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