ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2006 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  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 "color.h"
  31 #include "clifford.h"
  32 #include "power.h"
  33 #include "operators.h"
  34 #include "matrix.h"
  35 #include "indexed.h"
  36 #include "lst.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
  43   print_func<print_context>(&mul::do_print).
  44   print_func<print_latex>(&mul::do_print_latex).
  45   print_func<print_csrc>(&mul::do_print_csrc).
  46   print_func<print_tree>(&mul::do_print_tree).
  47   print_func<print_python_repr>(&mul::do_print_python_repr))
  48
  49
  50 //////////
  51 // default constructor
  52 //////////
  53
  54 mul::mul()
  55 {
  56         tinfo_key = &mul::tinfo_static;
  57 }
  58
  59 //////////
  60 // other constructors
  61 //////////
  62
  63 // public
  64
  65 mul::mul(const ex & lh, const ex & rh)
  66 {
  67         tinfo_key = &mul::tinfo_static;
  68         overall_coeff = _ex1;
  69         construct_from_2_ex(lh,rh);
  70         GINAC_ASSERT(is_canonical());
  71 }
  72
  73 mul::mul(const exvector & v)
  74 {
  75         tinfo_key = &mul::tinfo_static;
  76         overall_coeff = _ex1;
  77         construct_from_exvector(v);
  78         GINAC_ASSERT(is_canonical());
  79 }
  80
  81 mul::mul(const epvector & v)
  82 {
  83         tinfo_key = &mul::tinfo_static;
  84         overall_coeff = _ex1;
  85         construct_from_epvector(v);
  86         GINAC_ASSERT(is_canonical());
  87 }
  88
  89 mul::mul(const epvector & v, const ex & oc)
  90 {
  91         tinfo_key = &mul::tinfo_static;
  92         overall_coeff = oc;
  93         construct_from_epvector(v);
  94         GINAC_ASSERT(is_canonical());
  95 }
  96
  97 mul::mul(std::auto_ptr<epvector> vp, const ex & oc)
  98 {
  99         tinfo_key = &mul::tinfo_static;
 100         GINAC_ASSERT(vp.get()!=0);
 101         overall_coeff = oc;
 102         construct_from_epvector(*vp);
 103         GINAC_ASSERT(is_canonical());
 104 }
 105
 106 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 107 {
 108         tinfo_key = &mul::tinfo_static;
 109         exvector factors;
 110         factors.reserve(3);
 111         factors.push_back(lh);
 112         factors.push_back(mh);
 113         factors.push_back(rh);
 114         overall_coeff = _ex1;
 115         construct_from_exvector(factors);
 116         GINAC_ASSERT(is_canonical());
 117 }
 118
 119 //////////
 120 // archiving
 121 //////////
 122
 123 DEFAULT_ARCHIVING(mul)
 124
 125 //////////
 126 // functions overriding virtual functions from base classes
 127 //////////
 128
 129 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
 130 {
 131         const numeric &coeff = ex_to<numeric>(overall_coeff);
 132         if (coeff.csgn() == -1)
 133                 c.s << '-';
 134         if (!coeff.is_equal(*_num1_p) &&
 135                 !coeff.is_equal(*_num_1_p)) {
 136                 if (coeff.is_rational()) {
 137                         if (coeff.is_negative())
 138                                 (-coeff).print(c);
 139                         else
 140                                 coeff.print(c);
 141                 } else {
 142                         if (coeff.csgn() == -1)
 143                                 (-coeff).print(c, precedence());
 144                         else
 145                                 coeff.print(c, precedence());
 146                 }
 147                 c.s << mul_sym;
 148         }
 149 }
 150
 151 void mul::do_print(const print_context & c, unsigned level) const
 152 {
 153         if (precedence() <= level)
 154                 c.s << '(';
 155
 156         print_overall_coeff(c, "*");
 157
 158         epvector::const_iterator it = seq.begin(), itend = seq.end();
 159         bool first = true;
 160         while (it != itend) {
 161                 if (!first)
 162                         c.s << '*';
 163                 else
 164                         first = false;
 165                 recombine_pair_to_ex(*it).print(c, precedence());
 166                 ++it;
 167         }
 168
 169         if (precedence() <= level)
 170                 c.s << ')';
 171 }
 172
 173 void mul::do_print_latex(const print_latex & c, unsigned level) const
 174 {
 175         if (precedence() <= level)
 176                 c.s << "{(";
 177
 178         print_overall_coeff(c, " ");
 179
 180         // Separate factors into those with negative numeric exponent
 181         // and all others
 182         epvector::const_iterator it = seq.begin(), itend = seq.end();
 183         exvector neg_powers, others;
 184         while (it != itend) {
 185                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
 186                 if (ex_to<numeric>(it->coeff).is_negative())
 187                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
 188                 else
 189                         others.push_back(recombine_pair_to_ex(*it));
 190                 ++it;
 191         }
 192
 193         if (!neg_powers.empty()) {
 194
 195                 // Factors with negative exponent are printed as a fraction
 196                 c.s << "\\frac{";
 197                 mul(others).eval().print(c);
 198                 c.s << "}{";
 199                 mul(neg_powers).eval().print(c);
 200                 c.s << "}";
 201
 202         } else {
 203
 204                 // All other factors are printed in the ordinary way
 205                 exvector::const_iterator vit = others.begin(), vitend = others.end();
 206                 while (vit != vitend) {
 207                         c.s << ' ';
 208                         vit->print(c, precedence());
 209                         ++vit;
 210                 }
 211         }
 212
 213         if (precedence() <= level)
 214                 c.s << ")}";
 215 }
 216
 217 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
 218 {
 219         if (precedence() <= level)
 220                 c.s << "(";
 221
 222         if (!overall_coeff.is_equal(_ex1)) {
 223                 overall_coeff.print(c, precedence());
 224                 c.s << "*";
 225         }
 226
 227         // Print arguments, separated by "*" or "/"
 228         epvector::const_iterator it = seq.begin(), itend = seq.end();
 229         while (it != itend) {
 230
 231                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 232                 bool needclosingparenthesis = false;
 233                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 234                         if (is_a<print_csrc_cl_N>(c)) {
 235                                 c.s << "recip(";
 236                                 needclosingparenthesis = true;
 237                         } else
 238                                 c.s << "1.0/";
 239                 }
 240
 241                 // If the exponent is 1 or -1, it is left out
 242                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 243                         it->rest.print(c, precedence());
 244                 else if (it->coeff.info(info_flags::negint))
 245                         // Outer parens around ex needed for broken GCC parser:
 246                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 247                 else
 248                         // Outer parens around ex needed for broken GCC parser:
 249                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 250
 251                 if (needclosingparenthesis)
 252                         c.s << ")";
 253
 254                 // Separator is "/" for negative integer powers, "*" otherwise
 255                 ++it;
 256                 if (it != itend) {
 257                         if (it->coeff.info(info_flags::negint))
 258                                 c.s << "/";
 259                         else
 260                                 c.s << "*";
 261                 }
 262         }
 263
 264         if (precedence() <= level)
 265                 c.s << ")";
 266 }
 267
 268 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 269 {
 270         c.s << class_name() << '(';
 271         op(0).print(c);
 272         for (size_t i=1; i<nops(); ++i) {
 273                 c.s << ',';
 274                 op(i).print(c);
 275         }
 276         c.s << ')';
 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         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 385         if (evaled_seqp.get()) {
 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_p)) {
 428                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 429                 const add & addref = ex_to<add>((*seq.begin()).rest);
 430                 std::auto_ptr<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         std::auto_ptr<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() 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         std::auto_ptr<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 /** Checks wheter e matches to the pattern pat and the (possibly to be updated
 565   * list of replacements repls. This matching is in the sense of algebraic
 566   * substitutions. Matching starts with pat.op(factor) of the pattern because
 567   * the factors before this one have already been matched. The (possibly
 568   * updated) number of matches is in nummatches. subsed[i] is true for factors
 569   * that already have been replaced by previous substitutions and matched[i]
 570   * is true for factors that have been matched by the current match.
 571   */
 572 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
 573                 int factor, int &nummatches, const std::vector<bool> &subsed,
 574                 std::vector<bool> &matched)
 575 {
 576         if (factor == pat.nops())
 577                 return true;
 578
 579         for (size_t i=0; i<e.nops(); ++i) {
 580                 if(subsed[i] || matched[i])
 581                         continue;
 582                 lst newrepls = repls;
 583                 int newnummatches = nummatches;
 584                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 585                         matched[i] = true;
 586                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 587                                         newnummatches, subsed, matched)) {
 588                                 repls = newrepls;
 589                                 nummatches = newnummatches;
 590                                 return true;
 591                         }
 592                         else
 593                                 matched[i] = false;
 594                 }
 595         }
 596
 597         return false;
 598 }
 599
 600 bool mul::has(const ex & pattern, unsigned options) const
 601 {
 602         if(!(options&has_options::algebraic))
 603                 return basic::has(pattern,options);
 604         if(is_a<mul>(pattern)) {
 605                 lst repls;
 606                 int nummatches = std::numeric_limits<int>::max();
 607                 std::vector<bool> subsed(seq.size(), false);
 608                 std::vector<bool> matched(seq.size(), false);
 609                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 610                                 subsed, matched))
 611                         return true;
 612         }
 613         return basic::has(pattern, options);
 614 }
 615
 616 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 617 {
 618         std::vector<bool> subsed(seq.size(), false);
 619         exvector subsresult(seq.size());
 620
 621         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 622
 623                 if (is_exactly_a<mul>(it->first)) {
 624 retry1:
 625                         int nummatches = std::numeric_limits<int>::max();
 626                         std::vector<bool> currsubsed(seq.size(), false);
 627                         bool succeed = true;
 628                         lst repls;
 629
 630                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 631                                 continue;
 632
 633                         bool foundfirstsubsedfactor = false;
 634                         for (size_t j=0; j<subsed.size(); j++) {
 635                                 if (currsubsed[j]) {
 636                                         if (foundfirstsubsedfactor)
 637                                                 subsresult[j] = op(j);
 638                                         else {
 639                                                 foundfirstsubsedfactor = true;
 640                                                 subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
 641                                         }
 642                                         subsed[j] = true;
 643                                 }
 644                         }
 645                         goto retry1;
 646
 647                 } else {
 648 retry2:
 649                         int nummatches = std::numeric_limits<int>::max();
 650                         lst repls;
 651
 652                         for (size_t j=0; j<this->nops(); j++) {
 653                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)) {
 654                                         subsed[j] = true;
 655                                         subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
 656                                         goto retry2;
 657                                 }
 658                         }
 659                 }
 660         }
 661
 662         bool subsfound = false;
 663         for (size_t i=0; i<subsed.size(); i++) {
 664                 if (subsed[i]) {
 665                         subsfound = true;
 666                         break;
 667                 }
 668         }
 669         if (!subsfound)
 670                 return subs_one_level(m, options | subs_options::algebraic);
 671
 672         exvector ev; ev.reserve(nops());
 673         for (size_t i=0; i<nops(); i++) {
 674                 if (subsed[i])
 675                         ev.push_back(subsresult[i]);
 676                 else
 677                         ev.push_back(op(i));
 678         }
 679
 680         return (new mul(ev))->setflag(status_flags::dynallocated);
 681 }
 682
 683 // protected
 684
 685 /** Implementation of ex::diff() for a product.  It applies the product rule.
 686  *  @see ex::diff */
 687 ex mul::derivative(const symbol & s) const
 688 {
 689         size_t num = seq.size();
 690         exvector addseq;
 691         addseq.reserve(num);
 692
 693         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 694         epvector mulseq = seq;
 695         epvector::const_iterator i = seq.begin(), end = seq.end();
 696         epvector::iterator i2 = mulseq.begin();
 697         while (i != end) {
 698                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 699                                              i->rest.diff(s));
 700                 ep.swap(*i2);
 701                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 702                 ep.swap(*i2);
 703                 ++i; ++i2;
 704         }
 705         return (new add(addseq))->setflag(status_flags::dynallocated);
 706 }
 707
 708 int mul::compare_same_type(const basic & other) const
 709 {
 710         return inherited::compare_same_type(other);
 711 }
 712
 713 unsigned mul::return_type() const
 714 {
 715         if (seq.empty()) {
 716                 // mul without factors: should not happen, but commutates
 717                 return return_types::commutative;
 718         }
 719
 720         bool all_commutative = true;
 721         epvector::const_iterator noncommutative_element; // point to first found nc element
 722
 723         epvector::const_iterator i = seq.begin(), end = seq.end();
 724         while (i != end) {
 725                 unsigned rt = i->rest.return_type();
 726                 if (rt == return_types::noncommutative_composite)
 727                         return rt; // one ncc -> mul also ncc
 728                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 729                         // first nc element found, remember position
 730                         noncommutative_element = i;
 731                         all_commutative = false;
 732                 }
 733                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 734                         // another nc element found, compare type_infos
 735                         if (noncommutative_element->rest.return_type_tinfo()->tinfo() == &clifford::tinfo_static) {
 736                                 if (i->rest.return_type_tinfo()->tinfo() != &clifford::tinfo_static ||
 737                                     ((clifford*)(noncommutative_element->rest.return_type_tinfo()))->get_representation_label() !=
 738                                     ((clifford*)(i->rest.return_type_tinfo()))->get_representation_label()) {
 739                                         // diffent types -> mul is ncc
 740                                         return return_types::noncommutative_composite;
 741                                 }
 742                         } else if (noncommutative_element->rest.return_type_tinfo()->tinfo() == &color::tinfo_static) {
 743                                 if (i->rest.return_type_tinfo()->tinfo() != &color::tinfo_static ||
 744                                     ((color*)(noncommutative_element->rest.return_type_tinfo()))->get_representation_label() !=
 745                                     ((color*)(i->rest.return_type_tinfo()))->get_representation_label()) {
 746                                         // diffent types -> mul is ncc
 747                                         return return_types::noncommutative_composite;
 748                                 }
 749                         } else if (noncommutative_element->rest.return_type_tinfo()->tinfo() != i->rest.return_type_tinfo()->tinfo()) {
 750                                         return return_types::noncommutative_composite;
 751                         }
 752                 }
 753                 ++i;
 754         }
 755         // all factors checked
 756         return all_commutative ? return_types::commutative : return_types::noncommutative;
 757 }
 758
 759 const basic* mul::return_type_tinfo() const
 760 {
 761         if (seq.empty())
 762                 return this;  // mul without factors: should not happen
 763
 764         // return type_info of first noncommutative element
 765         epvector::const_iterator i = seq.begin(), end = seq.end();
 766         while (i != end) {
 767                 if (i->rest.return_type() == return_types::noncommutative)
 768                         return i->rest.return_type_tinfo();
 769                 ++i;
 770         }
 771         // no noncommutative element found, should not happen
 772         return this;
 773 }
 774
 775 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
 776 {
 777         return (new mul(v, oc))->setflag(status_flags::dynallocated);
 778 }
 779
 780 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc) const
 781 {
 782         return (new mul(vp, oc))->setflag(status_flags::dynallocated);
 783 }
 784
 785 expair mul::split_ex_to_pair(const ex & e) const
 786 {
 787         if (is_exactly_a<power>(e)) {
 788                 const power & powerref = ex_to<power>(e);
 789                 if (is_exactly_a<numeric>(powerref.exponent))
 790                         return expair(powerref.basis,powerref.exponent);
 791         }
 792         return expair(e,_ex1);
 793 }
 794
 795 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 796                                           const ex & c) const
 797 {
 798         // to avoid duplication of power simplification rules,
 799         // we create a temporary power object
 800         // otherwise it would be hard to correctly evaluate
 801         // expression like (4^(1/3))^(3/2)
 802         if (c.is_equal(_ex1))
 803                 return split_ex_to_pair(e);
 804
 805         return split_ex_to_pair(power(e,c));
 806 }
 807
 808 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 809                                             const ex & c) const
 810 {
 811         // to avoid duplication of power simplification rules,
 812         // we create a temporary power object
 813         // otherwise it would be hard to correctly evaluate
 814         // expression like (4^(1/3))^(3/2)
 815         if (c.is_equal(_ex1))
 816                 return p;
 817
 818         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 819 }
 820
 821 ex mul::recombine_pair_to_ex(const expair & p) const
 822 {
 823         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 824                 return p.rest;
 825         else
 826                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 827 }
 828
 829 bool mul::expair_needs_further_processing(epp it)
 830 {
 831         if (is_exactly_a<mul>(it->rest) &&
 832                 ex_to<numeric>(it->coeff).is_integer()) {
 833                 // combined pair is product with integer power -> expand it
 834                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 835                 return true;
 836         }
 837         if (is_exactly_a<numeric>(it->rest)) {
 838                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 839                 if (!ep.is_equal(*it)) {
 840                         // combined pair is a numeric power which can be simplified
 841                         *it = ep;
 842                         return true;
 843                 }
 844                 if (it->coeff.is_equal(_ex1)) {
 845                         // combined pair has coeff 1 and must be moved to the end
 846                         return true;
 847                 }
 848         }
 849         return false;
 850 }
 851
 852 ex mul::default_overall_coeff() const
 853 {
 854         return _ex1;
 855 }
 856
 857 void mul::combine_overall_coeff(const ex & c)
 858 {
 859         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 860         GINAC_ASSERT(is_exactly_a<numeric>(c));
 861         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 862 }
 863
 864 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 865 {
 866         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 867         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 868         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 869         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 870 }
 871
 872 bool mul::can_make_flat(const expair & p) const
 873 {
 874         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 875         // this assertion will probably fail somewhere
 876         // it would require a more careful make_flat, obeying the power laws
 877         // probably should return true only if p.coeff is integer
 878         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 879 }
 880
 881 bool mul::can_be_further_expanded(const ex & e)
 882 {
 883         if (is_exactly_a<mul>(e)) {
 884                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 885                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 886                                 return true;
 887                 }
 888         } else if (is_exactly_a<power>(e)) {
 889                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 890                         return true;
 891         }
 892         return false;
 893 }
 894
 895 ex mul::expand(unsigned options) const
 896 {
 897         // First, expand the children
 898         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
 899         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
 900
 901         // Now, look for all the factors that are sums and multiply each one out
 902         // with the next one that is found while collecting the factors which are
 903         // not sums
 904         ex last_expanded = _ex1;
 905
 906         epvector non_adds;
 907         non_adds.reserve(expanded_seq.size());
 908
 909         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
 910                 if (is_exactly_a<add>(cit->rest) &&
 911                         (cit->coeff.is_equal(_ex1))) {
 912                         if (is_exactly_a<add>(last_expanded)) {
 913
 914                                 // Expand a product of two sums, aggressive version.
 915                                 // Caring for the overall coefficients in separate loops can
 916                                 // sometimes give a performance gain of up to 15%!
 917
 918                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
 919                                 // add2 is for the inner loop and should be the bigger of the two sums
 920                                 // in the presence of asymptotically good sorting:
 921                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
 922                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
 923                                 const epvector::const_iterator add1begin = add1.seq.begin();
 924                                 const epvector::const_iterator add1end   = add1.seq.end();
 925                                 const epvector::const_iterator add2begin = add2.seq.begin();
 926                                 const epvector::const_iterator add2end   = add2.seq.end();
 927                                 epvector distrseq;
 928                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
 929
 930                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
 931                                 if (!add1.overall_coeff.is_zero()) {
 932                                         if (add1.overall_coeff.is_equal(_ex1))
 933                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
 934                                         else
 935                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
 936                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
 937                                 }
 938
 939                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
 940                                 if (!add2.overall_coeff.is_zero()) {
 941                                         if (add2.overall_coeff.is_equal(_ex1))
 942                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
 943                                         else
 944                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
 945                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
 946                                 }
 947
 948                                 // Compute the new overall coefficient and put it together:
 949                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
 950
 951                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
 952
 953                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
 954                                         add_indices = get_all_dummy_indices(i->rest);
 955                                         add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
 956                                 }
 957                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
 958                                         add_indices = get_all_dummy_indices(i->rest);
 959                                         add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
 960                                 }
 961
 962                                 sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
 963                                 sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
 964                                 lst dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
 965
 966                                 // Multiply explicitly all non-numeric terms of add1 and add2:
 967                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
 968                                         // We really have to combine terms here in order to compactify
 969                                         // the result.  Otherwise it would become waayy tooo bigg.
 970                                         numeric oc;
 971                                         distrseq.clear();
 972                                         ex i2_new = (dummy_subs.op(0).nops()>0?
 973                                                                  i2->rest.subs((lst)dummy_subs.op(0), (lst)dummy_subs.op(1), subs_options::no_pattern) : i2->rest);
 974                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
 975                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
 976                                                 // since that would violate an invariant of expairseq:
 977                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
 978                                                 if (is_exactly_a<numeric>(rest)) {
 979                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
 980                                                 } else {
 981                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
 982                                                 }
 983                                         }
 984                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
 985                                 }
 986                                 last_expanded = tmp_accu;
 987
 988                         } else {
 989                                 if (!last_expanded.is_equal(_ex1))
 990                                         non_adds.push_back(split_ex_to_pair(last_expanded));
 991                                 last_expanded = cit->rest;
 992                         }
 993
 994                 } else {
 995                         non_adds.push_back(*cit);
 996                 }
 997         }
 998
 999         // Now the only remaining thing to do is to multiply the factors which
1000         // were not sums into the "last_expanded" sum
1001         if (is_exactly_a<add>(last_expanded)) {
1002                 size_t n = last_expanded.nops();
1003                 exvector distrseq;
1004                 distrseq.reserve(n);
1005                 exvector va = get_all_dummy_indices(mul(non_adds));
1006                 sort(va.begin(), va.end(), ex_is_less());
1007
1008                 for (size_t i=0; i<n; ++i) {
1009                         epvector factors = non_adds;
1010                         factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1011                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1012                         if (can_be_further_expanded(term)) {
1013                                 distrseq.push_back(term.expand());
1014                         } else {
1015                                 if (options == 0)
1016                                         ex_to<basic>(term).setflag(status_flags::expanded);
1017                                 distrseq.push_back(term);
1018                         }
1019                 }
1020
1021                 return ((new add(distrseq))->
1022                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1023         }
1024
1025         non_adds.push_back(split_ex_to_pair(last_expanded));
1026         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1027         if (can_be_further_expanded(result)) {
1028                 return result.expand();
1029         } else {
1030                 if (options == 0)
1031                         ex_to<basic>(result).setflag(status_flags::expanded);
1032                 return result;
1033         }
1034 }
1035
1036
1037 //////////
1038 // new virtual functions which can be overridden by derived classes
1039 //////////
1040
1041 // none
1042
1043 //////////
1044 // non-virtual functions in this class
1045 //////////
1046
1047
1048 /** Member-wise expand the expairs representing this sequence.  This must be
1049  *  overridden from expairseq::expandchildren() and done iteratively in order
1050  *  to allow for early cancallations and thus safe memory.
1051  *
1052  *  @see mul::expand()
1053  *  @return pointer to epvector containing expanded representation or zero
1054  *  pointer, if sequence is unchanged. */
1055 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1056 {
1057         const epvector::const_iterator last = seq.end();
1058         epvector::const_iterator cit = seq.begin();
1059         while (cit!=last) {
1060                 const ex & factor = recombine_pair_to_ex(*cit);
1061                 const ex & expanded_factor = factor.expand(options);
1062                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1063
1064                         // something changed, copy seq, eval and return it
1065                         std::auto_ptr<epvector> s(new epvector);
1066                         s->reserve(seq.size());
1067
1068                         // copy parts of seq which are known not to have changed
1069                         epvector::const_iterator cit2 = seq.begin();
1070                         while (cit2!=cit) {
1071                                 s->push_back(*cit2);
1072                                 ++cit2;
1073                         }
1074
1075                         // copy first changed element
1076                         s->push_back(split_ex_to_pair(expanded_factor));
1077                         ++cit2;
1078
1079                         // copy rest
1080                         while (cit2!=last) {
1081                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1082                                 ++cit2;
1083                         }
1084                         return s;
1085                 }
1086                 ++cit;
1087         }
1088
1089         return std::auto_ptr<epvector>(0); // nothing has changed
1090 }
1091
1092 } // namespace GiNaC