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