1 | // emacs edit mode for this file is -*- C++ -*- |
---|
2 | // $Id: cf_gcd.cc,v 1.7 1997-04-07 15:05:38 schmidt Exp $ |
---|
3 | |
---|
4 | /* |
---|
5 | $Log: not supported by cvs2svn $ |
---|
6 | Revision 1.6 1997/03/26 16:39:06 schmidt |
---|
7 | debug output changed to DEBOUT |
---|
8 | |
---|
9 | Revision 1.5 1996/12/05 18:24:53 schmidt |
---|
10 | ``Unconditional'' check-in. |
---|
11 | Now it is my turn to develop factory. |
---|
12 | |
---|
13 | Revision 1.4 1996/07/08 08:21:10 stobbe |
---|
14 | "gcd_poly: now uses ezgcd if the switch SW_USE_EZGCD is on. |
---|
15 | " |
---|
16 | |
---|
17 | Revision 1.3 1996/06/18 12:22:54 stobbe |
---|
18 | "gcd_poly_univar0: now uses getSmallPrimes (due to changes in the handling |
---|
19 | of prime numbers in cf_primes. |
---|
20 | " |
---|
21 | |
---|
22 | Revision 1.2 1996/06/13 08:18:34 stobbe |
---|
23 | "balance: Now balaces polynomials even if the coefficient sizes are higher |
---|
24 | than the bound. |
---|
25 | gcd: Now returns the results with positive leading coefficient. |
---|
26 | The isOne test is now performed if pi or pi1 is multivariate. |
---|
27 | " |
---|
28 | |
---|
29 | Revision 1.1 1996/06/03 08:32:56 stobbe |
---|
30 | "gcd_poly: now uses new function gcd_poly_univar0 to compute univariate |
---|
31 | polynomial gcd's over Z. |
---|
32 | gcd_poly_univar0: computes univariate polynomial gcd's in characteristic 0 |
---|
33 | via chinese remaindering. |
---|
34 | maxnorm: computes the maximum norm of all coefficients of a polynomial. |
---|
35 | " |
---|
36 | |
---|
37 | Revision 1.0 1996/05/17 11:56:37 stobbe |
---|
38 | Initial revision |
---|
39 | |
---|
40 | */ |
---|
41 | |
---|
42 | #include <config.h> |
---|
43 | |
---|
44 | #include "assert.h" |
---|
45 | #include "debug.h" |
---|
46 | |
---|
47 | #include "cf_defs.h" |
---|
48 | #include "canonicalform.h" |
---|
49 | #include "cf_iter.h" |
---|
50 | #include "cf_reval.h" |
---|
51 | #include "cf_primes.h" |
---|
52 | #include "cf_chinese.h" |
---|
53 | #include "cf_map.h" |
---|
54 | #include "fac_util.h" |
---|
55 | #include "templates/functions.h" |
---|
56 | |
---|
57 | static CanonicalForm gcd_poly( const CanonicalForm & f, const CanonicalForm& g, bool modularflag ); |
---|
58 | |
---|
59 | |
---|
60 | static int |
---|
61 | isqrt ( int a ) |
---|
62 | { |
---|
63 | int h, x0, x1 = a; |
---|
64 | do { |
---|
65 | x0 = x1; |
---|
66 | h = x0 * x0 + a - 1; |
---|
67 | if ( h % (2 * x0) == 0 ) |
---|
68 | x1 = h / (2 * x0); |
---|
69 | else |
---|
70 | x1 = (h - 1) / (2 * x0); |
---|
71 | } while ( x1 < x0 ); |
---|
72 | return x1; |
---|
73 | } |
---|
74 | |
---|
75 | bool |
---|
76 | gcd_test_one ( const CanonicalForm & f, const CanonicalForm & g, bool swap ) |
---|
77 | { |
---|
78 | int count = 0; |
---|
79 | // assume polys have same level; |
---|
80 | CFRandom * sample = CFRandomFactory::generate(); |
---|
81 | REvaluation e( 2, tmax( f.level(), g.level() ), *sample ); |
---|
82 | delete sample; |
---|
83 | CanonicalForm lcf, lcg; |
---|
84 | if ( swap ) { |
---|
85 | lcf = swapvar( LC( f ), Variable(1), f.mvar() ); |
---|
86 | lcg = swapvar( LC( g ), Variable(1), f.mvar() ); |
---|
87 | } |
---|
88 | else { |
---|
89 | lcf = LC( f, Variable(1) ); |
---|
90 | lcg = LC( g, Variable(1) ); |
---|
91 | } |
---|
92 | while ( ( e( lcf ).isZero() || e( lcg ).isZero() ) && count < 100 ) { |
---|
93 | e.nextpoint(); |
---|
94 | count++; |
---|
95 | } |
---|
96 | if ( count == 100 ) |
---|
97 | return false; |
---|
98 | CanonicalForm F, G; |
---|
99 | if ( swap ) { |
---|
100 | F=swapvar( f, Variable(1), f.mvar() ); |
---|
101 | G=swapvar( g, Variable(1), g.mvar() ); |
---|
102 | } |
---|
103 | else { |
---|
104 | F = f; |
---|
105 | G = g; |
---|
106 | } |
---|
107 | if ( e(F).taildegree() > 0 && e(G).taildegree() > 0 ) |
---|
108 | return false; |
---|
109 | return gcd( e( F ), e( G ) ).degree() < 1; |
---|
110 | } |
---|
111 | |
---|
112 | static CanonicalForm |
---|
113 | maxnorm ( const CanonicalForm & f ) |
---|
114 | { |
---|
115 | CanonicalForm m = 0, h; |
---|
116 | CFIterator i; |
---|
117 | for ( i = f; i.hasTerms(); i++ ) |
---|
118 | m = tmax( m, abs( i.coeff() ) ); |
---|
119 | return m; |
---|
120 | } |
---|
121 | |
---|
122 | static void |
---|
123 | chinesePoly ( const CanonicalForm & f1, const CanonicalForm & q1, const CanonicalForm & f2, const CanonicalForm & q2, CanonicalForm & f, CanonicalForm & q ) |
---|
124 | { |
---|
125 | CFIterator i1 = f1, i2 = f2; |
---|
126 | CanonicalForm c; |
---|
127 | Variable x = f1.mvar(); |
---|
128 | f = 0; |
---|
129 | while ( i1.hasTerms() && i2.hasTerms() ) { |
---|
130 | if ( i1.exp() == i2.exp() ) { |
---|
131 | chineseRemainder( i1.coeff(), q1, i2.coeff(), q2, c, q ); |
---|
132 | f += power( x, i1.exp() ) * c; |
---|
133 | i1++; i2++; |
---|
134 | } |
---|
135 | else if ( i1.exp() > i2.exp() ) { |
---|
136 | chineseRemainder( 0, q1, i2.coeff(), q2, c, q ); |
---|
137 | f += power( x, i2.exp() ) * c; |
---|
138 | i2++; |
---|
139 | } |
---|
140 | else { |
---|
141 | chineseRemainder( i1.coeff(), q1, 0, q2, c, q ); |
---|
142 | f += power( x, i1.exp() ) * c; |
---|
143 | i1++; |
---|
144 | } |
---|
145 | } |
---|
146 | while ( i1.hasTerms() ) { |
---|
147 | chineseRemainder( i1.coeff(), q1, 0, q2, c, q ); |
---|
148 | f += power( x, i1.exp() ) * c; |
---|
149 | i1++; |
---|
150 | } |
---|
151 | while ( i2.hasTerms() ) { |
---|
152 | chineseRemainder( 0, q1, i2.coeff(), q2, c, q ); |
---|
153 | f += power( x, i2.exp() ) * c; |
---|
154 | i2++; |
---|
155 | } |
---|
156 | } |
---|
157 | |
---|
158 | static CanonicalForm |
---|
159 | balance ( const CanonicalForm & f, const CanonicalForm & q ) |
---|
160 | { |
---|
161 | CFIterator i; |
---|
162 | CanonicalForm result = 0, qh = q / 2; |
---|
163 | CanonicalForm c; |
---|
164 | Variable x = f.mvar(); |
---|
165 | for ( i = f; i.hasTerms(); i++ ) { |
---|
166 | c = i.coeff() % q; |
---|
167 | if ( c > qh ) |
---|
168 | result += power( x, i.exp() ) * (c - q); |
---|
169 | else |
---|
170 | result += power( x, i.exp() ) * c; |
---|
171 | } |
---|
172 | return result; |
---|
173 | } |
---|
174 | |
---|
175 | CanonicalForm |
---|
176 | igcd ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
177 | { |
---|
178 | CanonicalForm a, b, c, dummy; |
---|
179 | |
---|
180 | if ( f.inZ() && g.inZ() && ! isOn( SW_RATIONAL ) ) { |
---|
181 | if ( f.sign() < 0 ) a = -f; else a = f; |
---|
182 | if ( g.sign() < 0 ) b = -g; else b = g; |
---|
183 | while ( ! b.isZero() ) { |
---|
184 | divrem( a, b, dummy, c ); |
---|
185 | a = b; |
---|
186 | b = c; |
---|
187 | } |
---|
188 | return a; |
---|
189 | } |
---|
190 | else |
---|
191 | return 1; |
---|
192 | } |
---|
193 | |
---|
194 | static CanonicalForm |
---|
195 | icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
---|
196 | { |
---|
197 | if ( f.inCoeffDomain() ) |
---|
198 | return gcd( f, c ); |
---|
199 | else { |
---|
200 | CanonicalForm g = c; |
---|
201 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
---|
202 | g = icontent( i.coeff(), g ); |
---|
203 | return g; |
---|
204 | } |
---|
205 | } |
---|
206 | |
---|
207 | CanonicalForm |
---|
208 | icontent ( const CanonicalForm & f ) |
---|
209 | { |
---|
210 | return icontent( f, 0 ); |
---|
211 | } |
---|
212 | |
---|
213 | CanonicalForm |
---|
214 | iextgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
---|
215 | { |
---|
216 | CanonicalForm p0 = f, p1 = g; |
---|
217 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
---|
218 | |
---|
219 | while ( ! p1.isZero() ) { |
---|
220 | divrem( p0, p1, q, r ); |
---|
221 | p0 = p1; p1 = r; |
---|
222 | r = g0 - g1 * q; |
---|
223 | g0 = g1; g1 = r; |
---|
224 | r = f0 - f1 * q; |
---|
225 | f0 = f1; f1 = r; |
---|
226 | } |
---|
227 | a = f0; |
---|
228 | b = g0; |
---|
229 | if ( p0.sign() < 0 ) { |
---|
230 | p0 = -p0; |
---|
231 | a = -a; |
---|
232 | b = -b; |
---|
233 | } |
---|
234 | return p0; |
---|
235 | } |
---|
236 | |
---|
237 | CanonicalForm |
---|
238 | extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
---|
239 | { |
---|
240 | CanonicalForm contf = content( f ); |
---|
241 | CanonicalForm contg = content( g ); |
---|
242 | |
---|
243 | CanonicalForm p0 = f / contf, p1 = g / contg; |
---|
244 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
---|
245 | |
---|
246 | while ( ! p1.isZero() ) { |
---|
247 | divrem( p0, p1, q, r ); |
---|
248 | p0 = p1; p1 = r; |
---|
249 | r = g0 - g1 * q; |
---|
250 | g0 = g1; g1 = r; |
---|
251 | r = f0 - f1 * q; |
---|
252 | f0 = f1; f1 = r; |
---|
253 | } |
---|
254 | CanonicalForm contp0 = content( p0 ); |
---|
255 | a = f0 / ( contf * contp0 ); |
---|
256 | b = g0 / ( contg * contp0 ); |
---|
257 | p0 /= contp0; |
---|
258 | if ( p0.sign() < 0 ) { |
---|
259 | p0 = -p0; |
---|
260 | a = -a; |
---|
261 | b = -b; |
---|
262 | } |
---|
263 | return p0; |
---|
264 | } |
---|
265 | |
---|
266 | static CanonicalForm |
---|
267 | gcd_poly_univar0( const CanonicalForm & F, const CanonicalForm & G, bool primitive ) |
---|
268 | { |
---|
269 | CanonicalForm f, g, c, cg, cl, BB, B, M, q, Dp, newD, D, newq; |
---|
270 | int p, i, n; |
---|
271 | |
---|
272 | if ( primitive ) { |
---|
273 | f = F; |
---|
274 | g = G; |
---|
275 | c = 1; |
---|
276 | } |
---|
277 | else { |
---|
278 | CanonicalForm cF = content( F ), cG = content( G ); |
---|
279 | f = F / cF; |
---|
280 | g = G / cG; |
---|
281 | c = igcd( cF, cG ); |
---|
282 | } |
---|
283 | cg = gcd( f.lc(), g.lc() ); |
---|
284 | cl = ( f.lc() / cg ) * g.lc(); |
---|
285 | // B = 2 * cg * tmin( |
---|
286 | // maxnorm(f)*power(CanonicalForm(2),f.degree())*isqrt(f.degree()+1), |
---|
287 | // maxnorm(g)*power(CanonicalForm(2),g.degree())*isqrt(g.degree()+1) |
---|
288 | // )+1; |
---|
289 | M = tmin( maxnorm(f), maxnorm(g) ); |
---|
290 | BB = power(CanonicalForm(2),tmin(f.degree(),g.degree()))*M; |
---|
291 | q = 0; |
---|
292 | i = cf_getNumSmallPrimes() - 1; |
---|
293 | while ( true ) { |
---|
294 | B = BB; |
---|
295 | while ( i >= 0 && q < B ) { |
---|
296 | p = cf_getSmallPrime( i ); |
---|
297 | i--; |
---|
298 | while ( i >= 0 && mod( cl, p ) == 0 ) { |
---|
299 | p = cf_getSmallPrime( i ); |
---|
300 | i--; |
---|
301 | } |
---|
302 | setCharacteristic( p ); |
---|
303 | Dp = gcd( mapinto( f ), mapinto( g ) ); |
---|
304 | Dp = ( Dp / Dp.lc() ) * mapinto( cg ); |
---|
305 | setCharacteristic( 0 ); |
---|
306 | if ( Dp.degree() == 0 ) return c; |
---|
307 | if ( q.isZero() ) { |
---|
308 | D = mapinto( Dp ); |
---|
309 | q = p; |
---|
310 | B = power(CanonicalForm(2),D.degree())*M+1; |
---|
311 | } |
---|
312 | else { |
---|
313 | if ( Dp.degree() == D.degree() ) { |
---|
314 | chinesePoly( D, q, mapinto( Dp ), p, newD, newq ); |
---|
315 | q = newq; |
---|
316 | D = newD; |
---|
317 | } |
---|
318 | else if ( Dp.degree() < D.degree() ) { |
---|
319 | // all previous p's are bad primes |
---|
320 | q = p; |
---|
321 | D = mapinto( Dp ); |
---|
322 | B = power(CanonicalForm(2),D.degree())*M+1; |
---|
323 | } |
---|
324 | // else p is a bad prime |
---|
325 | } |
---|
326 | } |
---|
327 | if ( i >= 0 ) { |
---|
328 | // now balance D mod q |
---|
329 | D = pp( balance( D, q ) ); |
---|
330 | if ( divides( D, f ) && divides( D, g ) ) |
---|
331 | return D * c; |
---|
332 | else |
---|
333 | q = 0; |
---|
334 | } |
---|
335 | else { |
---|
336 | return gcd_poly( F, G, false ); |
---|
337 | } |
---|
338 | DEBOUTLN( cerr, "another try ...", ' ' ); |
---|
339 | } |
---|
340 | } |
---|
341 | |
---|
342 | |
---|
343 | static CanonicalForm |
---|
344 | gcd_poly1( const CanonicalForm & f, const CanonicalForm & g, bool modularflag ) |
---|
345 | { |
---|
346 | CanonicalForm C, Ci, Ci1, Hi, bi, pi, pi1, pi2; |
---|
347 | int delta; |
---|
348 | Variable v = f.mvar(); |
---|
349 | |
---|
350 | if ( f.degree( v ) >= g.degree( v ) ) { |
---|
351 | pi = f; pi1 = g; |
---|
352 | } |
---|
353 | else { |
---|
354 | pi = g; pi1 = f; |
---|
355 | } |
---|
356 | Ci = content( pi ); Ci1 = content( pi1 ); |
---|
357 | C = gcd( Ci, Ci1 ); |
---|
358 | pi1 = pi1 / Ci1; pi = pi / Ci; |
---|
359 | if ( pi.isUnivariate() && pi1.isUnivariate() ) { |
---|
360 | if ( modularflag ) |
---|
361 | return gcd_poly_univar0( pi, pi1, true ) * C; |
---|
362 | } |
---|
363 | else |
---|
364 | if ( gcd_test_one( pi1, pi, true ) ) |
---|
365 | return C; |
---|
366 | delta = degree( pi, v ) - degree( pi1, v ); |
---|
367 | Hi = power( LC( pi1, v ), delta ); |
---|
368 | if ( (delta+1) % 2 ) |
---|
369 | bi = 1; |
---|
370 | else |
---|
371 | bi = -1; |
---|
372 | while ( degree( pi1, v ) > 0 ) { |
---|
373 | pi2 = psr( pi, pi1, v ); |
---|
374 | pi2 = pi2 / bi; |
---|
375 | pi = pi1; pi1 = pi2; |
---|
376 | if ( degree( pi1, v ) > 0 ) { |
---|
377 | delta = degree( pi, v ) - degree( pi1, v ); |
---|
378 | if ( (delta+1) % 2 ) |
---|
379 | bi = LC( pi, v ) * power( Hi, delta ); |
---|
380 | else |
---|
381 | bi = -LC( pi, v ) * power( Hi, delta ); |
---|
382 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
---|
383 | } |
---|
384 | } |
---|
385 | if ( degree( pi1, v ) == 0 ) |
---|
386 | return C; |
---|
387 | else { |
---|
388 | return C * pp( pi ); |
---|
389 | } |
---|
390 | } |
---|
391 | |
---|
392 | |
---|
393 | static CanonicalForm |
---|
394 | gcd_poly( const CanonicalForm & f, const CanonicalForm & g, bool modularflag ) |
---|
395 | { |
---|
396 | if ( getCharacteristic() != 0 ) { |
---|
397 | return gcd_poly1( f, g, false ); |
---|
398 | } |
---|
399 | else if ( isOn( SW_USE_EZGCD ) && ! ( f.isUnivariate() && g.isUnivariate() ) ) { |
---|
400 | CFMap M, N; |
---|
401 | compress( f, g, M, N ); |
---|
402 | return N( ezgcd( M(f), M(g) ) ); |
---|
403 | } |
---|
404 | else { |
---|
405 | return gcd_poly1( f, g, modularflag ); |
---|
406 | } |
---|
407 | } |
---|
408 | |
---|
409 | |
---|
410 | static CanonicalForm |
---|
411 | cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
412 | { |
---|
413 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) { |
---|
414 | CFIterator i = f; |
---|
415 | CanonicalForm result = g; |
---|
416 | while ( i.hasTerms() && ! result.isOne() ) { |
---|
417 | result = gcd( result, i.coeff() ); |
---|
418 | i++; |
---|
419 | } |
---|
420 | return result; |
---|
421 | } |
---|
422 | else |
---|
423 | if ( f.sign() < 0 ) |
---|
424 | return -f; |
---|
425 | else |
---|
426 | return f; |
---|
427 | } |
---|
428 | |
---|
429 | CanonicalForm |
---|
430 | content ( const CanonicalForm & f ) |
---|
431 | { |
---|
432 | return cf_content( f, 0 ); |
---|
433 | } |
---|
434 | |
---|
435 | CanonicalForm |
---|
436 | content ( const CanonicalForm & f, const Variable & x ) |
---|
437 | { |
---|
438 | if ( f.mvar() == x ) |
---|
439 | return cf_content( f, 0 ); |
---|
440 | else if ( f.mvar() < x ) |
---|
441 | return f; |
---|
442 | else |
---|
443 | return swapvar( content( swapvar( f, f.mvar(), x ), f.mvar() ), f.mvar(), x ); |
---|
444 | } |
---|
445 | |
---|
446 | CanonicalForm |
---|
447 | vcontent ( const CanonicalForm & f, const Variable & x ) |
---|
448 | { |
---|
449 | if ( f.mvar() <= x ) |
---|
450 | return content( f, x ); |
---|
451 | else { |
---|
452 | CFIterator i; |
---|
453 | CanonicalForm d = 0; |
---|
454 | for ( i = f; i.hasTerms() && ! d.isOne(); i++ ) |
---|
455 | d = gcd( d, vcontent( i.coeff(), x ) ); |
---|
456 | return d; |
---|
457 | } |
---|
458 | } |
---|
459 | |
---|
460 | CanonicalForm |
---|
461 | pp ( const CanonicalForm & f ) |
---|
462 | { |
---|
463 | if ( f.isZero() ) |
---|
464 | return f; |
---|
465 | else |
---|
466 | return f / content( f ); |
---|
467 | } |
---|
468 | |
---|
469 | CanonicalForm |
---|
470 | gcd ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
471 | { |
---|
472 | if ( f.isZero() ) |
---|
473 | if ( g.lc().sign() < 0 ) |
---|
474 | return -g; |
---|
475 | else |
---|
476 | return g; |
---|
477 | else if ( g.isZero() ) |
---|
478 | if ( f.lc().sign() < 0 ) |
---|
479 | return -f; |
---|
480 | else |
---|
481 | return f; |
---|
482 | else if ( f.inBaseDomain() ) |
---|
483 | if ( g.inBaseDomain() ) |
---|
484 | return igcd( f, g ); |
---|
485 | else |
---|
486 | return cf_content( g, f ); |
---|
487 | else if ( g.inBaseDomain() ) |
---|
488 | return cf_content( f, g ); |
---|
489 | else if ( f.mvar() == g.mvar() ) |
---|
490 | if ( f.inExtension() && getReduce( f.mvar() ) ) |
---|
491 | return 1; |
---|
492 | else { |
---|
493 | if ( divides( f, g ) ) |
---|
494 | if ( f.lc().sign() < 0 ) |
---|
495 | return -f; |
---|
496 | else |
---|
497 | return f; |
---|
498 | else if ( divides( g, f ) ) |
---|
499 | if ( g.lc().sign() < 0 ) |
---|
500 | return -g; |
---|
501 | else |
---|
502 | return g; |
---|
503 | if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) { |
---|
504 | Off( SW_RATIONAL ); |
---|
505 | CanonicalForm l = lcm( common_den( f ), common_den( g ) ); |
---|
506 | On( SW_RATIONAL ); |
---|
507 | CanonicalForm F = f * l, G = g * l; |
---|
508 | Off( SW_RATIONAL ); |
---|
509 | l = gcd_poly( F, G, true ); |
---|
510 | On( SW_RATIONAL ); |
---|
511 | if ( l.lc().sign() < 0 ) |
---|
512 | return -l; |
---|
513 | else |
---|
514 | return l; |
---|
515 | } |
---|
516 | else { |
---|
517 | CanonicalForm d = gcd_poly( f, g, getCharacteristic()==0 ); |
---|
518 | if ( d.lc().sign() < 0 ) |
---|
519 | return -d; |
---|
520 | else |
---|
521 | return d; |
---|
522 | } |
---|
523 | } |
---|
524 | else if ( f.mvar() > g.mvar() ) |
---|
525 | return cf_content( f, g ); |
---|
526 | else |
---|
527 | return cf_content( g, f ); |
---|
528 | } |
---|
529 | |
---|
530 | CanonicalForm |
---|
531 | lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
532 | { |
---|
533 | return ( f / gcd( f, g ) ) * g; |
---|
534 | } |
---|