ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2020 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 "mul.h"
  24 #include "add.h"
  25 #include "power.h"
  26 #include "operators.h"
  27 #include "matrix.h"
  28 #include "indexed.h"
  29 #include "lst.h"
  30 #include "archive.h"
  31 #include "utils.h"
  32 #include "symbol.h"
  33 #include "compiler.h"
  34
  35 #include <iostream>
  36 #include <limits>
  37 #include <stdexcept>
  38 #include <vector>
  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 }
  57
  58 //////////
  59 // other constructors
  60 //////////
  61
  62 // public
  63
  64 mul::mul(const ex & lh, const ex & rh)
  65 {
  66         overall_coeff = _ex1;
  67         construct_from_2_ex(lh,rh);
  68         GINAC_ASSERT(is_canonical());
  69 }
  70
  71 mul::mul(const exvector & v)
  72 {
  73         overall_coeff = _ex1;
  74         construct_from_exvector(v);
  75         GINAC_ASSERT(is_canonical());
  76 }
  77
  78 mul::mul(const epvector & v)
  79 {
  80         overall_coeff = _ex1;
  81         construct_from_epvector(v);
  82         GINAC_ASSERT(is_canonical());
  83 }
  84
  85 mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
  86 {
  87         overall_coeff = oc;
  88         construct_from_epvector(v, do_index_renaming);
  89         GINAC_ASSERT(is_canonical());
  90 }
  91
  92 mul::mul(epvector && vp)
  93 {
  94         overall_coeff = _ex1;
  95         construct_from_epvector(std::move(vp));
  96         GINAC_ASSERT(is_canonical());
  97 }
  98
  99 mul::mul(epvector && vp, const ex & oc, bool do_index_renaming)
 100 {
 101         overall_coeff = oc;
 102         construct_from_epvector(std::move(vp), do_index_renaming);
 103         GINAC_ASSERT(is_canonical());
 104 }
 105
 106 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 107 {
 108         exvector factors;
 109         factors.reserve(3);
 110         factors.push_back(lh);
 111         factors.push_back(mh);
 112         factors.push_back(rh);
 113         overall_coeff = _ex1;
 114         construct_from_exvector(factors);
 115         GINAC_ASSERT(is_canonical());
 116 }
 117
 118 //////////
 119 // archiving
 120 //////////
 121
 122 //////////
 123 // functions overriding virtual functions from base classes
 124 //////////
 125
 126 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
 127 {
 128         const numeric &coeff = ex_to<numeric>(overall_coeff);
 129         if (coeff.csgn() == -1)
 130                 c.s << '-';
 131         if (!coeff.is_equal(*_num1_p) &&
 132                 !coeff.is_equal(*_num_1_p)) {
 133                 if (coeff.is_rational()) {
 134                         if (coeff.is_negative())
 135                                 (-coeff).print(c);
 136                         else
 137                                 coeff.print(c);
 138                 } else {
 139                         if (coeff.csgn() == -1)
 140                                 (-coeff).print(c, precedence());
 141                         else
 142                                 coeff.print(c, precedence());
 143                 }
 144                 c.s << mul_sym;
 145         }
 146 }
 147
 148 void mul::do_print(const print_context & c, unsigned level) const
 149 {
 150         if (precedence() <= level)
 151                 c.s << '(';
 152
 153         print_overall_coeff(c, "*");
 154
 155         bool first = true;
 156         for (auto & it : seq) {
 157                 if (!first)
 158                         c.s << '*';
 159                 else
 160                         first = false;
 161                 recombine_pair_to_ex(it).print(c, precedence());
 162         }
 163
 164         if (precedence() <= level)
 165                 c.s << ')';
 166 }
 167
 168 void mul::do_print_latex(const print_latex & c, unsigned level) const
 169 {
 170         if (precedence() <= level)
 171                 c.s << "{(";
 172
 173         print_overall_coeff(c, " ");
 174
 175         // Separate factors into those with negative numeric exponent
 176         // and all others
 177         exvector neg_powers, others;
 178         for (auto & it : seq) {
 179                 GINAC_ASSERT(is_exactly_a<numeric>(it.coeff));
 180                 if (ex_to<numeric>(it.coeff).is_negative())
 181                         neg_powers.push_back(recombine_pair_to_ex(expair(it.rest, -it.coeff)));
 182                 else
 183                         others.push_back(recombine_pair_to_ex(it));
 184         }
 185
 186         if (!neg_powers.empty()) {
 187
 188                 // Factors with negative exponent are printed as a fraction
 189                 c.s << "\\frac{";
 190                 mul(others).eval().print(c);
 191                 c.s << "}{";
 192                 mul(neg_powers).eval().print(c);
 193                 c.s << "}";
 194
 195         } else {
 196
 197                 // All other factors are printed in the ordinary way
 198                 for (auto & vit : others) {
 199                         c.s << ' ';
 200                         vit.print(c, precedence());
 201                 }
 202         }
 203
 204         if (precedence() <= level)
 205                 c.s << ")}";
 206 }
 207
 208 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
 209 {
 210         if (precedence() <= level)
 211                 c.s << "(";
 212
 213         if (!overall_coeff.is_equal(_ex1)) {
 214                 if (overall_coeff.is_equal(_ex_1))
 215                         c.s << "-";
 216                 else {
 217                         overall_coeff.print(c, precedence());
 218                         c.s << "*";
 219                 }
 220         }
 221
 222         // Print arguments, separated by "*" or "/"
 223         auto it = seq.begin(), itend = seq.end();
 224         while (it != itend) {
 225
 226                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 227                 bool needclosingparenthesis = false;
 228                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 229                         if (is_a<print_csrc_cl_N>(c)) {
 230                                 c.s << "recip(";
 231                                 needclosingparenthesis = true;
 232                         } else
 233                                 c.s << "1.0/";
 234                 }
 235
 236                 // If the exponent is 1 or -1, it is left out
 237                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 238                         it->rest.print(c, precedence());
 239                 else if (it->coeff.info(info_flags::negint))
 240                         ex(power(it->rest, -ex_to<numeric>(it->coeff))).print(c, level);
 241                 else
 242                         ex(power(it->rest, ex_to<numeric>(it->coeff))).print(c, level);
 243
 244                 if (needclosingparenthesis)
 245                         c.s << ")";
 246
 247                 // Separator is "/" for negative integer powers, "*" otherwise
 248                 ++it;
 249                 if (it != itend) {
 250                         if (it->coeff.info(info_flags::negint))
 251                                 c.s << "/";
 252                         else
 253                                 c.s << "*";
 254                 }
 255         }
 256
 257         if (precedence() <= level)
 258                 c.s << ")";
 259 }
 260
 261 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 262 {
 263         c.s << class_name() << '(';
 264         op(0).print(c);
 265         for (size_t i=1; i<nops(); ++i) {
 266                 c.s << ',';
 267                 op(i).print(c);
 268         }
 269         c.s << ')';
 270 }
 271
 272 bool mul::info(unsigned inf) const
 273 {
 274         switch (inf) {
 275                 case info_flags::polynomial:
 276                 case info_flags::integer_polynomial:
 277                 case info_flags::cinteger_polynomial:
 278                 case info_flags::rational_polynomial:
 279                 case info_flags::real:
 280                 case info_flags::rational:
 281                 case info_flags::integer:
 282                 case info_flags::crational:
 283                 case info_flags::cinteger:
 284                 case info_flags::even:
 285                 case info_flags::crational_polynomial:
 286                 case info_flags::rational_function: {
 287                         for (auto & it : seq) {
 288                                 if (!recombine_pair_to_ex(it).info(inf))
 289                                         return false;
 290                         }
 291                         if (overall_coeff.is_equal(*_num1_p) && inf == info_flags::even)
 292                                 return true;
 293                         return overall_coeff.info(inf);
 294                 }
 295                 case info_flags::positive:
 296                 case info_flags::negative: {
 297                         if ((inf==info_flags::positive) && (flags & status_flags::is_positive))
 298                                 return true;
 299                         else if ((inf==info_flags::negative) && (flags & status_flags::is_negative))
 300                                 return true;
 301                         if (flags & status_flags::purely_indefinite)
 302                                 return false;
 303
 304                         bool pos = true;
 305                         for (auto & it : seq) {
 306                                 const ex& factor = recombine_pair_to_ex(it);
 307                                 if (factor.info(info_flags::positive))
 308                                         continue;
 309                                 else if (factor.info(info_flags::negative))
 310                                         pos = !pos;
 311                                 else
 312                                         return false;
 313                         }
 314                         if (overall_coeff.info(info_flags::negative))
 315                                 pos = !pos;
 316                         setflag(pos ? status_flags::is_positive : status_flags::is_negative);
 317                         return (inf == info_flags::positive? pos : !pos);
 318                 }
 319                 case info_flags::nonnegative: {
 320                         if  (flags & status_flags::is_positive)
 321                                 return true;
 322                         bool pos = true;
 323                         for (auto & it : seq) {
 324                                 const ex& factor = recombine_pair_to_ex(it);
 325                                 if (factor.info(info_flags::nonnegative) || factor.info(info_flags::positive))
 326                                         continue;
 327                                 else if (factor.info(info_flags::negative))
 328                                         pos = !pos;
 329                                 else
 330                                         return false;
 331                         }
 332                         return (overall_coeff.info(info_flags::negative)? !pos : pos);
 333                 }
 334                 case info_flags::posint:
 335                 case info_flags::negint: {
 336                         bool pos = true;
 337                         for (auto & it : seq) {
 338                                 const ex& factor = recombine_pair_to_ex(it);
 339                                 if (factor.info(info_flags::posint))
 340                                         continue;
 341                                 else if (factor.info(info_flags::negint))
 342                                         pos = !pos;
 343                                 else
 344                                         return false;
 345                         }
 346                         if (overall_coeff.info(info_flags::negint))
 347                                 pos = !pos;
 348                         else if (!overall_coeff.info(info_flags::posint))
 349                                 return false;
 350                         return (inf ==info_flags::posint? pos : !pos);
 351                 }
 352                 case info_flags::nonnegint: {
 353                         bool pos = true;
 354                         for (auto & it : seq) {
 355                                 const ex& factor = recombine_pair_to_ex(it);
 356                                 if (factor.info(info_flags::nonnegint) || factor.info(info_flags::posint))
 357                                         continue;
 358                                 else if (factor.info(info_flags::negint))
 359                                         pos = !pos;
 360                                 else
 361                                         return false;
 362                         }
 363                         if (overall_coeff.info(info_flags::negint))
 364                                 pos = !pos;
 365                         else if (!overall_coeff.info(info_flags::posint))
 366                                 return false;
 367                         return pos;
 368                 }
 369                 case info_flags::indefinite: {
 370                         if (flags & status_flags::purely_indefinite)
 371                                 return true;
 372                         if (flags & (status_flags::is_positive | status_flags::is_negative))
 373                                 return false;
 374                         for (auto & it : seq) {
 375                                 const ex& term = recombine_pair_to_ex(it);
 376                                 if (term.info(info_flags::positive) || term.info(info_flags::negative))
 377                                         return false;
 378                         }
 379                         setflag(status_flags::purely_indefinite);
 380                         return true;
 381                 }
 382         }
 383         return inherited::info(inf);
 384 }
 385
 386 bool mul::is_polynomial(const ex & var) const
 387 {
 388         for (auto & it : seq) {
 389                 if (!it.rest.is_polynomial(var) ||
 390                     (it.rest.has(var) && !it.coeff.info(info_flags::nonnegint))) {
 391                         return false;
 392                 }
 393         }
 394         return true;
 395 }
 396
 397 int mul::degree(const ex & s) const
 398 {
 399         // Sum up degrees of factors
 400         int deg_sum = 0;
 401         for (auto & it : seq) {
 402                 if (ex_to<numeric>(it.coeff).is_integer())
 403                         deg_sum += recombine_pair_to_ex(it).degree(s);
 404                 else {
 405                         if (it.rest.has(s))
 406                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
 407                 }
 408         }
 409         return deg_sum;
 410 }
 411
 412 int mul::ldegree(const ex & s) const
 413 {
 414         // Sum up degrees of factors
 415         int deg_sum = 0;
 416         for (auto & it : seq) {
 417                 if (ex_to<numeric>(it.coeff).is_integer())
 418                         deg_sum += recombine_pair_to_ex(it).ldegree(s);
 419                 else {
 420                         if (it.rest.has(s))
 421                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
 422                 }
 423         }
 424         return deg_sum;
 425 }
 426
 427 ex mul::coeff(const ex & s, int n) const
 428 {
 429         exvector coeffseq;
 430         coeffseq.reserve(seq.size()+1);
 431
 432         if (n==0) {
 433                 // product of individual coeffs
 434                 // if a non-zero power of s is found, the resulting product will be 0
 435                 for (auto & it : seq)
 436                         coeffseq.push_back(recombine_pair_to_ex(it).coeff(s,n));
 437                 coeffseq.push_back(overall_coeff);
 438                 return dynallocate<mul>(coeffseq);
 439         }
 440
 441         bool coeff_found = false;
 442         for (auto & it : seq) {
 443                 ex t = recombine_pair_to_ex(it);
 444                 ex c = t.coeff(s, n);
 445                 if (!c.is_zero()) {
 446                         coeffseq.push_back(c);
 447                         coeff_found = 1;
 448                 } else {
 449                         coeffseq.push_back(t);
 450                 }
 451         }
 452         if (coeff_found) {
 453                 coeffseq.push_back(overall_coeff);
 454                 return dynallocate<mul>(coeffseq);
 455         }
 456
 457         return _ex0;
 458 }
 459
 460 /** Perform automatic term rewriting rules in this class.  In the following
 461  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 462  *  stand for such expressions that contain a plain number.
 463  *  - *(...,x;0) -> 0
 464  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 465  *  - *(x;1) -> x
 466  *  - *(;c) -> c
 467  */
 468 ex mul::eval() const
 469 {
 470         if (flags & status_flags::evaluated) {
 471                 GINAC_ASSERT(seq.size()>0);
 472                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 473                 return *this;
 474         }
 475
 476         const epvector evaled = evalchildren();
 477         if (unlikely(!evaled.empty())) {
 478                 // start over evaluating a new object
 479                 return dynallocate<mul>(std::move(evaled), overall_coeff);
 480         }
 481
 482         size_t seq_size = seq.size();
 483         if (overall_coeff.is_zero()) {
 484                 // *(...,x;0) -> 0
 485                 return _ex0;
 486         } else if (seq_size==0) {
 487                 // *(;c) -> c
 488                 return overall_coeff;
 489         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 490                 // *(x;1) -> x
 491                 return recombine_pair_to_ex(*(seq.begin()));
 492         } else if ((seq_size==1) &&
 493                    is_exactly_a<add>((*seq.begin()).rest) &&
 494                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 495                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 496                 const add & addref = ex_to<add>((*seq.begin()).rest);
 497                 epvector distrseq;
 498                 distrseq.reserve(addref.seq.size());
 499                 for (auto & it : addref.seq) {
 500                         distrseq.push_back(addref.combine_pair_with_coeff_to_pair(it, overall_coeff));
 501                 }
 502                 return dynallocate<add>(std::move(distrseq),
 503                                         ex_to<numeric>(addref.overall_coeff).mul_dyn(ex_to<numeric>(overall_coeff)))
 504                         .setflag(status_flags::evaluated);
 505         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
 506                 // Strip the content and the unit part from each term. Thus
 507                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)^2
 508
 509                 auto i = seq.begin(), last = seq.end();
 510                 auto j = seq.begin();
 511                 epvector s;
 512                 numeric oc = *_num1_p;
 513                 bool something_changed = false;
 514                 while (i!=last) {
 515                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
 516                                 // power::eval has such a rule, no need to handle powers here
 517                                 ++i;
 518                                 continue;
 519                         }
 520
 521                         // XXX: What is the best way to check if the polynomial is a primitive?
 522                         numeric c = i->rest.integer_content();
 523                         const numeric lead_coeff =
 524                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
 525                         const bool canonicalizable = lead_coeff.is_integer();
 526
 527                         // XXX: The main variable is chosen in a random way, so this code
 528                         // does NOT transform the term into the canonical form (thus, in some
 529                         // very unlucky event it can even loop forever). Hopefully the main
 530                         // variable will be the same for all terms in *this
 531                         const bool unit_normal = lead_coeff.is_pos_integer();
 532                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
 533                                 ++i;
 534                                 continue;
 535                         }
 536
 537                         if (! something_changed) {
 538                                 s.reserve(seq_size);
 539                                 something_changed = true;
 540                         }
 541
 542                         while ((j!=i) && (j!=last)) {
 543                                 s.push_back(*j);
 544                                 ++j;
 545                         }
 546
 547                         if (! unit_normal)
 548                                 c = c.mul(*_num_1_p);
 549
 550                         oc = oc.mul(c);
 551
 552                         // divide add by the number in place to save at least 2 .eval() calls
 553                         const add& addref = ex_to<add>(i->rest);
 554                         add & primitive = dynallocate<add>(addref);
 555                         primitive.clearflag(status_flags::hash_calculated);
 556                         primitive.overall_coeff = ex_to<numeric>(primitive.overall_coeff).div_dyn(c);
 557                         for (auto & ai : primitive.seq)
 558                                 ai.coeff = ex_to<numeric>(ai.coeff).div_dyn(c);
 559
 560                         s.push_back(expair(primitive, _ex1));
 561
 562                         ++i;
 563                         ++j;
 564                 }
 565                 if (something_changed) {
 566                         while (j!=last) {
 567                                 s.push_back(*j);
 568                                 ++j;
 569                         }
 570                         return dynallocate<mul>(std::move(s), ex_to<numeric>(overall_coeff).mul_dyn(oc));
 571                 }
 572         }
 573
 574         return this->hold();
 575 }
 576
 577 ex mul::evalf() const
 578 {
 579         epvector s;
 580         s.reserve(seq.size());
 581
 582         for (auto & it : seq)
 583                 s.push_back(expair(it.rest.evalf(), it.coeff));
 584         return dynallocate<mul>(std::move(s), overall_coeff.evalf());
 585 }
 586
 587 void mul::find_real_imag(ex & rp, ex & ip) const
 588 {
 589         rp = overall_coeff.real_part();
 590         ip = overall_coeff.imag_part();
 591         for (auto & it : seq) {
 592                 ex factor = recombine_pair_to_ex(it);
 593                 ex new_rp = factor.real_part();
 594                 ex new_ip = factor.imag_part();
 595                 if (new_ip.is_zero()) {
 596                         rp *= new_rp;
 597                         ip *= new_rp;
 598                 } else {
 599                         ex temp = rp*new_rp - ip*new_ip;
 600                         ip = ip*new_rp + rp*new_ip;
 601                         rp = temp;
 602                 }
 603         }
 604         rp = rp.expand();
 605         ip = ip.expand();
 606 }
 607
 608 ex mul::real_part() const
 609 {
 610         ex rp, ip;
 611         find_real_imag(rp, ip);
 612         return rp;
 613 }
 614
 615 ex mul::imag_part() const
 616 {
 617         ex rp, ip;
 618         find_real_imag(rp, ip);
 619         return ip;
 620 }
 621
 622 ex mul::evalm() const
 623 {
 624         // numeric*matrix
 625         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 626          && is_a<matrix>(seq[0].rest))
 627                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 628
 629         // Evaluate children first, look whether there are any matrices at all
 630         // (there can be either no matrices or one matrix; if there were more
 631         // than one matrix, it would be a non-commutative product)
 632         epvector s;
 633         s.reserve(seq.size());
 634
 635         bool have_matrix = false;
 636         epvector::iterator the_matrix;
 637
 638         for (auto & it : seq) {
 639                 const ex &m = recombine_pair_to_ex(it).evalm();
 640                 s.push_back(split_ex_to_pair(m));
 641                 if (is_a<matrix>(m)) {
 642                         have_matrix = true;
 643                         the_matrix = s.end() - 1;
 644                 }
 645         }
 646
 647         if (have_matrix) {
 648
 649                 // The product contained a matrix. We will multiply all other factors
 650                 // into that matrix.
 651                 matrix m = ex_to<matrix>(the_matrix->rest);
 652                 s.erase(the_matrix);
 653                 ex scalar = dynallocate<mul>(std::move(s), overall_coeff);
 654                 return m.mul_scalar(scalar);
 655
 656         } else
 657                 return dynallocate<mul>(std::move(s), overall_coeff);
 658 }
 659
 660 ex mul::eval_ncmul(const exvector & v) const
 661 {
 662         if (seq.empty())
 663                 return inherited::eval_ncmul(v);
 664
 665         // Find first noncommutative element and call its eval_ncmul()
 666         for (auto & it : seq)
 667                 if (it.rest.return_type() == return_types::noncommutative)
 668                         return it.rest.eval_ncmul(v);
 669         return inherited::eval_ncmul(v);
 670 }
 671
 672 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
 673 {
 674         ex origbase;
 675         int origexponent;
 676         int origexpsign;
 677
 678         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 679                 origbase = origfactor.op(0);
 680                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 681                 origexponent = expon > 0 ? expon : -expon;
 682                 origexpsign = expon > 0 ? 1 : -1;
 683         } else {
 684                 origbase = origfactor;
 685                 origexponent = 1;
 686                 origexpsign = 1;
 687         }
 688
 689         ex patternbase;
 690         int patternexponent;
 691         int patternexpsign;
 692
 693         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 694                 patternbase = patternfactor.op(0);
 695                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 696                 patternexponent = expon > 0 ? expon : -expon;
 697                 patternexpsign = expon > 0 ? 1 : -1;
 698         } else {
 699                 patternbase = patternfactor;
 700                 patternexponent = 1;
 701                 patternexpsign = 1;
 702         }
 703
 704         exmap saverepls = repls;
 705         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 706                 return false;
 707         repls = saverepls;
 708
 709         int newnummatches = origexponent / patternexponent;
 710         if (newnummatches < nummatches)
 711                 nummatches = newnummatches;
 712         return true;
 713 }
 714
 715 /** Checks whether e matches to the pattern pat and the (possibly to be updated)
 716   * list of replacements repls. This matching is in the sense of algebraic
 717   * substitutions. Matching starts with pat.op(factor) of the pattern because
 718   * the factors before this one have already been matched. The (possibly
 719   * updated) number of matches is in nummatches. subsed[i] is true for factors
 720   * that already have been replaced by previous substitutions and matched[i]
 721   * is true for factors that have been matched by the current match.
 722   */
 723 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
 724                                   int factor, int &nummatches, const std::vector<bool> &subsed,
 725                                   std::vector<bool> &matched)
 726 {
 727         GINAC_ASSERT(subsed.size() == e.nops());
 728         GINAC_ASSERT(matched.size() == e.nops());
 729
 730         if (factor == (int)pat.nops())
 731                 return true;
 732
 733         for (size_t i=0; i<e.nops(); ++i) {
 734                 if(subsed[i] || matched[i])
 735                         continue;
 736                 exmap newrepls = repls;
 737                 int newnummatches = nummatches;
 738                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 739                         matched[i] = true;
 740                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 741                                         newnummatches, subsed, matched)) {
 742                                 repls = newrepls;
 743                                 nummatches = newnummatches;
 744                                 return true;
 745                         }
 746                         else
 747                                 matched[i] = false;
 748                 }
 749         }
 750
 751         return false;
 752 }
 753
 754 bool mul::has(const ex & pattern, unsigned options) const
 755 {
 756         if(!(options & has_options::algebraic))
 757                 return basic::has(pattern,options);
 758         if(is_a<mul>(pattern)) {
 759                 exmap repls;
 760                 int nummatches = std::numeric_limits<int>::max();
 761                 std::vector<bool> subsed(nops(), false);
 762                 std::vector<bool> matched(nops(), false);
 763                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 764                                 subsed, matched))
 765                         return true;
 766         }
 767         return basic::has(pattern, options);
 768 }
 769
 770 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 771 {
 772         std::vector<bool> subsed(nops(), false);
 773         ex divide_by = 1;
 774         ex multiply_by = 1;
 775
 776         for (auto & it : m) {
 777
 778                 if (is_exactly_a<mul>(it.first)) {
 779 retry1:
 780                         int nummatches = std::numeric_limits<int>::max();
 781                         std::vector<bool> currsubsed(nops(), false);
 782                         exmap repls;
 783
 784                         if (!algebraic_match_mul_with_mul(*this, it.first, repls, 0, nummatches, subsed, currsubsed))
 785                                 continue;
 786
 787                         for (size_t j=0; j<subsed.size(); j++)
 788                                 if (currsubsed[j])
 789                                         subsed[j] = true;
 790                         ex subsed_pattern
 791                                 = it.first.subs(repls, subs_options::no_pattern);
 792                         divide_by *= pow(subsed_pattern, nummatches);
 793                         ex subsed_result
 794                                 = it.second.subs(repls, subs_options::no_pattern);
 795                         multiply_by *= pow(subsed_result, nummatches);
 796                         goto retry1;
 797
 798                 } else {
 799
 800                         for (size_t j=0; j<this->nops(); j++) {
 801                                 int nummatches = std::numeric_limits<int>::max();
 802                                 exmap repls;
 803                                 if (!subsed[j] && tryfactsubs(op(j), it.first, nummatches, repls)){
 804                                         subsed[j] = true;
 805                                         ex subsed_pattern
 806                                                 = it.first.subs(repls, subs_options::no_pattern);
 807                                         divide_by *= pow(subsed_pattern, nummatches);
 808                                         ex subsed_result
 809                                                 = it.second.subs(repls, subs_options::no_pattern);
 810                                         multiply_by *= pow(subsed_result, nummatches);
 811                                 }
 812                         }
 813                 }
 814         }
 815
 816         bool subsfound = false;
 817         for (size_t i=0; i<subsed.size(); i++) {
 818                 if (subsed[i]) {
 819                         subsfound = true;
 820                         break;
 821                 }
 822         }
 823         if (!subsfound)
 824                 return subs_one_level(m, options | subs_options::algebraic);
 825
 826         return ((*this)/divide_by)*multiply_by;
 827 }
 828
 829 ex mul::conjugate() const
 830 {
 831         // The base class' method is wrong here because we have to be careful at
 832         // branch cuts. power::conjugate takes care of that already, so use it.
 833         std::unique_ptr<epvector> newepv(nullptr);
 834         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 835                 if (newepv) {
 836                         newepv->push_back(split_ex_to_pair(recombine_pair_to_ex(*i).conjugate()));
 837                         continue;
 838                 }
 839                 ex x = recombine_pair_to_ex(*i);
 840                 ex c = x.conjugate();
 841                 if (c.is_equal(x)) {
 842                         continue;
 843                 }
 844                 newepv.reset(new epvector);
 845                 newepv->reserve(seq.size());
 846                 for (auto j=seq.begin(); j!=i; ++j) {
 847                         newepv->push_back(*j);
 848                 }
 849                 newepv->push_back(split_ex_to_pair(c));
 850         }
 851         ex x = overall_coeff.conjugate();
 852         if (!newepv && are_ex_trivially_equal(x, overall_coeff)) {
 853                 return *this;
 854         }
 855         return thisexpairseq(newepv ? std::move(*newepv) : seq, x);
 856 }
 857
 858
 859 // protected
 860
 861 /** Implementation of ex::diff() for a product.  It applies the product rule.
 862  *  @see ex::diff */
 863 ex mul::derivative(const symbol & s) const
 864 {
 865         size_t num = seq.size();
 866         exvector addseq;
 867         addseq.reserve(num);
 868
 869         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 870         epvector mulseq = seq;
 871         auto i = seq.begin(), end = seq.end();
 872         auto i2 = mulseq.begin();
 873         while (i != end) {
 874                 expair ep = split_ex_to_pair(pow(i->rest, i->coeff - _ex1) *
 875                                              i->rest.diff(s));
 876                 ep.swap(*i2);
 877                 addseq.push_back(dynallocate<mul>(mulseq, overall_coeff * i->coeff));
 878                 ep.swap(*i2);
 879                 ++i; ++i2;
 880         }
 881         return dynallocate<add>(addseq);
 882 }
 883
 884 int mul::compare_same_type(const basic & other) const
 885 {
 886         return inherited::compare_same_type(other);
 887 }
 888
 889 unsigned mul::return_type() const
 890 {
 891         if (seq.empty()) {
 892                 // mul without factors: should not happen, but commutates
 893                 return return_types::commutative;
 894         }
 895
 896         bool all_commutative = true;
 897         epvector::const_iterator noncommutative_element; // point to first found nc element
 898
 899         epvector::const_iterator i = seq.begin(), end = seq.end();
 900         while (i != end) {
 901                 unsigned rt = i->rest.return_type();
 902                 if (rt == return_types::noncommutative_composite)
 903                         return rt; // one ncc -> mul also ncc
 904                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 905                         // first nc element found, remember position
 906                         noncommutative_element = i;
 907                         all_commutative = false;
 908                 }
 909                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 910                         // another nc element found, compare type_infos
 911                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 912                                         // different types -> mul is ncc
 913                                         return return_types::noncommutative_composite;
 914                         }
 915                 }
 916                 ++i;
 917         }
 918         // all factors checked
 919         return all_commutative ? return_types::commutative : return_types::noncommutative;
 920 }
 921
 922 return_type_t mul::return_type_tinfo() const
 923 {
 924         if (seq.empty())
 925                 return make_return_type_t<mul>(); // mul without factors: should not happen
 926
 927         // return type_info of first noncommutative element
 928         for (auto & it : seq)
 929                 if (it.rest.return_type() == return_types::noncommutative)
 930                         return it.rest.return_type_tinfo();
 931
 932         // no noncommutative element found, should not happen
 933         return make_return_type_t<mul>();
 934 }
 935
 936 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 937 {
 938         return dynallocate<mul>(v, oc, do_index_renaming);
 939 }
 940
 941 ex mul::thisexpairseq(epvector && vp, const ex & oc, bool do_index_renaming) const
 942 {
 943         return dynallocate<mul>(std::move(vp), oc, do_index_renaming);
 944 }
 945
 946 expair mul::split_ex_to_pair(const ex & e) const
 947 {
 948         if (is_exactly_a<power>(e)) {
 949                 const power & powerref = ex_to<power>(e);
 950                 if (is_exactly_a<numeric>(powerref.exponent))
 951                         return expair(powerref.basis,powerref.exponent);
 952         }
 953         return expair(e,_ex1);
 954 }
 955
 956 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 957                                           const ex & c) const
 958 {
 959         GINAC_ASSERT(is_exactly_a<numeric>(c));
 960
 961         // First, try a common shortcut:
 962         if (is_exactly_a<symbol>(e))
 963                 return expair(e, c);
 964
 965         // trivial case: exponent 1
 966         if (c.is_equal(_ex1))
 967                 return split_ex_to_pair(e);
 968
 969         // to avoid duplication of power simplification rules,
 970         // we create a temporary power object
 971         // otherwise it would be hard to correctly evaluate
 972         // expression like (4^(1/3))^(3/2)
 973         return split_ex_to_pair(pow(e,c));
 974 }
 975
 976 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 977                                             const ex & c) const
 978 {
 979         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 980         GINAC_ASSERT(is_exactly_a<numeric>(c));
 981
 982         // First, try a common shortcut:
 983         if (is_exactly_a<symbol>(p.rest))
 984                 return expair(p.rest, p.coeff * c);
 985
 986         // trivial case: exponent 1
 987         if (c.is_equal(_ex1))
 988                 return p;
 989         if (p.coeff.is_equal(_ex1))
 990                 return expair(p.rest, c);
 991
 992         // to avoid duplication of power simplification rules,
 993         // we create a temporary power object
 994         // otherwise it would be hard to correctly evaluate
 995         // expression like (4^(1/3))^(3/2)
 996         return split_ex_to_pair(pow(recombine_pair_to_ex(p),c));
 997 }
 998
 999 ex mul::recombine_pair_to_ex(const expair & p) const
1000 {
1001         if (p.coeff.is_equal(_ex1))
1002                 return p.rest;
1003         else
1004                 return dynallocate<power>(p.rest, p.coeff);
1005 }
1006
1007 bool mul::expair_needs_further_processing(epp it)
1008 {
1009         if (is_exactly_a<mul>(it->rest) &&
1010             ex_to<numeric>(it->coeff).is_integer()) {
1011                 // combined pair is product with integer power -> expand it
1012                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
1013                 return true;
1014         }
1015         if (is_exactly_a<numeric>(it->rest)) {
1016                 if (it->coeff.is_equal(_ex1)) {
1017                         // pair has coeff 1 and must be moved to the end
1018                         return true;
1019                 }
1020                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
1021                 if (!ep.is_equal(*it)) {
1022                         // combined pair is a numeric power which can be simplified
1023                         *it = ep;
1024                         return true;
1025                 }
1026         }
1027         return false;
1028 }
1029
1030 ex mul::default_overall_coeff() const
1031 {
1032         return _ex1;
1033 }
1034
1035 void mul::combine_overall_coeff(const ex & c)
1036 {
1037         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1038         GINAC_ASSERT(is_exactly_a<numeric>(c));
1039         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
1040 }
1041
1042 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
1043 {
1044         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1045         GINAC_ASSERT(is_exactly_a<numeric>(c1));
1046         GINAC_ASSERT(is_exactly_a<numeric>(c2));
1047         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
1048 }
1049
1050 bool mul::can_make_flat(const expair & p) const
1051 {
1052         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
1053
1054         // (x*y)^c == x^c*y^c  if c ∈ ℤ
1055         return p.coeff.info(info_flags::integer);
1056 }
1057
1058 bool mul::can_be_further_expanded(const ex & e)
1059 {
1060         if (is_exactly_a<mul>(e)) {
1061                 for (auto & it : ex_to<mul>(e).seq) {
1062                         if (is_exactly_a<add>(it.rest) && it.coeff.info(info_flags::posint))
1063                                 return true;
1064                 }
1065         } else if (is_exactly_a<power>(e)) {
1066                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
1067                         return true;
1068         }
1069         return false;
1070 }
1071
1072 ex mul::expand(unsigned options) const
1073 {
1074         // Check for trivial case: expanding the monomial (~ 30% of all calls)
1075         bool monomial_case = true;
1076         for (const auto & i : seq) {
1077                 if (!is_a<symbol>(i.rest) || !i.coeff.info(info_flags::integer)) {
1078                         monomial_case = false;
1079                         break;
1080                 }
1081         }
1082         if (monomial_case) {
1083                 setflag(status_flags::expanded);
1084                 return *this;
1085         }
1086
1087         // do not rename indices if the object has no indices at all
1088         if ((!(options & expand_options::expand_rename_idx)) &&
1089             this->info(info_flags::has_indices))
1090                 options |= expand_options::expand_rename_idx;
1091
1092         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1093
1094         // First, expand the children
1095         epvector expanded = expandchildren(options);
1096         const epvector & expanded_seq = (expanded.empty() ? seq : expanded);
1097
1098         // Now, look for all the factors that are sums and multiply each one out
1099         // with the next one that is found while collecting the factors which are
1100         // not sums
1101         ex last_expanded = _ex1;
1102
1103         epvector non_adds;
1104         non_adds.reserve(expanded_seq.size());
1105
1106         for (const auto & cit : expanded_seq) {
1107                 if (is_exactly_a<add>(cit.rest) &&
1108                         (cit.coeff.is_equal(_ex1))) {
1109                         if (is_exactly_a<add>(last_expanded)) {
1110
1111                                 // Expand a product of two sums, aggressive version.
1112                                 // Caring for the overall coefficients in separate loops can
1113                                 // sometimes give a performance gain of up to 15%!
1114
1115                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit.rest).seq.size();
1116                                 // add2 is for the inner loop and should be the bigger of the two sums
1117                                 // in the presence of asymptotically good sorting:
1118                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit.rest));
1119                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit.rest) : ex_to<add>(last_expanded));
1120                                 epvector distrseq;
1121                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
1122
1123                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1124                                 if (!add1.overall_coeff.is_zero()) {
1125                                         if (add1.overall_coeff.is_equal(_ex1))
1126                                                 distrseq.insert(distrseq.end(), add2.seq.begin(), add2.seq.end());
1127                                         else
1128                                                 for (const auto & i : add2.seq)
1129                                                         distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1130                                 }
1131
1132                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1133                                 if (!add2.overall_coeff.is_zero()) {
1134                                         if (add2.overall_coeff.is_equal(_ex1))
1135                                                 distrseq.insert(distrseq.end(), add1.seq.begin(), add1.seq.end());
1136                                         else
1137                                                 for (const auto & i : add1.seq)
1138                                                         distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1139                                 }
1140
1141                                 // Compute the new overall coefficient and put it together:
1142                                 ex tmp_accu = dynallocate<add>(distrseq, add1.overall_coeff*add2.overall_coeff);
1143
1144                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1145                                 lst dummy_subs;
1146
1147                                 if (!skip_idx_rename) {
1148                                         for (const auto & i : add1.seq) {
1149                                                 add_indices = get_all_dummy_indices_safely(i.rest);
1150                                                 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1151                                         }
1152                                         for (const auto & i : add2.seq) {
1153                                                 add_indices = get_all_dummy_indices_safely(i.rest);
1154                                                 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1155                                         }
1156
1157                                         sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1158                                         sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1159                                         dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1160                                 }
1161
1162                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1163                                 for (const auto & i2 : add2.seq) {
1164                                         // We really have to combine terms here in order to compactify
1165                                         // the result.  Otherwise it would become waayy tooo bigg.
1166                                         numeric oc(*_num0_p);
1167                                         epvector distrseq2;
1168                                         distrseq2.reserve(add1.seq.size());
1169                                         const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1170                                                         i2.rest :
1171                                                         i2.rest.subs(ex_to<lst>(dummy_subs.op(0)),
1172                                                                      ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1173                                         for (const auto & i1 : add1.seq) {
1174                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1175                                                 // since that would violate an invariant of expairseq:
1176                                                 const ex rest = dynallocate<mul>(i1.rest, i2_new);
1177                                                 if (is_exactly_a<numeric>(rest)) {
1178                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1.coeff).mul(ex_to<numeric>(i2.coeff)));
1179                                                 } else {
1180                                                         distrseq2.push_back(expair(rest, ex_to<numeric>(i1.coeff).mul_dyn(ex_to<numeric>(i2.coeff))));
1181                                                 }
1182                                         }
1183                                         tmp_accu += dynallocate<add>(std::move(distrseq2), oc);
1184                                 }
1185                                 last_expanded = tmp_accu;
1186                         } else {
1187                                 if (!last_expanded.is_equal(_ex1))
1188                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1189                                 last_expanded = cit.rest;
1190                         }
1191
1192                 } else {
1193                         non_adds.push_back(cit);
1194                 }
1195         }
1196
1197         // Now the only remaining thing to do is to multiply the factors which
1198         // were not sums into the "last_expanded" sum
1199         if (is_exactly_a<add>(last_expanded)) {
1200                 size_t n = last_expanded.nops();
1201                 exvector distrseq;
1202                 distrseq.reserve(n);
1203                 exvector va;
1204                 if (! skip_idx_rename) {
1205                         va = get_all_dummy_indices_safely(mul(non_adds));
1206                         sort(va.begin(), va.end(), ex_is_less());
1207                 }
1208
1209                 for (size_t i=0; i<n; ++i) {
1210                         epvector factors = non_adds;
1211                         if (skip_idx_rename)
1212                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1213                         else
1214                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1215                         ex term = dynallocate<mul>(factors, overall_coeff);
1216                         if (can_be_further_expanded(term)) {
1217                                 distrseq.push_back(term.expand());
1218                         } else {
1219                                 if (options == 0)
1220                                         ex_to<basic>(term).setflag(status_flags::expanded);
1221                                 distrseq.push_back(term);
1222                         }
1223                 }
1224
1225                 return dynallocate<add>(distrseq).setflag(options == 0 ? status_flags::expanded : 0);
1226         }
1227
1228         non_adds.push_back(split_ex_to_pair(last_expanded));
1229         ex result = dynallocate<mul>(non_adds, overall_coeff);
1230         if (can_be_further_expanded(result)) {
1231                 return result.expand();
1232         } else {
1233                 if (options == 0)
1234                         ex_to<basic>(result).setflag(status_flags::expanded);
1235                 return result;
1236         }
1237 }
1238
1239
1240 //////////
1241 // new virtual functions which can be overridden by derived classes
1242 //////////
1243
1244 // none
1245
1246 //////////
1247 // non-virtual functions in this class
1248 //////////
1249
1250
1251 /** Member-wise expand the expairs representing this sequence.  This must be
1252  *  overridden from expairseq::expandchildren() and done iteratively in order
1253  *  to allow for early cancellations and thus safe memory.
1254  *
1255  *  @see mul::expand()
1256  *  @return epvector containing expanded pairs, empty if no members
1257  *    had to be changed. */
1258 epvector mul::expandchildren(unsigned options) const
1259 {
1260         auto cit = seq.begin(), last = seq.end();
1261         while (cit!=last) {
1262                 const ex & factor = recombine_pair_to_ex(*cit);
1263                 const ex & expanded_factor = factor.expand(options);
1264                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1265
1266                         // something changed, copy seq, eval and return it
1267                         epvector s;
1268                         s.reserve(seq.size());
1269
1270                         // copy parts of seq which are known not to have changed
1271                         auto cit2 = seq.begin();
1272                         while (cit2!=cit) {
1273                                 s.push_back(*cit2);
1274                                 ++cit2;
1275                         }
1276
1277                         // copy first changed element
1278                         s.push_back(split_ex_to_pair(expanded_factor));
1279                         ++cit2;
1280
1281                         // copy rest
1282                         while (cit2!=last) {
1283                                 s.push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1284                                 ++cit2;
1285                         }
1286                         return s;
1287                 }
1288                 ++cit;
1289         }
1290
1291         return epvector(); // nothing has changed
1292 }
1293
1294 GINAC_BIND_UNARCHIVER(mul);
1295
1296 } // namespace GiNaC