ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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::algebraic: {
 296                         for (auto & it : seq) {
 297                                 if (recombine_pair_to_ex(it).info(inf))
 298                                         return true;
 299                         }
 300                         return false;
 301                 }
 302                 case info_flags::positive:
 303                 case info_flags::negative: {
 304                         if ((inf==info_flags::positive) && (flags & status_flags::is_positive))
 305                                 return true;
 306                         else if ((inf==info_flags::negative) && (flags & status_flags::is_negative))
 307                                 return true;
 308                         if (flags & status_flags::purely_indefinite)
 309                                 return false;
 310
 311                         bool pos = true;
 312                         for (auto & it : seq) {
 313                                 const ex& factor = recombine_pair_to_ex(it);
 314                                 if (factor.info(info_flags::positive))
 315                                         continue;
 316                                 else if (factor.info(info_flags::negative))
 317                                         pos = !pos;
 318                                 else
 319                                         return false;
 320                         }
 321                         if (overall_coeff.info(info_flags::negative))
 322                                 pos = !pos;
 323                         setflag(pos ? status_flags::is_positive : status_flags::is_negative);
 324                         return (inf == info_flags::positive? pos : !pos);
 325                 }
 326                 case info_flags::nonnegative: {
 327                         if  (flags & status_flags::is_positive)
 328                                 return true;
 329                         bool pos = true;
 330                         for (auto & it : seq) {
 331                                 const ex& factor = recombine_pair_to_ex(it);
 332                                 if (factor.info(info_flags::nonnegative) || factor.info(info_flags::positive))
 333                                         continue;
 334                                 else if (factor.info(info_flags::negative))
 335                                         pos = !pos;
 336                                 else
 337                                         return false;
 338                         }
 339                         return (overall_coeff.info(info_flags::negative)? !pos : pos);
 340                 }
 341                 case info_flags::posint:
 342                 case info_flags::negint: {
 343                         bool pos = true;
 344                         for (auto & it : seq) {
 345                                 const ex& factor = recombine_pair_to_ex(it);
 346                                 if (factor.info(info_flags::posint))
 347                                         continue;
 348                                 else if (factor.info(info_flags::negint))
 349                                         pos = !pos;
 350                                 else
 351                                         return false;
 352                         }
 353                         if (overall_coeff.info(info_flags::negint))
 354                                 pos = !pos;
 355                         else if (!overall_coeff.info(info_flags::posint))
 356                                 return false;
 357                         return (inf ==info_flags::posint? pos : !pos);
 358                 }
 359                 case info_flags::nonnegint: {
 360                         bool pos = true;
 361                         for (auto & it : seq) {
 362                                 const ex& factor = recombine_pair_to_ex(it);
 363                                 if (factor.info(info_flags::nonnegint) || factor.info(info_flags::posint))
 364                                         continue;
 365                                 else if (factor.info(info_flags::negint))
 366                                         pos = !pos;
 367                                 else
 368                                         return false;
 369                         }
 370                         if (overall_coeff.info(info_flags::negint))
 371                                 pos = !pos;
 372                         else if (!overall_coeff.info(info_flags::posint))
 373                                 return false;
 374                         return pos;
 375                 }
 376                 case info_flags::indefinite: {
 377                         if (flags & status_flags::purely_indefinite)
 378                                 return true;
 379                         if (flags & (status_flags::is_positive | status_flags::is_negative))
 380                                 return false;
 381                         for (auto & it : seq) {
 382                                 const ex& term = recombine_pair_to_ex(it);
 383                                 if (term.info(info_flags::positive) || term.info(info_flags::negative))
 384                                         return false;
 385                         }
 386                         setflag(status_flags::purely_indefinite);
 387                         return true;
 388                 }
 389         }
 390         return inherited::info(inf);
 391 }
 392
 393 bool mul::is_polynomial(const ex & var) const
 394 {
 395         for (auto & it : seq) {
 396                 if (!it.rest.is_polynomial(var) ||
 397                     (it.rest.has(var) && !it.coeff.info(info_flags::nonnegint))) {
 398                         return false;
 399                 }
 400         }
 401         return true;
 402 }
 403
 404 int mul::degree(const ex & s) const
 405 {
 406         // Sum up degrees of factors
 407         int deg_sum = 0;
 408         for (auto & it : seq) {
 409                 if (ex_to<numeric>(it.coeff).is_integer())
 410                         deg_sum += recombine_pair_to_ex(it).degree(s);
 411                 else {
 412                         if (it.rest.has(s))
 413                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
 414                 }
 415         }
 416         return deg_sum;
 417 }
 418
 419 int mul::ldegree(const ex & s) const
 420 {
 421         // Sum up degrees of factors
 422         int deg_sum = 0;
 423         for (auto & it : seq) {
 424                 if (ex_to<numeric>(it.coeff).is_integer())
 425                         deg_sum += recombine_pair_to_ex(it).ldegree(s);
 426                 else {
 427                         if (it.rest.has(s))
 428                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
 429                 }
 430         }
 431         return deg_sum;
 432 }
 433
 434 ex mul::coeff(const ex & s, int n) const
 435 {
 436         exvector coeffseq;
 437         coeffseq.reserve(seq.size()+1);
 438
 439         if (n==0) {
 440                 // product of individual coeffs
 441                 // if a non-zero power of s is found, the resulting product will be 0
 442                 for (auto & it : seq)
 443                         coeffseq.push_back(recombine_pair_to_ex(it).coeff(s,n));
 444                 coeffseq.push_back(overall_coeff);
 445                 return dynallocate<mul>(coeffseq);
 446         }
 447
 448         bool coeff_found = false;
 449         for (auto & it : seq) {
 450                 ex t = recombine_pair_to_ex(it);
 451                 ex c = t.coeff(s, n);
 452                 if (!c.is_zero()) {
 453                         coeffseq.push_back(c);
 454                         coeff_found = 1;
 455                 } else {
 456                         coeffseq.push_back(t);
 457                 }
 458         }
 459         if (coeff_found) {
 460                 coeffseq.push_back(overall_coeff);
 461                 return dynallocate<mul>(coeffseq);
 462         }
 463
 464         return _ex0;
 465 }
 466
 467 /** Perform automatic term rewriting rules in this class.  In the following
 468  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 469  *  stand for such expressions that contain a plain number.
 470  *  - *(...,x;0) -> 0
 471  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 472  *  - *(x;1) -> x
 473  *  - *(;c) -> c
 474  */
 475 ex mul::eval() const
 476 {
 477         if (flags & status_flags::evaluated) {
 478                 GINAC_ASSERT(seq.size()>0);
 479                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 480                 return *this;
 481         }
 482
 483         const epvector evaled = evalchildren();
 484         if (unlikely(!evaled.empty())) {
 485                 // start over evaluating a new object
 486                 return dynallocate<mul>(std::move(evaled), overall_coeff);
 487         }
 488
 489         size_t seq_size = seq.size();
 490         if (overall_coeff.is_zero()) {
 491                 // *(...,x;0) -> 0
 492                 return _ex0;
 493         } else if (seq_size==0) {
 494                 // *(;c) -> c
 495                 return overall_coeff;
 496         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 497                 // *(x;1) -> x
 498                 return recombine_pair_to_ex(*(seq.begin()));
 499         } else if ((seq_size==1) &&
 500                    is_exactly_a<add>((*seq.begin()).rest) &&
 501                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 502                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 503                 const add & addref = ex_to<add>((*seq.begin()).rest);
 504                 epvector distrseq;
 505                 distrseq.reserve(addref.seq.size());
 506                 for (auto & it : addref.seq) {
 507                         distrseq.push_back(addref.combine_pair_with_coeff_to_pair(it, overall_coeff));
 508                 }
 509                 return dynallocate<add>(std::move(distrseq),
 510                                         ex_to<numeric>(addref.overall_coeff).mul_dyn(ex_to<numeric>(overall_coeff)))
 511                         .setflag(status_flags::evaluated);
 512         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
 513                 // Strip the content and the unit part from each term. Thus
 514                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)^2
 515
 516                 auto i = seq.begin(), last = seq.end();
 517                 auto j = seq.begin();
 518                 epvector s;
 519                 numeric oc = *_num1_p;
 520                 bool something_changed = false;
 521                 while (i!=last) {
 522                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
 523                                 // power::eval has such a rule, no need to handle powers here
 524                                 ++i;
 525                                 continue;
 526                         }
 527
 528                         // XXX: What is the best way to check if the polynomial is a primitive?
 529                         numeric c = i->rest.integer_content();
 530                         const numeric lead_coeff =
 531                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
 532                         const bool canonicalizable = lead_coeff.is_integer();
 533
 534                         // XXX: The main variable is chosen in a random way, so this code
 535                         // does NOT transform the term into the canonical form (thus, in some
 536                         // very unlucky event it can even loop forever). Hopefully the main
 537                         // variable will be the same for all terms in *this
 538                         const bool unit_normal = lead_coeff.is_pos_integer();
 539                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
 540                                 ++i;
 541                                 continue;
 542                         }
 543
 544                         if (! something_changed) {
 545                                 s.reserve(seq_size);
 546                                 something_changed = true;
 547                         }
 548
 549                         while ((j!=i) && (j!=last)) {
 550                                 s.push_back(*j);
 551                                 ++j;
 552                         }
 553
 554                         if (! unit_normal)
 555                                 c = c.mul(*_num_1_p);
 556
 557                         oc = oc.mul(c);
 558
 559                         // divide add by the number in place to save at least 2 .eval() calls
 560                         const add& addref = ex_to<add>(i->rest);
 561                         add & primitive = dynallocate<add>(addref);
 562                         primitive.clearflag(status_flags::hash_calculated);
 563                         primitive.overall_coeff = ex_to<numeric>(primitive.overall_coeff).div_dyn(c);
 564                         for (auto & ai : primitive.seq)
 565                                 ai.coeff = ex_to<numeric>(ai.coeff).div_dyn(c);
 566
 567                         s.push_back(expair(primitive, _ex1));
 568
 569                         ++i;
 570                         ++j;
 571                 }
 572                 if (something_changed) {
 573                         while (j!=last) {
 574                                 s.push_back(*j);
 575                                 ++j;
 576                         }
 577                         return dynallocate<mul>(std::move(s), ex_to<numeric>(overall_coeff).mul_dyn(oc));
 578                 }
 579         }
 580
 581         return this->hold();
 582 }
 583
 584 ex mul::evalf(int level) const
 585 {
 586         if (level==1)
 587                 return mul(seq, overall_coeff);
 588
 589         if (level==-max_recursion_level)
 590                 throw(std::runtime_error("max recursion level reached"));
 591
 592         epvector s;
 593         s.reserve(seq.size());
 594
 595         --level;
 596         for (auto & it : seq) {
 597                 s.push_back(expair(it.rest.evalf(level), it.coeff));
 598         }
 599         return dynallocate<mul>(std::move(s), overall_coeff.evalf(level));
 600 }
 601
 602 void mul::find_real_imag(ex & rp, ex & ip) const
 603 {
 604         rp = overall_coeff.real_part();
 605         ip = overall_coeff.imag_part();
 606         for (auto & it : seq) {
 607                 ex factor = recombine_pair_to_ex(it);
 608                 ex new_rp = factor.real_part();
 609                 ex new_ip = factor.imag_part();
 610                 if (new_ip.is_zero()) {
 611                         rp *= new_rp;
 612                         ip *= new_rp;
 613                 } else {
 614                         ex temp = rp*new_rp - ip*new_ip;
 615                         ip = ip*new_rp + rp*new_ip;
 616                         rp = temp;
 617                 }
 618         }
 619         rp = rp.expand();
 620         ip = ip.expand();
 621 }
 622
 623 ex mul::real_part() const
 624 {
 625         ex rp, ip;
 626         find_real_imag(rp, ip);
 627         return rp;
 628 }
 629
 630 ex mul::imag_part() const
 631 {
 632         ex rp, ip;
 633         find_real_imag(rp, ip);
 634         return ip;
 635 }
 636
 637 ex mul::evalm() const
 638 {
 639         // numeric*matrix
 640         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 641          && is_a<matrix>(seq[0].rest))
 642                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 643
 644         // Evaluate children first, look whether there are any matrices at all
 645         // (there can be either no matrices or one matrix; if there were more
 646         // than one matrix, it would be a non-commutative product)
 647         epvector s;
 648         s.reserve(seq.size());
 649
 650         bool have_matrix = false;
 651         epvector::iterator the_matrix;
 652
 653         for (auto & it : seq) {
 654                 const ex &m = recombine_pair_to_ex(it).evalm();
 655                 s.push_back(split_ex_to_pair(m));
 656                 if (is_a<matrix>(m)) {
 657                         have_matrix = true;
 658                         the_matrix = s.end() - 1;
 659                 }
 660         }
 661
 662         if (have_matrix) {
 663
 664                 // The product contained a matrix. We will multiply all other factors
 665                 // into that matrix.
 666                 matrix m = ex_to<matrix>(the_matrix->rest);
 667                 s.erase(the_matrix);
 668                 ex scalar = dynallocate<mul>(std::move(s), overall_coeff);
 669                 return m.mul_scalar(scalar);
 670
 671         } else
 672                 return dynallocate<mul>(std::move(s), overall_coeff);
 673 }
 674
 675 ex mul::eval_ncmul(const exvector & v) const
 676 {
 677         if (seq.empty())
 678                 return inherited::eval_ncmul(v);
 679
 680         // Find first noncommutative element and call its eval_ncmul()
 681         for (auto & it : seq)
 682                 if (it.rest.return_type() == return_types::noncommutative)
 683                         return it.rest.eval_ncmul(v);
 684         return inherited::eval_ncmul(v);
 685 }
 686
 687 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
 688 {
 689         ex origbase;
 690         int origexponent;
 691         int origexpsign;
 692
 693         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 694                 origbase = origfactor.op(0);
 695                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 696                 origexponent = expon > 0 ? expon : -expon;
 697                 origexpsign = expon > 0 ? 1 : -1;
 698         } else {
 699                 origbase = origfactor;
 700                 origexponent = 1;
 701                 origexpsign = 1;
 702         }
 703
 704         ex patternbase;
 705         int patternexponent;
 706         int patternexpsign;
 707
 708         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 709                 patternbase = patternfactor.op(0);
 710                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 711                 patternexponent = expon > 0 ? expon : -expon;
 712                 patternexpsign = expon > 0 ? 1 : -1;
 713         } else {
 714                 patternbase = patternfactor;
 715                 patternexponent = 1;
 716                 patternexpsign = 1;
 717         }
 718
 719         exmap saverepls = repls;
 720         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 721                 return false;
 722         repls = saverepls;
 723
 724         int newnummatches = origexponent / patternexponent;
 725         if (newnummatches < nummatches)
 726                 nummatches = newnummatches;
 727         return true;
 728 }
 729
 730 /** Checks whether e matches to the pattern pat and the (possibly to be updated)
 731   * list of replacements repls. This matching is in the sense of algebraic
 732   * substitutions. Matching starts with pat.op(factor) of the pattern because
 733   * the factors before this one have already been matched. The (possibly
 734   * updated) number of matches is in nummatches. subsed[i] is true for factors
 735   * that already have been replaced by previous substitutions and matched[i]
 736   * is true for factors that have been matched by the current match.
 737   */
 738 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
 739                                   int factor, int &nummatches, const std::vector<bool> &subsed,
 740                                   std::vector<bool> &matched)
 741 {
 742         GINAC_ASSERT(subsed.size() == e.nops());
 743         GINAC_ASSERT(matched.size() == e.nops());
 744
 745         if (factor == (int)pat.nops())
 746                 return true;
 747
 748         for (size_t i=0; i<e.nops(); ++i) {
 749                 if(subsed[i] || matched[i])
 750                         continue;
 751                 exmap newrepls = repls;
 752                 int newnummatches = nummatches;
 753                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 754                         matched[i] = true;
 755                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 756                                         newnummatches, subsed, matched)) {
 757                                 repls = newrepls;
 758                                 nummatches = newnummatches;
 759                                 return true;
 760                         }
 761                         else
 762                                 matched[i] = false;
 763                 }
 764         }
 765
 766         return false;
 767 }
 768
 769 bool mul::has(const ex & pattern, unsigned options) const
 770 {
 771         if(!(options & has_options::algebraic))
 772                 return basic::has(pattern,options);
 773         if(is_a<mul>(pattern)) {
 774                 exmap repls;
 775                 int nummatches = std::numeric_limits<int>::max();
 776                 std::vector<bool> subsed(nops(), false);
 777                 std::vector<bool> matched(nops(), false);
 778                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 779                                 subsed, matched))
 780                         return true;
 781         }
 782         return basic::has(pattern, options);
 783 }
 784
 785 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 786 {
 787         std::vector<bool> subsed(nops(), false);
 788         ex divide_by = 1;
 789         ex multiply_by = 1;
 790
 791         for (auto & it : m) {
 792
 793                 if (is_exactly_a<mul>(it.first)) {
 794 retry1:
 795                         int nummatches = std::numeric_limits<int>::max();
 796                         std::vector<bool> currsubsed(nops(), false);
 797                         exmap repls;
 798
 799                         if (!algebraic_match_mul_with_mul(*this, it.first, repls, 0, nummatches, subsed, currsubsed))
 800                                 continue;
 801
 802                         for (size_t j=0; j<subsed.size(); j++)
 803                                 if (currsubsed[j])
 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                         goto retry1;
 812
 813                 } else {
 814
 815                         for (size_t j=0; j<this->nops(); j++) {
 816                                 int nummatches = std::numeric_limits<int>::max();
 817                                 exmap repls;
 818                                 if (!subsed[j] && tryfactsubs(op(j), it.first, nummatches, repls)){
 819                                         subsed[j] = true;
 820                                         ex subsed_pattern
 821                                                 = it.first.subs(repls, subs_options::no_pattern);
 822                                         divide_by *= pow(subsed_pattern, nummatches);
 823                                         ex subsed_result
 824                                                 = it.second.subs(repls, subs_options::no_pattern);
 825                                         multiply_by *= pow(subsed_result, nummatches);
 826                                 }
 827                         }
 828                 }
 829         }
 830
 831         bool subsfound = false;
 832         for (size_t i=0; i<subsed.size(); i++) {
 833                 if (subsed[i]) {
 834                         subsfound = true;
 835                         break;
 836                 }
 837         }
 838         if (!subsfound)
 839                 return subs_one_level(m, options | subs_options::algebraic);
 840
 841         return ((*this)/divide_by)*multiply_by;
 842 }
 843
 844 ex mul::conjugate() const
 845 {
 846         // The base class' method is wrong here because we have to be careful at
 847         // branch cuts. power::conjugate takes care of that already, so use it.
 848         std::unique_ptr<epvector> newepv(nullptr);
 849         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 850                 if (newepv) {
 851                         newepv->push_back(split_ex_to_pair(recombine_pair_to_ex(*i).conjugate()));
 852                         continue;
 853                 }
 854                 ex x = recombine_pair_to_ex(*i);
 855                 ex c = x.conjugate();
 856                 if (c.is_equal(x)) {
 857                         continue;
 858                 }
 859                 newepv.reset(new epvector);
 860                 newepv->reserve(seq.size());
 861                 for (auto j=seq.begin(); j!=i; ++j) {
 862                         newepv->push_back(*j);
 863                 }
 864                 newepv->push_back(split_ex_to_pair(c));
 865         }
 866         ex x = overall_coeff.conjugate();
 867         if (!newepv && are_ex_trivially_equal(x, overall_coeff)) {
 868                 return *this;
 869         }
 870         return thisexpairseq(newepv ? std::move(*newepv) : seq, x);
 871 }
 872
 873
 874 // protected
 875
 876 /** Implementation of ex::diff() for a product.  It applies the product rule.
 877  *  @see ex::diff */
 878 ex mul::derivative(const symbol & s) const
 879 {
 880         size_t num = seq.size();
 881         exvector addseq;
 882         addseq.reserve(num);
 883
 884         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 885         epvector mulseq = seq;
 886         auto i = seq.begin(), end = seq.end();
 887         auto i2 = mulseq.begin();
 888         while (i != end) {
 889                 expair ep = split_ex_to_pair(pow(i->rest, i->coeff - _ex1) *
 890                                              i->rest.diff(s));
 891                 ep.swap(*i2);
 892                 addseq.push_back(dynallocate<mul>(mulseq, overall_coeff * i->coeff));
 893                 ep.swap(*i2);
 894                 ++i; ++i2;
 895         }
 896         return dynallocate<add>(addseq);
 897 }
 898
 899 int mul::compare_same_type(const basic & other) const
 900 {
 901         return inherited::compare_same_type(other);
 902 }
 903
 904 unsigned mul::return_type() const
 905 {
 906         if (seq.empty()) {
 907                 // mul without factors: should not happen, but commutates
 908                 return return_types::commutative;
 909         }
 910
 911         bool all_commutative = true;
 912         epvector::const_iterator noncommutative_element; // point to first found nc element
 913
 914         epvector::const_iterator i = seq.begin(), end = seq.end();
 915         while (i != end) {
 916                 unsigned rt = i->rest.return_type();
 917                 if (rt == return_types::noncommutative_composite)
 918                         return rt; // one ncc -> mul also ncc
 919                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 920                         // first nc element found, remember position
 921                         noncommutative_element = i;
 922                         all_commutative = false;
 923                 }
 924                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 925                         // another nc element found, compare type_infos
 926                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 927                                         // different types -> mul is ncc
 928                                         return return_types::noncommutative_composite;
 929                         }
 930                 }
 931                 ++i;
 932         }
 933         // all factors checked
 934         return all_commutative ? return_types::commutative : return_types::noncommutative;
 935 }
 936
 937 return_type_t mul::return_type_tinfo() const
 938 {
 939         if (seq.empty())
 940                 return make_return_type_t<mul>(); // mul without factors: should not happen
 941
 942         // return type_info of first noncommutative element
 943         for (auto & it : seq)
 944                 if (it.rest.return_type() == return_types::noncommutative)
 945                         return it.rest.return_type_tinfo();
 946
 947         // no noncommutative element found, should not happen
 948         return make_return_type_t<mul>();
 949 }
 950
 951 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 952 {
 953         return dynallocate<mul>(v, oc, do_index_renaming);
 954 }
 955
 956 ex mul::thisexpairseq(epvector && vp, const ex & oc, bool do_index_renaming) const
 957 {
 958         return dynallocate<mul>(std::move(vp), oc, do_index_renaming);
 959 }
 960
 961 expair mul::split_ex_to_pair(const ex & e) const
 962 {
 963         if (is_exactly_a<power>(e)) {
 964                 const power & powerref = ex_to<power>(e);
 965                 if (is_exactly_a<numeric>(powerref.exponent))
 966                         return expair(powerref.basis,powerref.exponent);
 967         }
 968         return expair(e,_ex1);
 969 }
 970
 971 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 972                                           const ex & c) const
 973 {
 974         GINAC_ASSERT(is_exactly_a<numeric>(c));
 975
 976         // First, try a common shortcut:
 977         if (is_exactly_a<symbol>(e))
 978                 return expair(e, c);
 979
 980         // to avoid duplication of power simplification rules,
 981         // we create a temporary power object
 982         // otherwise it would be hard to correctly evaluate
 983         // expression like (4^(1/3))^(3/2)
 984         if (c.is_equal(_ex1))
 985                 return split_ex_to_pair(e);
 986
 987         return split_ex_to_pair(pow(e,c));
 988 }
 989
 990 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 991                                             const ex & c) const
 992 {
 993         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 994         GINAC_ASSERT(is_exactly_a<numeric>(c));
 995
 996         // to avoid duplication of power simplification rules,
 997         // we create a temporary power object
 998         // otherwise it would be hard to correctly evaluate
 999         // expression like (4^(1/3))^(3/2)
1000         if (c.is_equal(_ex1))
1001                 return p;
1002
1003         return split_ex_to_pair(pow(recombine_pair_to_ex(p),c));
1004 }
1005
1006 ex mul::recombine_pair_to_ex(const expair & p) const
1007 {
1008         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
1009                 return p.rest;
1010         else
1011                 return dynallocate<power>(p.rest, p.coeff);
1012 }
1013
1014 bool mul::expair_needs_further_processing(epp it)
1015 {
1016         if (is_exactly_a<mul>(it->rest) &&
1017             ex_to<numeric>(it->coeff).is_integer()) {
1018                 // combined pair is product with integer power -> expand it
1019                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
1020                 return true;
1021         }
1022         if (is_exactly_a<numeric>(it->rest)) {
1023                 if (it->coeff.is_equal(_ex1)) {
1024                         // pair has coeff 1 and must be moved to the end
1025                         return true;
1026                 }
1027                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
1028                 if (!ep.is_equal(*it)) {
1029                         // combined pair is a numeric power which can be simplified
1030                         *it = ep;
1031                         return true;
1032                 }
1033         }
1034         return false;
1035 }
1036
1037 ex mul::default_overall_coeff() const
1038 {
1039         return _ex1;
1040 }
1041
1042 void mul::combine_overall_coeff(const ex & c)
1043 {
1044         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1045         GINAC_ASSERT(is_exactly_a<numeric>(c));
1046         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
1047 }
1048
1049 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
1050 {
1051         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1052         GINAC_ASSERT(is_exactly_a<numeric>(c1));
1053         GINAC_ASSERT(is_exactly_a<numeric>(c2));
1054         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
1055 }
1056
1057 bool mul::can_make_flat(const expair & p) const
1058 {
1059         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
1060
1061         // (x*y)^c == x^c*y^c  if c ∈ ℤ
1062         return p.coeff.info(info_flags::integer);
1063 }
1064
1065 bool mul::can_be_further_expanded(const ex & e)
1066 {
1067         if (is_exactly_a<mul>(e)) {
1068                 for (auto & it : ex_to<mul>(e).seq) {
1069                         if (is_exactly_a<add>(it.rest) && it.coeff.info(info_flags::posint))
1070                                 return true;
1071                 }
1072         } else if (is_exactly_a<power>(e)) {
1073                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
1074                         return true;
1075         }
1076         return false;
1077 }
1078
1079 ex mul::expand(unsigned options) const
1080 {
1081         // Check for trivial case: expanding the monomial (~ 30% of all calls)
1082         bool monomial_case = true;
1083         for (const auto & i : seq) {
1084                 if (!is_a<symbol>(i.rest) || !i.coeff.info(info_flags::integer)) {
1085                         monomial_case = false;
1086                         break;
1087                 }
1088         }
1089         if (monomial_case) {
1090                 setflag(status_flags::expanded);
1091                 return *this;
1092         }
1093
1094         // do not rename indices if the object has no indices at all
1095         if ((!(options & expand_options::expand_rename_idx)) &&
1096             this->info(info_flags::has_indices))
1097                 options |= expand_options::expand_rename_idx;
1098
1099         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1100
1101         // First, expand the children
1102         epvector expanded = expandchildren(options);
1103         const epvector & expanded_seq = (expanded.empty() ? seq : expanded);
1104
1105         // Now, look for all the factors that are sums and multiply each one out
1106         // with the next one that is found while collecting the factors which are
1107         // not sums
1108         ex last_expanded = _ex1;
1109
1110         epvector non_adds;
1111         non_adds.reserve(expanded_seq.size());
1112
1113         for (const auto & cit : expanded_seq) {
1114                 if (is_exactly_a<add>(cit.rest) &&
1115                         (cit.coeff.is_equal(_ex1))) {
1116                         if (is_exactly_a<add>(last_expanded)) {
1117
1118                                 // Expand a product of two sums, aggressive version.
1119                                 // Caring for the overall coefficients in separate loops can
1120                                 // sometimes give a performance gain of up to 15%!
1121
1122                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit.rest).seq.size();
1123                                 // add2 is for the inner loop and should be the bigger of the two sums
1124                                 // in the presence of asymptotically good sorting:
1125                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit.rest));
1126                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit.rest) : ex_to<add>(last_expanded));
1127                                 epvector distrseq;
1128                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
1129
1130                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1131                                 if (!add1.overall_coeff.is_zero()) {
1132                                         if (add1.overall_coeff.is_equal(_ex1))
1133                                                 distrseq.insert(distrseq.end(), add2.seq.begin(), add2.seq.end());
1134                                         else
1135                                                 for (const auto & i : add2.seq)
1136                                                         distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1137                                 }
1138
1139                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1140                                 if (!add2.overall_coeff.is_zero()) {
1141                                         if (add2.overall_coeff.is_equal(_ex1))
1142                                                 distrseq.insert(distrseq.end(), add1.seq.begin(), add1.seq.end());
1143                                         else
1144                                                 for (const auto & i : add1.seq)
1145                                                         distrseq.push_back(expair(i.rest, ex_to<numeric>(i.coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1146                                 }
1147
1148                                 // Compute the new overall coefficient and put it together:
1149                                 ex tmp_accu = dynallocate<add>(distrseq, add1.overall_coeff*add2.overall_coeff);
1150
1151                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1152                                 lst dummy_subs;
1153
1154                                 if (!skip_idx_rename) {
1155                                         for (const auto & i : add1.seq) {
1156                                                 add_indices = get_all_dummy_indices_safely(i.rest);
1157                                                 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1158                                         }
1159                                         for (const auto & i : add2.seq) {
1160                                                 add_indices = get_all_dummy_indices_safely(i.rest);
1161                                                 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1162                                         }
1163
1164                                         sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1165                                         sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1166                                         dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1167                                 }
1168
1169                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1170                                 for (const auto & i2 : add2.seq) {
1171                                         // We really have to combine terms here in order to compactify
1172                                         // the result.  Otherwise it would become waayy tooo bigg.
1173                                         numeric oc(*_num0_p);
1174                                         epvector distrseq2;
1175                                         distrseq2.reserve(add1.seq.size());
1176                                         const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1177                                                         i2.rest :
1178                                                         i2.rest.subs(ex_to<lst>(dummy_subs.op(0)),
1179                                                                      ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1180                                         for (const auto & i1 : add1.seq) {
1181                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1182                                                 // since that would violate an invariant of expairseq:
1183                                                 const ex rest = dynallocate<mul>(i1.rest, i2_new);
1184                                                 if (is_exactly_a<numeric>(rest)) {
1185                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1.coeff).mul(ex_to<numeric>(i2.coeff)));
1186                                                 } else {
1187                                                         distrseq2.push_back(expair(rest, ex_to<numeric>(i1.coeff).mul_dyn(ex_to<numeric>(i2.coeff))));
1188                                                 }
1189                                         }
1190                                         tmp_accu += dynallocate<add>(std::move(distrseq2), oc);
1191                                 }
1192                                 last_expanded = tmp_accu;
1193                         } else {
1194                                 if (!last_expanded.is_equal(_ex1))
1195                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1196                                 last_expanded = cit.rest;
1197                         }
1198
1199                 } else {
1200                         non_adds.push_back(cit);
1201                 }
1202         }
1203
1204         // Now the only remaining thing to do is to multiply the factors which
1205         // were not sums into the "last_expanded" sum
1206         if (is_exactly_a<add>(last_expanded)) {
1207                 size_t n = last_expanded.nops();
1208                 exvector distrseq;
1209                 distrseq.reserve(n);
1210                 exvector va;
1211                 if (! skip_idx_rename) {
1212                         va = get_all_dummy_indices_safely(mul(non_adds));
1213                         sort(va.begin(), va.end(), ex_is_less());
1214                 }
1215
1216                 for (size_t i=0; i<n; ++i) {
1217                         epvector factors = non_adds;
1218                         if (skip_idx_rename)
1219                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1220                         else
1221                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1222                         ex term = dynallocate<mul>(factors, overall_coeff);
1223                         if (can_be_further_expanded(term)) {
1224                                 distrseq.push_back(term.expand());
1225                         } else {
1226                                 if (options == 0)
1227                                         ex_to<basic>(term).setflag(status_flags::expanded);
1228                                 distrseq.push_back(term);
1229                         }
1230                 }
1231
1232                 return dynallocate<add>(distrseq).setflag(options == 0 ? status_flags::expanded : 0);
1233         }
1234
1235         non_adds.push_back(split_ex_to_pair(last_expanded));
1236         ex result = dynallocate<mul>(non_adds, overall_coeff);
1237         if (can_be_further_expanded(result)) {
1238                 return result.expand();
1239         } else {
1240                 if (options == 0)
1241                         ex_to<basic>(result).setflag(status_flags::expanded);
1242                 return result;
1243         }
1244 }
1245
1246
1247 //////////
1248 // new virtual functions which can be overridden by derived classes
1249 //////////
1250
1251 // none
1252
1253 //////////
1254 // non-virtual functions in this class
1255 //////////
1256
1257
1258 /** Member-wise expand the expairs representing this sequence.  This must be
1259  *  overridden from expairseq::expandchildren() and done iteratively in order
1260  *  to allow for early cancellations and thus safe memory.
1261  *
1262  *  @see mul::expand()
1263  *  @return epvector containing expanded pairs, empty if no members
1264  *    had to be changed. */
1265 epvector mul::expandchildren(unsigned options) const
1266 {
1267         auto cit = seq.begin(), last = seq.end();
1268         while (cit!=last) {
1269                 const ex & factor = recombine_pair_to_ex(*cit);
1270                 const ex & expanded_factor = factor.expand(options);
1271                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1272
1273                         // something changed, copy seq, eval and return it
1274                         epvector s;
1275                         s.reserve(seq.size());
1276
1277                         // copy parts of seq which are known not to have changed
1278                         auto cit2 = seq.begin();
1279                         while (cit2!=cit) {
1280                                 s.push_back(*cit2);
1281                                 ++cit2;
1282                         }
1283
1284                         // copy first changed element
1285                         s.push_back(split_ex_to_pair(expanded_factor));
1286                         ++cit2;
1287
1288                         // copy rest
1289                         while (cit2!=last) {
1290                                 s.push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1291                                 ++cit2;
1292                         }
1293                         return s;
1294                 }
1295                 ++cit;
1296         }
1297
1298         return epvector(); // nothing has changed
1299 }
1300
1301 GINAC_BIND_UNARCHIVER(mul);
1302
1303 } // namespace GiNaC