[17b1f3] | 1 | /*****************************************************************************\ |
---|
| 2 | * Computer Algebra System SINGULAR |
---|
| 3 | \*****************************************************************************/ |
---|
| 4 | /** @file cfModResultant.cc |
---|
| 5 | * |
---|
| 6 | * modular resultant algorithm |
---|
| 7 | * |
---|
| 8 | * @author Martin Lee |
---|
| 9 | * |
---|
| 10 | * @internal @version \$Id$ |
---|
| 11 | * |
---|
| 12 | **/ |
---|
| 13 | /*****************************************************************************/ |
---|
| 14 | |
---|
| 15 | |
---|
| 16 | #include "assert.h" |
---|
| 17 | |
---|
| 18 | #include "cfModResultant.h" |
---|
| 19 | #include "cf_primes.h" |
---|
| 20 | #include "templates/ftmpl_functions.h" |
---|
| 21 | #include "cf_map.h" |
---|
| 22 | #include "cf_algorithm.h" |
---|
| 23 | |
---|
| 24 | #ifdef HAVE_NTL |
---|
| 25 | #include "NTLconvert.h" |
---|
| 26 | #endif |
---|
| 27 | |
---|
| 28 | //TODO arrange by bound= deg (F,xlevel)*deg (G,i)+deg (G,xlevel)*deg (F, i) |
---|
| 29 | static inline |
---|
| 30 | void myCompress (const CanonicalForm& F, const CanonicalForm& G, CFMap & M, |
---|
| 31 | CFMap & N, const Variable& x) |
---|
| 32 | { |
---|
| 33 | int n= tmax (F.level(), G.level()); |
---|
| 34 | int * degsf= new int [n + 1]; |
---|
| 35 | int * degsg= new int [n + 1]; |
---|
| 36 | |
---|
| 37 | for (int i = 0; i <= n; i++) |
---|
| 38 | degsf[i]= degsg[i]= 0; |
---|
| 39 | |
---|
| 40 | degsf= degrees (F, degsf); |
---|
| 41 | degsg= degrees (G, degsg); |
---|
| 42 | |
---|
| 43 | int both_non_zero= 0; |
---|
| 44 | int f_zero= 0; |
---|
| 45 | int g_zero= 0; |
---|
| 46 | int both_zero= 0; |
---|
| 47 | int degsfx, degsgx; |
---|
| 48 | |
---|
| 49 | if (x.level() != 1) |
---|
| 50 | { |
---|
| 51 | int xlevel= x.level(); |
---|
| 52 | |
---|
| 53 | for (int i= 1; i <= n; i++) |
---|
| 54 | { |
---|
| 55 | if (degsf[i] != 0 && degsg[i] != 0) |
---|
| 56 | { |
---|
| 57 | both_non_zero++; |
---|
| 58 | continue; |
---|
| 59 | } |
---|
| 60 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
---|
| 61 | { |
---|
| 62 | f_zero++; |
---|
| 63 | continue; |
---|
| 64 | } |
---|
| 65 | if (degsg[i] == 0 && degsf[i] && i <= F.level()) |
---|
| 66 | { |
---|
| 67 | g_zero++; |
---|
| 68 | continue; |
---|
| 69 | } |
---|
| 70 | } |
---|
| 71 | |
---|
| 72 | M.newpair (Variable (xlevel), Variable (1)); |
---|
| 73 | N.newpair (Variable (1), Variable (xlevel)); |
---|
| 74 | degsfx= degsf [xlevel]; |
---|
| 75 | degsgx= degsg [xlevel]; |
---|
| 76 | degsf [xlevel]= 0; |
---|
| 77 | degsg [xlevel]= 0; |
---|
| 78 | if (getNumVars (F) == 2 || getNumVars (G) == 2) |
---|
| 79 | { |
---|
| 80 | int pos= 2; |
---|
| 81 | for (int i= 1; i <= n; i++) |
---|
| 82 | { |
---|
| 83 | if (i != xlevel) |
---|
| 84 | { |
---|
| 85 | if (i != pos && (degsf[i] != 0 || degsg [i] != 0)) |
---|
| 86 | { |
---|
| 87 | M.newpair (Variable (i), Variable (pos)); |
---|
| 88 | N.newpair (Variable (pos), Variable (i)); |
---|
| 89 | pos++; |
---|
| 90 | } |
---|
| 91 | } |
---|
| 92 | } |
---|
| 93 | delete [] degsf; |
---|
| 94 | delete [] degsg; |
---|
| 95 | return; |
---|
| 96 | } |
---|
| 97 | |
---|
| 98 | if (both_non_zero == 0) |
---|
| 99 | { |
---|
| 100 | delete [] degsf; |
---|
| 101 | delete [] degsg; |
---|
| 102 | return; |
---|
| 103 | } |
---|
| 104 | |
---|
| 105 | // map Variables which do not occur in both polynomials to higher levels |
---|
| 106 | int k= 1; |
---|
| 107 | int l= 1; |
---|
| 108 | for (int i= 1; i <= n; i++) |
---|
| 109 | { |
---|
| 110 | if (i == xlevel) |
---|
| 111 | continue; |
---|
| 112 | if (degsf[i] != 0 && degsg[i] == 0 && i <= F.level()) |
---|
| 113 | { |
---|
| 114 | if (k + both_non_zero != i) |
---|
| 115 | { |
---|
| 116 | M.newpair (Variable (i), Variable (k + both_non_zero)); |
---|
| 117 | N.newpair (Variable (k + both_non_zero), Variable (i)); |
---|
| 118 | } |
---|
| 119 | k++; |
---|
| 120 | } |
---|
| 121 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
---|
| 122 | { |
---|
| 123 | if (l + g_zero + both_non_zero != i) |
---|
| 124 | { |
---|
| 125 | M.newpair (Variable (i), Variable (l + g_zero + both_non_zero)); |
---|
| 126 | N.newpair (Variable (l + g_zero + both_non_zero), Variable (i)); |
---|
| 127 | } |
---|
| 128 | l++; |
---|
| 129 | } |
---|
| 130 | } |
---|
| 131 | |
---|
| 132 | int m= tmax (F.level(), G.level()); |
---|
| 133 | int min_max_deg; |
---|
| 134 | k= both_non_zero; |
---|
| 135 | l= 0; |
---|
| 136 | int i= 1; |
---|
| 137 | while (k > 0) |
---|
| 138 | { |
---|
| 139 | if (degsf [i] != 0 && degsg [i] != 0) |
---|
| 140 | min_max_deg= degsgx*degsf[i] + degsfx*degsg[i]; |
---|
| 141 | else |
---|
| 142 | min_max_deg= 0; |
---|
| 143 | while (min_max_deg == 0) |
---|
| 144 | { |
---|
| 145 | i++; |
---|
| 146 | if (degsf [i] != 0 && degsg [i] != 0) |
---|
| 147 | min_max_deg= degsgx*degsf[i] + degsfx*degsg[i]; |
---|
| 148 | else |
---|
| 149 | min_max_deg= 0; |
---|
| 150 | } |
---|
| 151 | for (int j= i + 1; j <= m; j++) |
---|
| 152 | { |
---|
| 153 | if (degsgx*degsf[j] + degsfx*degsg[j] <= min_max_deg && |
---|
| 154 | degsf[j] != 0 && degsg [j] != 0) |
---|
| 155 | { |
---|
| 156 | min_max_deg= degsgx*degsf[j] + degsfx*degsg[j]; |
---|
| 157 | l= j; |
---|
| 158 | } |
---|
| 159 | } |
---|
| 160 | if (l != 0) |
---|
| 161 | { |
---|
| 162 | if (l != k && l != xlevel && k != 1) |
---|
| 163 | { |
---|
| 164 | M.newpair (Variable (l), Variable(k)); |
---|
| 165 | N.newpair (Variable (k), Variable(l)); |
---|
| 166 | degsf[l]= 0; |
---|
| 167 | degsg[l]= 0; |
---|
| 168 | l= 0; |
---|
| 169 | } |
---|
| 170 | else |
---|
| 171 | { |
---|
| 172 | degsf[l]= 0; |
---|
| 173 | degsg[l]= 0; |
---|
| 174 | l= 0; |
---|
| 175 | } |
---|
| 176 | } |
---|
| 177 | else if (l == 0) |
---|
| 178 | { |
---|
| 179 | if (i != k && i != xlevel && k != 1) |
---|
| 180 | { |
---|
| 181 | M.newpair (Variable (i), Variable (k)); |
---|
| 182 | N.newpair (Variable (k), Variable (i)); |
---|
| 183 | degsf[i]= 0; |
---|
| 184 | degsg[i]= 0; |
---|
| 185 | } |
---|
| 186 | else |
---|
| 187 | { |
---|
| 188 | degsf[i]= 0; |
---|
| 189 | degsg[i]= 0; |
---|
| 190 | } |
---|
| 191 | i++; |
---|
| 192 | } |
---|
| 193 | k--; |
---|
| 194 | } |
---|
| 195 | } |
---|
| 196 | else |
---|
| 197 | { |
---|
| 198 | //arrange Variables such that no gaps occur |
---|
| 199 | for (int i= 1; i <= n; i++) |
---|
| 200 | { |
---|
| 201 | if (degsf[i] == 0 && degsg[i] == 0) |
---|
| 202 | { |
---|
| 203 | both_zero++; |
---|
| 204 | continue; |
---|
| 205 | } |
---|
| 206 | else |
---|
| 207 | { |
---|
| 208 | if (both_zero != 0 && i != 1) |
---|
| 209 | { |
---|
| 210 | M.newpair (Variable (i), Variable (i - both_zero)); |
---|
| 211 | N.newpair (Variable (i - both_zero), Variable (i)); |
---|
| 212 | } |
---|
| 213 | } |
---|
| 214 | } |
---|
| 215 | } |
---|
| 216 | |
---|
| 217 | delete [] degsf; |
---|
| 218 | delete [] degsg; |
---|
| 219 | } |
---|
| 220 | |
---|
| 221 | static inline |
---|
| 222 | CanonicalForm oneNorm (const CanonicalForm& F) |
---|
| 223 | { |
---|
| 224 | if (F.inZ()) |
---|
| 225 | return abs (F); |
---|
| 226 | |
---|
| 227 | CanonicalForm result= 0; |
---|
| 228 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
| 229 | result += oneNorm (i.coeff()); |
---|
| 230 | return result; |
---|
| 231 | } |
---|
| 232 | |
---|
| 233 | // if F and G are both non constant, make sure their level is equal |
---|
| 234 | static inline |
---|
| 235 | CanonicalForm uniResultant (const CanonicalForm& F, const CanonicalForm& G) |
---|
| 236 | { |
---|
| 237 | #ifdef HAVE_NTL |
---|
| 238 | ASSERT (getCharacteristic() > 0, "characteristic > 0 expected"); |
---|
| 239 | if (F.inCoeffDomain() && G.inCoeffDomain()) |
---|
| 240 | return 1; |
---|
| 241 | if (F.isZero() || G.isZero()) |
---|
| 242 | return 0; |
---|
| 243 | |
---|
[35eb6c] | 244 | zz_pBak bak; |
---|
| 245 | bak.save(); |
---|
[17b1f3] | 246 | zz_p::init (getCharacteristic()); |
---|
| 247 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
---|
| 248 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
---|
| 249 | |
---|
| 250 | zz_p NTLResult= resultant (NTLF, NTLG); |
---|
| 251 | |
---|
[35eb6c] | 252 | bak.restore(); |
---|
[17b1f3] | 253 | return CanonicalForm (to_long (rep (NTLResult))); |
---|
| 254 | #else |
---|
| 255 | return resultant (F, G, F.mvar()); |
---|
| 256 | #endif |
---|
| 257 | } |
---|
| 258 | |
---|
| 259 | static inline |
---|
| 260 | void evalPoint (const CanonicalForm& F, const CanonicalForm& G, |
---|
| 261 | CanonicalForm& FEval, CanonicalForm& GEval, int& evalPoint) |
---|
| 262 | { |
---|
| 263 | int degF, degG; |
---|
| 264 | Variable x= Variable (1); |
---|
| 265 | degF= degree (F, x); |
---|
| 266 | degG= degree (G, x); |
---|
| 267 | do |
---|
| 268 | { |
---|
| 269 | evalPoint++; |
---|
| 270 | if (evalPoint >= getCharacteristic()) |
---|
| 271 | break; |
---|
| 272 | FEval= F (evalPoint, 2); |
---|
| 273 | GEval= G (evalPoint, 2); |
---|
| 274 | if (degree (FEval, 1) < degF || degree (GEval, 1) < degG) |
---|
| 275 | continue; |
---|
| 276 | else |
---|
| 277 | return; |
---|
| 278 | } |
---|
| 279 | while (evalPoint < getCharacteristic()); |
---|
| 280 | } |
---|
| 281 | |
---|
| 282 | static inline CanonicalForm |
---|
| 283 | newtonInterp (const CanonicalForm alpha, const CanonicalForm u, |
---|
| 284 | const CanonicalForm newtonPoly, const CanonicalForm oldInterPoly, |
---|
| 285 | const Variable & x) |
---|
| 286 | { |
---|
| 287 | CanonicalForm interPoly; |
---|
| 288 | |
---|
| 289 | interPoly= oldInterPoly+((u - oldInterPoly (alpha, x))/newtonPoly (alpha, x)) |
---|
| 290 | *newtonPoly; |
---|
| 291 | return interPoly; |
---|
| 292 | } |
---|
| 293 | |
---|
| 294 | CanonicalForm |
---|
| 295 | resultantFp (const CanonicalForm& A, const CanonicalForm& B, const Variable& x, |
---|
| 296 | bool prob) |
---|
| 297 | { |
---|
| 298 | ASSERT (getCharacteristic() > 0, "characteristic > 0 expected"); |
---|
| 299 | |
---|
| 300 | if (A.isZero() || B.isZero()) |
---|
| 301 | return 0; |
---|
| 302 | |
---|
| 303 | int degAx= degree (A, x); |
---|
| 304 | int degBx= degree (B, x); |
---|
| 305 | if (A.level() < x.level()) |
---|
| 306 | return power (A, degBx); |
---|
| 307 | if (B.level() < x.level()) |
---|
| 308 | return power (B, degAx); |
---|
| 309 | |
---|
| 310 | if (degAx == 0) |
---|
| 311 | return power (A, degBx); |
---|
| 312 | else if (degBx == 0) |
---|
| 313 | return power (B, degAx); |
---|
| 314 | |
---|
| 315 | CanonicalForm F= A; |
---|
| 316 | CanonicalForm G= B; |
---|
| 317 | |
---|
| 318 | CFMap M, N; |
---|
| 319 | myCompress (F, G, M, N, x); |
---|
| 320 | |
---|
| 321 | F= M (F); |
---|
| 322 | G= M (G); |
---|
| 323 | |
---|
| 324 | Variable y= Variable (2); |
---|
| 325 | |
---|
| 326 | if (F.isUnivariate() && G.isUnivariate() && F.level() == G.level()) |
---|
| 327 | return N(uniResultant (F, G)); |
---|
| 328 | |
---|
| 329 | int i= -1; |
---|
| 330 | CanonicalForm GEval, FEval, recResult, H; |
---|
| 331 | CanonicalForm newtonPoly= 1; |
---|
| 332 | CanonicalForm modResult= 0; |
---|
| 333 | |
---|
| 334 | Variable z= Variable (1); |
---|
| 335 | int bound= degAx*degree (G, 2) + degree (F, 2)*degBx; |
---|
| 336 | |
---|
| 337 | int count= 0; |
---|
| 338 | int equalCount= 0; |
---|
| 339 | do |
---|
| 340 | { |
---|
| 341 | evalPoint (F, G, FEval, GEval, i); |
---|
| 342 | |
---|
| 343 | ASSERT (i < getCharacteristic(), "ran out of points"); |
---|
| 344 | |
---|
| 345 | recResult= resultantFp (FEval, GEval, z); |
---|
| 346 | |
---|
| 347 | H= newtonInterp (i, recResult, newtonPoly, modResult, y); |
---|
| 348 | |
---|
| 349 | if (H == modResult) |
---|
| 350 | equalCount++; |
---|
| 351 | |
---|
| 352 | count++; |
---|
| 353 | if (count > bound || (prob && equalCount == 2)) |
---|
| 354 | break; |
---|
| 355 | |
---|
| 356 | modResult= H; |
---|
| 357 | newtonPoly *= (y - i); |
---|
| 358 | } while (1); |
---|
| 359 | |
---|
| 360 | return N (H); |
---|
| 361 | } |
---|
| 362 | |
---|
| 363 | static inline |
---|
| 364 | CanonicalForm |
---|
| 365 | balanceUni ( const CanonicalForm & f, const CanonicalForm & q ) |
---|
| 366 | { |
---|
| 367 | Variable x = f.mvar(); |
---|
| 368 | CanonicalForm result = 0, qh = q / 2; |
---|
| 369 | CanonicalForm c; |
---|
| 370 | CFIterator i; |
---|
| 371 | for ( i = f; i.hasTerms(); i++ ) { |
---|
| 372 | c = mod( i.coeff(), q ); |
---|
| 373 | if ( c > qh ) |
---|
| 374 | result += power( x, i.exp() ) * (c - q); |
---|
| 375 | else |
---|
| 376 | result += power( x, i.exp() ) * c; |
---|
| 377 | } |
---|
| 378 | return result; |
---|
| 379 | } |
---|
| 380 | |
---|
| 381 | static inline |
---|
| 382 | CanonicalForm |
---|
| 383 | symmetricRemainder (const CanonicalForm& f, const CanonicalForm& q) |
---|
| 384 | { |
---|
| 385 | CanonicalForm result= 0; |
---|
| 386 | if (f.isUnivariate() || f.inCoeffDomain()) |
---|
| 387 | return balanceUni (f, q); |
---|
| 388 | else |
---|
| 389 | { |
---|
| 390 | Variable x= f.mvar(); |
---|
| 391 | for (CFIterator i= f; i.hasTerms(); i++) |
---|
| 392 | result += power (x, i.exp())*symmetricRemainder (i.coeff(), q); |
---|
| 393 | } |
---|
| 394 | return result; |
---|
| 395 | } |
---|
| 396 | |
---|
| 397 | CanonicalForm |
---|
| 398 | resultantZ (const CanonicalForm& A, const CanonicalForm& B, const Variable& x, |
---|
| 399 | bool prob) |
---|
| 400 | { |
---|
| 401 | ASSERT (getCharacteristic() == 0, "characteristic > 0 expected"); |
---|
| 402 | |
---|
| 403 | int degAx= degree (A, x); |
---|
| 404 | int degBx= degree (B, x); |
---|
| 405 | if (A.level() < x.level()) |
---|
| 406 | return power (A, degBx); |
---|
| 407 | if (B.level() < x.level()) |
---|
| 408 | return power (B, degAx); |
---|
| 409 | |
---|
| 410 | if (degAx == 0) |
---|
| 411 | return power (A, degBx); |
---|
| 412 | else if (degBx == 0) |
---|
| 413 | return power (B, degAx); |
---|
| 414 | |
---|
| 415 | CanonicalForm F= A; |
---|
| 416 | CanonicalForm G= B; |
---|
| 417 | |
---|
| 418 | Variable X= x; |
---|
| 419 | if (F.level() != x.level() || G.level() != x.level()) |
---|
| 420 | { |
---|
| 421 | if (F.level() > G.level()) |
---|
| 422 | X= F.mvar(); |
---|
| 423 | else |
---|
| 424 | X= G.mvar(); |
---|
| 425 | F= swapvar (F, X, x); |
---|
| 426 | G= swapvar (G, X, x); |
---|
| 427 | } |
---|
| 428 | |
---|
| 429 | // now X is the main variable |
---|
| 430 | |
---|
| 431 | CanonicalForm d= 0; |
---|
| 432 | CanonicalForm dd= 0; |
---|
| 433 | CanonicalForm buf; |
---|
| 434 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
| 435 | { |
---|
| 436 | buf= oneNorm (i.coeff()); |
---|
| 437 | d= (buf > d) ? buf : d; |
---|
| 438 | } |
---|
| 439 | CanonicalForm e= 0, ee= 0; |
---|
| 440 | for (CFIterator i= G; i.hasTerms(); i++) |
---|
| 441 | { |
---|
| 442 | buf= oneNorm (i.coeff()); |
---|
| 443 | e= (buf > e) ? buf : e; |
---|
| 444 | } |
---|
| 445 | d= power (d, degBx); |
---|
| 446 | e= power (e, degAx); |
---|
| 447 | CanonicalForm bound= 1; |
---|
| 448 | for (int i= degBx + degAx; i > 1; i--) |
---|
| 449 | bound *= i; |
---|
| 450 | bound *= d*e; |
---|
| 451 | bound *= 2; |
---|
| 452 | |
---|
| 453 | bool onRational= isOn (SW_RATIONAL); |
---|
| 454 | if (onRational) |
---|
| 455 | Off (SW_RATIONAL); |
---|
| 456 | int i = cf_getNumBigPrimes() - 1; |
---|
| 457 | int p; |
---|
| 458 | CanonicalForm l= lc (F)*lc(G); |
---|
| 459 | CanonicalForm resultModP, q (0), newResult, newQ; |
---|
| 460 | CanonicalForm result; |
---|
| 461 | do |
---|
| 462 | { |
---|
| 463 | p = cf_getBigPrime( i ); |
---|
| 464 | i--; |
---|
| 465 | while ( i >= 0 && mod( l, p ) == 0) |
---|
| 466 | { |
---|
| 467 | p = cf_getBigPrime( i ); |
---|
| 468 | i--; |
---|
| 469 | } |
---|
| 470 | |
---|
| 471 | ASSERT (i >= 0, "ran out of primes"); //sic |
---|
| 472 | |
---|
| 473 | setCharacteristic (p); |
---|
| 474 | |
---|
| 475 | resultModP= resultantFp (mapinto (F), mapinto (G), X, prob); |
---|
| 476 | |
---|
| 477 | setCharacteristic (0); |
---|
| 478 | |
---|
| 479 | if ( q.isZero() ) |
---|
| 480 | { |
---|
| 481 | result= mapinto(resultModP); |
---|
| 482 | q= p; |
---|
| 483 | } |
---|
| 484 | else |
---|
| 485 | { |
---|
| 486 | chineseRemainder( result, q, mapinto (resultModP), p, newResult, newQ ); |
---|
| 487 | q= newQ; |
---|
| 488 | result= newResult; |
---|
| 489 | if (newQ > bound) |
---|
| 490 | { |
---|
| 491 | result= symmetricRemainder (result, q); |
---|
| 492 | break; |
---|
| 493 | } |
---|
| 494 | } |
---|
| 495 | } while (1); |
---|
| 496 | |
---|
| 497 | if (onRational) |
---|
| 498 | On (SW_RATIONAL); |
---|
| 499 | return swapvar (result, X, x); |
---|
| 500 | } |
---|
| 501 | |
---|