1, U2, .., UM are new variables. These variables
604will be added to every polynomial A and BI as the last M variables.
605Se also CHARACTERISTIC<MATH CLASS="INLINE">
606-
607</MATH>COMBINATION. </BLOCKQUOTE><H4><A NAME="SECTION000700280000000000000">
608<I>characteristic<MATH CLASS="INLINE">
609-
610</MATH>polynomial</I></A>
611</H4>
612<P><IMG WIDTH="463" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
613 SRC="img133.gif"
614 ALT="$\textstyle\parbox{\pboxargslen}{\em a {\sf \&optional} (order
615 \char93 'lex$\gt$) (b
616 (list
617 (identity$-$matrix
618 (length a)
619 (length
620 (caaaar a))))) \/}$"> [<EM>FUNCTION</EM>]
621<BLOCKQUOTE>
622Returns the generalized characteristic polynomial, i.e. the
623determinant DET(A <MATH CLASS="INLINE">
624-
625</MATH> U1 * B1 <MATH CLASS="INLINE">
626-
627</MATH> U2 * B2 <MATH CLASS="INLINE">
628-
629</MATH> ... <MATH CLASS="INLINE">
630-
631</MATH> UM * BM), where
632A and BI are square polynomial matrices in N variables. The resulting
633polynomial will have N+M variables, with U1, U2, ..., UM added as the
634last M variables. </BLOCKQUOTE><H4><A NAME="SECTION000700290000000000000">
635<I>identity<MATH CLASS="INLINE">
636-
637</MATH>matrix</I></A>
638</H4>
639<P><IMG WIDTH="503" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
640 SRC="img134.gif"
641 ALT="$\textstyle\parbox{\pboxargslen}{\em dim nvars \/}$"> [<EM>FUNCTION</EM>]
642<BLOCKQUOTE>
643Return the polynomial matrix which is the identity matrix. DIM is the
644requested dimension and NVARS is the number of variables of each
645entry. </BLOCKQUOTE><H4><A NAME="SECTION000700300000000000000">
646<I>print<MATH CLASS="INLINE">
647-
648</MATH>matrix</I></A>
649</H4>
650<P><IMG WIDTH="522" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
651 SRC="img135.gif"
652 ALT="$\textstyle\parbox{\pboxargslen}{\em f vars \/}$"> [<EM>FUNCTION</EM>]
653<BLOCKQUOTE>
654Prints a polynomial matrix F, using a list of symbols VARS as
655variable names. </BLOCKQUOTE><H4><A NAME="SECTION000700310000000000000">
656<I>jacobi<MATH CLASS="INLINE">
657-
658</MATH>matrix</I></A>
659</H4>
660<P><IMG WIDTH="514" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
661 SRC="img136.gif"
662 ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (m (length f)) (n (length (caaaar f))) \/}$"> [<EM>FUNCTION</EM>]
663<BLOCKQUOTE>
664Returns the Jacobi matrix of a polynomial list F over the first N
665variables. </BLOCKQUOTE><H4><A NAME="SECTION000700320000000000000">
666<I>jacobian</I></A>
667</H4>
668<P><IMG WIDTH="554" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
669 SRC="img137.gif"
670 ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (order \char93 'lex$\gt$) (m
671 (length f)) (n
672 (length
673 (caaaar f))) \/}$"> [<EM>FUNCTION</EM>]
674<BLOCKQUOTE>
675Returns the Jacobian (determinant) of a polynomial list F over the
676first N variables. </BLOCKQUOTE><HR>
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698<ADDRESS>
699<I>Marek Rychlik</I>
700<BR><I>3/21/1998</I>
701</ADDRESS>
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