ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <vector>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "mul.h"
  29 #include "add.h"
  30 #include "power.h"
  31 #include "operators.h"
  32 #include "matrix.h"
  33 #include "lst.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36
  37 namespace GiNaC {
  38
  39 GINAC_IMPLEMENT_REGISTERED_CLASS(mul, expairseq)
  40
  41 //////////
  42 // default ctor, dtor, copy ctor, assignment operator and helpers
  43 //////////
  44
  45 mul::mul()
  46 {
  47         tinfo_key = TINFO_mul;
  48 }
  49
  50 DEFAULT_COPY(mul)
  51 DEFAULT_DESTROY(mul)
  52
  53 //////////
  54 // other ctors
  55 //////////
  56
  57 // public
  58
  59 mul::mul(const ex & lh, const ex & rh)
  60 {
  61         tinfo_key = TINFO_mul;
  62         overall_coeff = _ex1;
  63         construct_from_2_ex(lh,rh);
  64         GINAC_ASSERT(is_canonical());
  65 }
  66
  67 mul::mul(const exvector & v)
  68 {
  69         tinfo_key = TINFO_mul;
  70         overall_coeff = _ex1;
  71         construct_from_exvector(v);
  72         GINAC_ASSERT(is_canonical());
  73 }
  74
  75 mul::mul(const epvector & v)
  76 {
  77         tinfo_key = TINFO_mul;
  78         overall_coeff = _ex1;
  79         construct_from_epvector(v);
  80         GINAC_ASSERT(is_canonical());
  81 }
  82
  83 mul::mul(const epvector & v, const ex & oc)
  84 {
  85         tinfo_key = TINFO_mul;
  86         overall_coeff = oc;
  87         construct_from_epvector(v);
  88         GINAC_ASSERT(is_canonical());
  89 }
  90
  91 mul::mul(epvector * vp, const ex & oc)
  92 {
  93         tinfo_key = TINFO_mul;
  94         GINAC_ASSERT(vp!=0);
  95         overall_coeff = oc;
  96         construct_from_epvector(*vp);
  97         delete vp;
  98         GINAC_ASSERT(is_canonical());
  99 }
 100
 101 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 102 {
 103         tinfo_key = TINFO_mul;
 104         exvector factors;
 105         factors.reserve(3);
 106         factors.push_back(lh);
 107         factors.push_back(mh);
 108         factors.push_back(rh);
 109         overall_coeff = _ex1;
 110         construct_from_exvector(factors);
 111         GINAC_ASSERT(is_canonical());
 112 }
 113
 114 //////////
 115 // archiving
 116 //////////
 117
 118 DEFAULT_ARCHIVING(mul)
 119
 120 //////////
 121 // functions overriding virtual functions from base classes
 122 //////////
 123
 124 // public
 125 void mul::print(const print_context & c, unsigned level) const
 126 {
 127         if (is_a<print_tree>(c)) {
 128
 129                 inherited::print(c, level);
 130
 131         } else if (is_a<print_csrc>(c)) {
 132
 133                 if (precedence() <= level)
 134                         c.s << "(";
 135
 136                 if (!overall_coeff.is_equal(_ex1)) {
 137                         overall_coeff.print(c, precedence());
 138                         c.s << "*";
 139                 }
 140
 141                 // Print arguments, separated by "*" or "/"
 142                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 143                 while (it != itend) {
 144
 145                         // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 146                         bool needclosingparenthesis = false;
 147                         if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 148                                 if (is_a<print_csrc_cl_N>(c)) {
 149                                         c.s << "recip(";
 150                                         needclosingparenthesis = true;
 151                                 } else
 152                                         c.s << "1.0/";
 153                         }
 154
 155                         // If the exponent is 1 or -1, it is left out
 156                         if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 157                                 it->rest.print(c, precedence());
 158                         else if (it->coeff.info(info_flags::negint))
 159                                 // Outer parens around ex needed for broken GCC parser:
 160                                 (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 161                         else
 162                                 // Outer parens around ex needed for broken GCC parser:
 163                                 (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 164
 165                         if (needclosingparenthesis)
 166                                 c.s << ")";
 167
 168                         // Separator is "/" for negative integer powers, "*" otherwise
 169                         ++it;
 170                         if (it != itend) {
 171                                 if (it->coeff.info(info_flags::negint))
 172                                         c.s << "/";
 173                                 else
 174                                         c.s << "*";
 175                         }
 176                 }
 177
 178                 if (precedence() <= level)
 179                         c.s << ")";
 180
 181         } else if (is_a<print_python_repr>(c)) {
 182                 c.s << class_name() << '(';
 183                 op(0).print(c);
 184                 for (size_t i=1; i<nops(); ++i) {
 185                         c.s << ',';
 186                         op(i).print(c);
 187                 }
 188                 c.s << ')';
 189         } else {
 190
 191                 if (precedence() <= level) {
 192                         if (is_a<print_latex>(c))
 193                                 c.s << "{(";
 194                         else
 195                                 c.s << "(";
 196                 }
 197
 198                 // First print the overall numeric coefficient
 199                 const numeric &coeff = ex_to<numeric>(overall_coeff);
 200                 if (coeff.csgn() == -1)
 201                         c.s << '-';
 202                 if (!coeff.is_equal(_num1) &&
 203                         !coeff.is_equal(_num_1)) {
 204                         if (coeff.is_rational()) {
 205                                 if (coeff.is_negative())
 206                                         (-coeff).print(c);
 207                                 else
 208                                         coeff.print(c);
 209                         } else {
 210                                 if (coeff.csgn() == -1)
 211                                         (-coeff).print(c, precedence());
 212                                 else
 213                                         coeff.print(c, precedence());
 214                         }
 215                         if (is_a<print_latex>(c))
 216                                 c.s << ' ';
 217                         else
 218                                 c.s << '*';
 219                 }
 220
 221                 // Then proceed with the remaining factors
 222                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 223                 if (is_a<print_latex>(c)) {
 224
 225                         // Separate factors into those with negative numeric exponent
 226                         // and all others
 227                         exvector neg_powers, others;
 228                         while (it != itend) {
 229                                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
 230                                 if (ex_to<numeric>(it->coeff).is_negative())
 231                                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
 232                                 else
 233                                         others.push_back(recombine_pair_to_ex(*it));
 234                                 ++it;
 235                         }
 236
 237                         if (!neg_powers.empty()) {
 238
 239                                 // Factors with negative exponent are printed as a fraction
 240                                 c.s << "\\frac{";
 241                                 mul(others).eval().print(c);
 242                                 c.s << "}{";
 243                                 mul(neg_powers).eval().print(c);
 244                                 c.s << "}";
 245
 246                         } else {
 247
 248                                 // All other factors are printed in the ordinary way
 249                                 exvector::const_iterator vit = others.begin(), vitend = others.end();
 250                                 while (vit != vitend) {
 251                                         c.s << ' ';
 252                                         vit->print(c, precedence());
 253                                         ++vit;
 254                                 }
 255                         }
 256
 257                 } else {
 258
 259                         bool first = true;
 260                         while (it != itend) {
 261                                 if (!first)
 262                                         c.s << '*';
 263                                 else
 264                                         first = false;
 265                                 recombine_pair_to_ex(*it).print(c, precedence());
 266                                 ++it;
 267                         }
 268                 }
 269
 270                 if (precedence() <= level) {
 271                         if (is_a<print_latex>(c))
 272                                 c.s << ")}";
 273                         else
 274                                 c.s << ")";
 275                 }
 276         }
 277 }
 278
 279 bool mul::info(unsigned inf) const
 280 {
 281         switch (inf) {
 282                 case info_flags::polynomial:
 283                 case info_flags::integer_polynomial:
 284                 case info_flags::cinteger_polynomial:
 285                 case info_flags::rational_polynomial:
 286                 case info_flags::crational_polynomial:
 287                 case info_flags::rational_function: {
 288                         epvector::const_iterator i = seq.begin(), end = seq.end();
 289                         while (i != end) {
 290                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 291                                         return false;
 292                                 ++i;
 293                         }
 294                         return overall_coeff.info(inf);
 295                 }
 296                 case info_flags::algebraic: {
 297                         epvector::const_iterator i = seq.begin(), end = seq.end();
 298                         while (i != end) {
 299                                 if ((recombine_pair_to_ex(*i).info(inf)))
 300                                         return true;
 301                                 ++i;
 302                         }
 303                         return false;
 304                 }
 305         }
 306         return inherited::info(inf);
 307 }
 308
 309 int mul::degree(const ex & s) const
 310 {
 311         // Sum up degrees of factors
 312         int deg_sum = 0;
 313         epvector::const_iterator i = seq.begin(), end = seq.end();
 314         while (i != end) {
 315                 if (ex_to<numeric>(i->coeff).is_integer())
 316                         deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
 317                 ++i;
 318         }
 319         return deg_sum;
 320 }
 321
 322 int mul::ldegree(const ex & s) const
 323 {
 324         // Sum up degrees of factors
 325         int deg_sum = 0;
 326         epvector::const_iterator i = seq.begin(), end = seq.end();
 327         while (i != end) {
 328                 if (ex_to<numeric>(i->coeff).is_integer())
 329                         deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
 330                 ++i;
 331         }
 332         return deg_sum;
 333 }
 334
 335 ex mul::coeff(const ex & s, int n) const
 336 {
 337         exvector coeffseq;
 338         coeffseq.reserve(seq.size()+1);
 339
 340         if (n==0) {
 341                 // product of individual coeffs
 342                 // if a non-zero power of s is found, the resulting product will be 0
 343                 epvector::const_iterator i = seq.begin(), end = seq.end();
 344                 while (i != end) {
 345                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 346                         ++i;
 347                 }
 348                 coeffseq.push_back(overall_coeff);
 349                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 350         }
 351
 352         epvector::const_iterator i = seq.begin(), end = seq.end();
 353         bool coeff_found = false;
 354         while (i != end) {
 355                 ex t = recombine_pair_to_ex(*i);
 356                 ex c = t.coeff(s, n);
 357                 if (!c.is_zero()) {
 358                         coeffseq.push_back(c);
 359                         coeff_found = 1;
 360                 } else {
 361                         coeffseq.push_back(t);
 362                 }
 363                 ++i;
 364         }
 365         if (coeff_found) {
 366                 coeffseq.push_back(overall_coeff);
 367                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 368         }
 369
 370         return _ex0;
 371 }
 372
 373 /** Perform automatic term rewriting rules in this class.  In the following
 374  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 375  *  stand for such expressions that contain a plain number.
 376  *  - *(...,x;0) -> 0
 377  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 378  *  - *(x;1) -> x
 379  *  - *(;c) -> c
 380  *
 381  *  @param level cut-off in recursive evaluation */
 382 ex mul::eval(int level) const
 383 {
 384         epvector *evaled_seqp = evalchildren(level);
 385         if (evaled_seqp) {
 386                 // do more evaluation later
 387                 return (new mul(evaled_seqp,overall_coeff))->
 388                            setflag(status_flags::dynallocated);
 389         }
 390
 391 #ifdef DO_GINAC_ASSERT
 392         epvector::const_iterator i = seq.begin(), end = seq.end();
 393         while (i != end) {
 394                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 395                              (!(ex_to<numeric>(i->coeff).is_integer())));
 396                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 397                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 398                     print(print_tree(std::cerr));
 399                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 400                 /* for paranoia */
 401                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 402                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 403                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 404                 /* end paranoia */
 405                 ++i;
 406         }
 407 #endif // def DO_GINAC_ASSERT
 408
 409         if (flags & status_flags::evaluated) {
 410                 GINAC_ASSERT(seq.size()>0);
 411                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 412                 return *this;
 413         }
 414
 415         int seq_size = seq.size();
 416         if (overall_coeff.is_zero()) {
 417                 // *(...,x;0) -> 0
 418                 return _ex0;
 419         } else if (seq_size==0) {
 420                 // *(;c) -> c
 421                 return overall_coeff;
 422         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 423                 // *(x;1) -> x
 424                 return recombine_pair_to_ex(*(seq.begin()));
 425         } else if ((seq_size==1) &&
 426                    is_exactly_a<add>((*seq.begin()).rest) &&
 427                    ex_to<numeric>((*seq.begin()).coeff).is_equal(_num1)) {
 428                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 429                 const add & addref = ex_to<add>((*seq.begin()).rest);
 430                 epvector *distrseq = new epvector();
 431                 distrseq->reserve(addref.seq.size());
 432                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 433                 while (i != end) {
 434                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 435                         ++i;
 436                 }
 437                 return (new add(distrseq,
 438                                 ex_to<numeric>(addref.overall_coeff).
 439                                 mul_dyn(ex_to<numeric>(overall_coeff))))
 440                       ->setflag(status_flags::dynallocated | status_flags::evaluated);
 441         }
 442         return this->hold();
 443 }
 444
 445 ex mul::evalf(int level) const
 446 {
 447         if (level==1)
 448                 return mul(seq,overall_coeff);
 449
 450         if (level==-max_recursion_level)
 451                 throw(std::runtime_error("max recursion level reached"));
 452
 453         epvector *s = new epvector();
 454         s->reserve(seq.size());
 455
 456         --level;
 457         epvector::const_iterator i = seq.begin(), end = seq.end();
 458         while (i != end) {
 459                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 460                                                            i->coeff));
 461                 ++i;
 462         }
 463         return mul(s, overall_coeff.evalf(level));
 464 }
 465
 466 ex mul::evalm(void) const
 467 {
 468         // numeric*matrix
 469         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 470          && is_a<matrix>(seq[0].rest))
 471                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 472
 473         // Evaluate children first, look whether there are any matrices at all
 474         // (there can be either no matrices or one matrix; if there were more
 475         // than one matrix, it would be a non-commutative product)
 476         epvector *s = new epvector;
 477         s->reserve(seq.size());
 478
 479         bool have_matrix = false;
 480         epvector::iterator the_matrix;
 481
 482         epvector::const_iterator i = seq.begin(), end = seq.end();
 483         while (i != end) {
 484                 const ex &m = recombine_pair_to_ex(*i).evalm();
 485                 s->push_back(split_ex_to_pair(m));
 486                 if (is_a<matrix>(m)) {
 487                         have_matrix = true;
 488                         the_matrix = s->end() - 1;
 489                 }
 490                 ++i;
 491         }
 492
 493         if (have_matrix) {
 494
 495                 // The product contained a matrix. We will multiply all other factors
 496                 // into that matrix.
 497                 matrix m = ex_to<matrix>(the_matrix->rest);
 498                 s->erase(the_matrix);
 499                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 500                 return m.mul_scalar(scalar);
 501
 502         } else
 503                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 504 }
 505
 506 ex mul::eval_ncmul(const exvector & v) const
 507 {
 508         if (seq.empty())
 509                 return inherited::eval_ncmul(v);
 510
 511         // Find first noncommutative element and call its eval_ncmul()
 512         epvector::const_iterator i = seq.begin(), end = seq.end();
 513         while (i != end) {
 514                 if (i->rest.return_type() == return_types::noncommutative)
 515                         return i->rest.eval_ncmul(v);
 516                 ++i;
 517         }
 518         return inherited::eval_ncmul(v);
 519 }
 520
 521 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
 522 {
 523         ex origbase;
 524         int origexponent;
 525         int origexpsign;
 526
 527         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 528                 origbase = origfactor.op(0);
 529                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 530                 origexponent = expon > 0 ? expon : -expon;
 531                 origexpsign = expon > 0 ? 1 : -1;
 532         } else {
 533                 origbase = origfactor;
 534                 origexponent = 1;
 535                 origexpsign = 1;
 536         }
 537
 538         ex patternbase;
 539         int patternexponent;
 540         int patternexpsign;
 541
 542         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 543                 patternbase = patternfactor.op(0);
 544                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 545                 patternexponent = expon > 0 ? expon : -expon;
 546                 patternexpsign = expon > 0 ? 1 : -1;
 547         } else {
 548                 patternbase = patternfactor;
 549                 patternexponent = 1;
 550                 patternexpsign = 1;
 551         }
 552
 553         lst saverepls = repls;
 554         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 555                 return false;
 556         repls = saverepls;
 557
 558         int newnummatches = origexponent / patternexponent;
 559         if (newnummatches < nummatches)
 560                 nummatches = newnummatches;
 561         return true;
 562 }
 563
 564 ex mul::algebraic_subs_mul(const lst & ls, const lst & lr, unsigned options) const
 565 {
 566         std::vector<bool> subsed(seq.size(), false);
 567         exvector subsresult(seq.size());
 568
 569         lst::const_iterator its, itr;
 570         for (its = ls.begin(), itr = lr.begin(); its != ls.end(); ++its, ++itr) {
 571
 572                 if (is_exactly_a<mul>(*its)) {
 573
 574                         int nummatches = std::numeric_limits<int>::max();
 575                         std::vector<bool> currsubsed(seq.size(), false);
 576                         bool succeed = true;
 577                         lst repls;
 578
 579                         for (size_t j=0; j<its->nops(); j++) {
 580                                 bool found=false;
 581                                 for (size_t k=0; k<nops(); k++) {
 582                                         if (currsubsed[k] || subsed[k])
 583                                                 continue;
 584                                         if (tryfactsubs(op(k), its->op(j), nummatches, repls)) {
 585                                                 currsubsed[k] = true;
 586                                                 found = true;
 587                                                 break;
 588                                         }
 589                                 }
 590                                 if (!found) {
 591                                         succeed = false;
 592                                         break;
 593                                 }
 594                         }
 595                         if (!succeed)
 596                                 continue;
 597
 598                         bool foundfirstsubsedfactor = false;
 599                         for (size_t j=0; j<subsed.size(); j++) {
 600                                 if (currsubsed[j]) {
 601                                         if (foundfirstsubsedfactor)
 602                                                 subsresult[j] = op(j);
 603                                         else {
 604                                                 foundfirstsubsedfactor = true;
 605                                                 subsresult[j] = op(j) * power(itr->subs(ex(repls), subs_options::no_pattern) / its->subs(ex(repls), subs_options::no_pattern), nummatches);
 606                                         }
 607                                         subsed[j] = true;
 608                                 }
 609                         }
 610
 611                 } else {
 612
 613                         int nummatches = std::numeric_limits<int>::max();
 614                         lst repls;
 615
 616                         for (size_t j=0; j<this->nops(); j++) {
 617                                 if (!subsed[j] && tryfactsubs(op(j), *its, nummatches, repls)) {
 618                                         subsed[j] = true;
 619                                         subsresult[j] = op(j) * power(itr->subs(ex(repls), subs_options::no_pattern) / its->subs(ex(repls), subs_options::no_pattern), nummatches);
 620                                 }
 621                         }
 622                 }
 623         }
 624
 625         bool subsfound = false;
 626         for (size_t i=0; i<subsed.size(); i++) {
 627                 if (subsed[i]) {
 628                         subsfound = true;
 629                         break;
 630                 }
 631         }
 632         if (!subsfound)
 633                 return basic::subs(ls, lr, options | subs_options::algebraic);
 634
 635         exvector ev; ev.reserve(nops());
 636         for (size_t i=0; i<nops(); i++) {
 637                 if (subsed[i])
 638                         ev.push_back(subsresult[i]);
 639                 else
 640                         ev.push_back(op(i));
 641         }
 642
 643         return (new mul(ev))->setflag(status_flags::dynallocated);
 644 }
 645
 646 // protected
 647
 648 /** Implementation of ex::diff() for a product.  It applies the product rule.
 649  *  @see ex::diff */
 650 ex mul::derivative(const symbol & s) const
 651 {
 652         size_t num = seq.size();
 653         exvector addseq;
 654         addseq.reserve(num);
 655
 656         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 657         epvector mulseq = seq;
 658         epvector::const_iterator i = seq.begin(), end = seq.end();
 659         epvector::iterator i2 = mulseq.begin();
 660         while (i != end) {
 661                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 662                                              i->rest.diff(s));
 663                 ep.swap(*i2);
 664                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 665                 ep.swap(*i2);
 666                 ++i; ++i2;
 667         }
 668         return (new add(addseq))->setflag(status_flags::dynallocated);
 669 }
 670
 671 int mul::compare_same_type(const basic & other) const
 672 {
 673         return inherited::compare_same_type(other);
 674 }
 675
 676 unsigned mul::return_type(void) const
 677 {
 678         if (seq.empty()) {
 679                 // mul without factors: should not happen, but commutes
 680                 return return_types::commutative;
 681         }
 682
 683         bool all_commutative = true;
 684         epvector::const_iterator noncommutative_element; // point to first found nc element
 685
 686         epvector::const_iterator i = seq.begin(), end = seq.end();
 687         while (i != end) {
 688                 unsigned rt = i->rest.return_type();
 689                 if (rt == return_types::noncommutative_composite)
 690                         return rt; // one ncc -> mul also ncc
 691                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 692                         // first nc element found, remember position
 693                         noncommutative_element = i;
 694                         all_commutative = false;
 695                 }
 696                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 697                         // another nc element found, compare type_infos
 698                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 699                                 // diffent types -> mul is ncc
 700                                 return return_types::noncommutative_composite;
 701                         }
 702                 }
 703                 ++i;
 704         }
 705         // all factors checked
 706         return all_commutative ? return_types::commutative : return_types::noncommutative;
 707 }
 708
 709 unsigned mul::return_type_tinfo(void) const
 710 {
 711         if (seq.empty())
 712                 return tinfo_key;  // mul without factors: should not happen
 713
 714         // return type_info of first noncommutative element
 715         epvector::const_iterator i = seq.begin(), end = seq.end();
 716         while (i != end) {
 717                 if (i->rest.return_type() == return_types::noncommutative)
 718                         return i->rest.return_type_tinfo();
 719                 ++i;
 720         }
 721         // no noncommutative element found, should not happen
 722         return tinfo_key;
 723 }
 724
 725 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
 726 {
 727         return (new mul(v, oc))->setflag(status_flags::dynallocated);
 728 }
 729
 730 ex mul::thisexpairseq(epvector * vp, const ex & oc) const
 731 {
 732         return (new mul(vp, oc))->setflag(status_flags::dynallocated);
 733 }
 734
 735 expair mul::split_ex_to_pair(const ex & e) const
 736 {
 737         if (is_exactly_a<power>(e)) {
 738                 const power & powerref = ex_to<power>(e);
 739                 if (is_exactly_a<numeric>(powerref.exponent))
 740                         return expair(powerref.basis,powerref.exponent);
 741         }
 742         return expair(e,_ex1);
 743 }
 744
 745 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 746                                           const ex & c) const
 747 {
 748         // to avoid duplication of power simplification rules,
 749         // we create a temporary power object
 750         // otherwise it would be hard to correctly evaluate
 751         // expression like (4^(1/3))^(3/2)
 752         if (c.is_equal(_ex1))
 753                 return split_ex_to_pair(e);
 754
 755         return split_ex_to_pair(power(e,c));
 756 }
 757
 758 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 759                                             const ex & c) const
 760 {
 761         // to avoid duplication of power simplification rules,
 762         // we create a temporary power object
 763         // otherwise it would be hard to correctly evaluate
 764         // expression like (4^(1/3))^(3/2)
 765         if (c.is_equal(_ex1))
 766                 return p;
 767
 768         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 769 }
 770
 771 ex mul::recombine_pair_to_ex(const expair & p) const
 772 {
 773         if (ex_to<numeric>(p.coeff).is_equal(_num1))
 774                 return p.rest;
 775         else
 776                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 777 }
 778
 779 bool mul::expair_needs_further_processing(epp it)
 780 {
 781         if (is_exactly_a<mul>(it->rest) &&
 782                 ex_to<numeric>(it->coeff).is_integer()) {
 783                 // combined pair is product with integer power -> expand it
 784                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 785                 return true;
 786         }
 787         if (is_exactly_a<numeric>(it->rest)) {
 788                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 789                 if (!ep.is_equal(*it)) {
 790                         // combined pair is a numeric power which can be simplified
 791                         *it = ep;
 792                         return true;
 793                 }
 794                 if (it->coeff.is_equal(_ex1)) {
 795                         // combined pair has coeff 1 and must be moved to the end
 796                         return true;
 797                 }
 798         }
 799         return false;
 800 }
 801
 802 ex mul::default_overall_coeff(void) const
 803 {
 804         return _ex1;
 805 }
 806
 807 void mul::combine_overall_coeff(const ex & c)
 808 {
 809         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 810         GINAC_ASSERT(is_exactly_a<numeric>(c));
 811         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 812 }
 813
 814 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 815 {
 816         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 817         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 818         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 819         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 820 }
 821
 822 bool mul::can_make_flat(const expair & p) const
 823 {
 824         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 825         // this assertion will probably fail somewhere
 826         // it would require a more careful make_flat, obeying the power laws
 827         // probably should return true only if p.coeff is integer
 828         return ex_to<numeric>(p.coeff).is_equal(_num1);
 829 }
 830
 831 ex mul::expand(unsigned options) const
 832 {
 833         // First, expand the children
 834         epvector * expanded_seqp = expandchildren(options);
 835         const epvector & expanded_seq = (expanded_seqp == NULL) ? seq : *expanded_seqp;
 836
 837         // Now, look for all the factors that are sums and multiply each one out
 838         // with the next one that is found while collecting the factors which are
 839         // not sums
 840         int number_of_adds = 0;
 841         ex last_expanded = _ex1;
 842         epvector non_adds;
 843         non_adds.reserve(expanded_seq.size());
 844         epvector::const_iterator cit = expanded_seq.begin(), last = expanded_seq.end();
 845         while (cit != last) {
 846                 if (is_exactly_a<add>(cit->rest) &&
 847                         (cit->coeff.is_equal(_ex1))) {
 848                         ++number_of_adds;
 849                         if (is_exactly_a<add>(last_expanded)) {
 850
 851                                 // Expand a product of two sums, aggressive version.
 852                                 // Caring for the overall coefficients in separate loops can
 853                                 // sometimes give a performance gain of up to 15%!
 854
 855                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
 856                                 // add2 is for the inner loop and should be the bigger of the two sums
 857                                 // in the presence of asymptotically good sorting:
 858                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
 859                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
 860                                 const epvector::const_iterator add1begin = add1.seq.begin();
 861                                 const epvector::const_iterator add1end   = add1.seq.end();
 862                                 const epvector::const_iterator add2begin = add2.seq.begin();
 863                                 const epvector::const_iterator add2end   = add2.seq.end();
 864                                 epvector distrseq;
 865                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
 866                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
 867                                 if (!add1.overall_coeff.is_zero()) {
 868                                         if (add1.overall_coeff.is_equal(_ex1))
 869                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
 870                                         else
 871                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
 872                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
 873                                 }
 874                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
 875                                 if (!add2.overall_coeff.is_zero()) {
 876                                         if (add2.overall_coeff.is_equal(_ex1))
 877                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
 878                                         else
 879                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
 880                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
 881                                 }
 882                                 // Compute the new overall coefficient and put it together:
 883                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
 884                                 // Multiply explicitly all non-numeric terms of add1 and add2:
 885                                 for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
 886                                         // We really have to combine terms here in order to compactify
 887                                         // the result.  Otherwise it would become waayy tooo bigg.
 888                                         numeric oc;
 889                                         distrseq.clear();
 890                                         for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
 891                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
 892                                                 // since that would violate an invariant of expairseq:
 893                                                 const ex rest = (new mul(i1->rest, i2->rest))->setflag(status_flags::dynallocated);
 894                                                 if (is_exactly_a<numeric>(rest))
 895                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
 896                                                 else
 897                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
 898                                         }
 899                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
 900                                 }
 901                                 last_expanded = tmp_accu;
 902
 903                         } else {
 904                                 non_adds.push_back(split_ex_to_pair(last_expanded));
 905                                 last_expanded = cit->rest;
 906                         }
 907                 } else {
 908                         non_adds.push_back(*cit);
 909                 }
 910                 ++cit;
 911         }
 912         if (expanded_seqp)
 913                 delete expanded_seqp;
 914
 915         // Now the only remaining thing to do is to multiply the factors which
 916         // were not sums into the "last_expanded" sum
 917         if (is_exactly_a<add>(last_expanded)) {
 918                 const add & finaladd = ex_to<add>(last_expanded);
 919                 exvector distrseq;
 920                 size_t n = finaladd.nops();
 921                 distrseq.reserve(n);
 922                 for (size_t i=0; i<n; ++i) {
 923                         epvector factors = non_adds;
 924                         factors.push_back(split_ex_to_pair(finaladd.op(i)));
 925                         distrseq.push_back((new mul(factors, overall_coeff))->
 926                                             setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
 927                 }
 928                 return ((new add(distrseq))->
 929                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
 930         }
 931         non_adds.push_back(split_ex_to_pair(last_expanded));
 932         return (new mul(non_adds, overall_coeff))->
 933                 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 934 }
 935
 936
 937 //////////
 938 // new virtual functions which can be overridden by derived classes
 939 //////////
 940
 941 // none
 942
 943 //////////
 944 // non-virtual functions in this class
 945 //////////
 946
 947
 948 /** Member-wise expand the expairs representing this sequence.  This must be
 949  *  overridden from expairseq::expandchildren() and done iteratively in order
 950  *  to allow for early cancallations and thus safe memory.
 951  *
 952  *  @see mul::expand()
 953  *  @return pointer to epvector containing expanded representation or zero
 954  *  pointer, if sequence is unchanged. */
 955 epvector * mul::expandchildren(unsigned options) const
 956 {
 957         const epvector::const_iterator last = seq.end();
 958         epvector::const_iterator cit = seq.begin();
 959         while (cit!=last) {
 960                 const ex & factor = recombine_pair_to_ex(*cit);
 961                 const ex & expanded_factor = factor.expand(options);
 962                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
 963
 964                         // something changed, copy seq, eval and return it
 965                         epvector *s = new epvector;
 966                         s->reserve(seq.size());
 967
 968                         // copy parts of seq which are known not to have changed
 969                         epvector::const_iterator cit2 = seq.begin();
 970                         while (cit2!=cit) {
 971                                 s->push_back(*cit2);
 972                                 ++cit2;
 973                         }
 974                         // copy first changed element
 975                         s->push_back(split_ex_to_pair(expanded_factor));
 976                         ++cit2;
 977                         // copy rest
 978                         while (cit2!=last) {
 979                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
 980                                 ++cit2;
 981                         }
 982                         return s;
 983                 }
 984                 ++cit;
 985         }
 986
 987         return 0; // nothing has changed
 988 }
 989
 990 } // namespace GiNaC