ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <iostream>
  24 #include <vector>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "mul.h"
  29 #include "add.h"
  30 #include "power.h"
  31 #include "operators.h"
  32 #include "matrix.h"
  33 #include "indexed.h"
  34 #include "lst.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
  41   print_func<print_context>(&mul::do_print).
  42   print_func<print_latex>(&mul::do_print_latex).
  43   print_func<print_csrc>(&mul::do_print_csrc).
  44   print_func<print_tree>(&mul::do_print_tree).
  45   print_func<print_python_repr>(&mul::do_print_python_repr))
  46
  47
  48 //////////
  49 // default constructor
  50 //////////
  51
  52 mul::mul()
  53 {
  54         tinfo_key = &mul::tinfo_static;
  55 }
  56
  57 //////////
  58 // other constructors
  59 //////////
  60
  61 // public
  62
  63 mul::mul(const ex & lh, const ex & rh)
  64 {
  65         tinfo_key = &mul::tinfo_static;
  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         tinfo_key = &mul::tinfo_static;
  74         overall_coeff = _ex1;
  75         construct_from_exvector(v);
  76         GINAC_ASSERT(is_canonical());
  77 }
  78
  79 mul::mul(const epvector & v)
  80 {
  81         tinfo_key = &mul::tinfo_static;
  82         overall_coeff = _ex1;
  83         construct_from_epvector(v);
  84         GINAC_ASSERT(is_canonical());
  85 }
  86
  87 mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
  88 {
  89         tinfo_key = &mul::tinfo_static;
  90         overall_coeff = oc;
  91         construct_from_epvector(v, do_index_renaming);
  92         GINAC_ASSERT(is_canonical());
  93 }
  94
  95 mul::mul(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming)
  96 {
  97         tinfo_key = &mul::tinfo_static;
  98         GINAC_ASSERT(vp.get()!=0);
  99         overall_coeff = oc;
 100         construct_from_epvector(*vp, do_index_renaming);
 101         GINAC_ASSERT(is_canonical());
 102 }
 103
 104 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 105 {
 106         tinfo_key = &mul::tinfo_static;
 107         exvector factors;
 108         factors.reserve(3);
 109         factors.push_back(lh);
 110         factors.push_back(mh);
 111         factors.push_back(rh);
 112         overall_coeff = _ex1;
 113         construct_from_exvector(factors);
 114         GINAC_ASSERT(is_canonical());
 115 }
 116
 117 //////////
 118 // archiving
 119 //////////
 120
 121 DEFAULT_ARCHIVING(mul)
 122
 123 //////////
 124 // functions overriding virtual functions from base classes
 125 //////////
 126
 127 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
 128 {
 129         const numeric &coeff = ex_to<numeric>(overall_coeff);
 130         if (coeff.csgn() == -1)
 131                 c.s << '-';
 132         if (!coeff.is_equal(*_num1_p) &&
 133                 !coeff.is_equal(*_num_1_p)) {
 134                 if (coeff.is_rational()) {
 135                         if (coeff.is_negative())
 136                                 (-coeff).print(c);
 137                         else
 138                                 coeff.print(c);
 139                 } else {
 140                         if (coeff.csgn() == -1)
 141                                 (-coeff).print(c, precedence());
 142                         else
 143                                 coeff.print(c, precedence());
 144                 }
 145                 c.s << mul_sym;
 146         }
 147 }
 148
 149 void mul::do_print(const print_context & c, unsigned level) const
 150 {
 151         if (precedence() <= level)
 152                 c.s << '(';
 153
 154         print_overall_coeff(c, "*");
 155
 156         epvector::const_iterator it = seq.begin(), itend = seq.end();
 157         bool first = true;
 158         while (it != itend) {
 159                 if (!first)
 160                         c.s << '*';
 161                 else
 162                         first = false;
 163                 recombine_pair_to_ex(*it).print(c, precedence());
 164                 ++it;
 165         }
 166
 167         if (precedence() <= level)
 168                 c.s << ')';
 169 }
 170
 171 void mul::do_print_latex(const print_latex & c, unsigned level) const
 172 {
 173         if (precedence() <= level)
 174                 c.s << "{(";
 175
 176         print_overall_coeff(c, " ");
 177
 178         // Separate factors into those with negative numeric exponent
 179         // and all others
 180         epvector::const_iterator it = seq.begin(), itend = seq.end();
 181         exvector neg_powers, others;
 182         while (it != itend) {
 183                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
 184                 if (ex_to<numeric>(it->coeff).is_negative())
 185                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
 186                 else
 187                         others.push_back(recombine_pair_to_ex(*it));
 188                 ++it;
 189         }
 190
 191         if (!neg_powers.empty()) {
 192
 193                 // Factors with negative exponent are printed as a fraction
 194                 c.s << "\\frac{";
 195                 mul(others).eval().print(c);
 196                 c.s << "}{";
 197                 mul(neg_powers).eval().print(c);
 198                 c.s << "}";
 199
 200         } else {
 201
 202                 // All other factors are printed in the ordinary way
 203                 exvector::const_iterator vit = others.begin(), vitend = others.end();
 204                 while (vit != vitend) {
 205                         c.s << ' ';
 206                         vit->print(c, precedence());
 207                         ++vit;
 208                 }
 209         }
 210
 211         if (precedence() <= level)
 212                 c.s << ")}";
 213 }
 214
 215 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
 216 {
 217         if (precedence() <= level)
 218                 c.s << "(";
 219
 220         if (!overall_coeff.is_equal(_ex1)) {
 221                 if (overall_coeff.is_equal(_ex_1))
 222                         c.s << "-";
 223                 else {
 224                         overall_coeff.print(c, precedence());
 225                         c.s << "*";
 226                 }
 227         }
 228
 229         // Print arguments, separated by "*" or "/"
 230         epvector::const_iterator it = seq.begin(), itend = seq.end();
 231         while (it != itend) {
 232
 233                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 234                 bool needclosingparenthesis = false;
 235                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 236                         if (is_a<print_csrc_cl_N>(c)) {
 237                                 c.s << "recip(";
 238                                 needclosingparenthesis = true;
 239                         } else
 240                                 c.s << "1.0/";
 241                 }
 242
 243                 // If the exponent is 1 or -1, it is left out
 244                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 245                         it->rest.print(c, precedence());
 246                 else if (it->coeff.info(info_flags::negint))
 247                         // Outer parens around ex needed for broken GCC parser:
 248                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 249                 else
 250                         // Outer parens around ex needed for broken GCC parser:
 251                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 252
 253                 if (needclosingparenthesis)
 254                         c.s << ")";
 255
 256                 // Separator is "/" for negative integer powers, "*" otherwise
 257                 ++it;
 258                 if (it != itend) {
 259                         if (it->coeff.info(info_flags::negint))
 260                                 c.s << "/";
 261                         else
 262                                 c.s << "*";
 263                 }
 264         }
 265
 266         if (precedence() <= level)
 267                 c.s << ")";
 268 }
 269
 270 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 271 {
 272         c.s << class_name() << '(';
 273         op(0).print(c);
 274         for (size_t i=1; i<nops(); ++i) {
 275                 c.s << ',';
 276                 op(i).print(c);
 277         }
 278         c.s << ')';
 279 }
 280
 281 bool mul::info(unsigned inf) const
 282 {
 283         switch (inf) {
 284                 case info_flags::polynomial:
 285                 case info_flags::integer_polynomial:
 286                 case info_flags::cinteger_polynomial:
 287                 case info_flags::rational_polynomial:
 288                 case info_flags::crational_polynomial:
 289                 case info_flags::rational_function: {
 290                         epvector::const_iterator i = seq.begin(), end = seq.end();
 291                         while (i != end) {
 292                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 293                                         return false;
 294                                 ++i;
 295                         }
 296                         return overall_coeff.info(inf);
 297                 }
 298                 case info_flags::algebraic: {
 299                         epvector::const_iterator i = seq.begin(), end = seq.end();
 300                         while (i != end) {
 301                                 if ((recombine_pair_to_ex(*i).info(inf)))
 302                                         return true;
 303                                 ++i;
 304                         }
 305                         return false;
 306                 }
 307         }
 308         return inherited::info(inf);
 309 }
 310
 311 int mul::degree(const ex & s) const
 312 {
 313         // Sum up degrees of factors
 314         int deg_sum = 0;
 315         epvector::const_iterator i = seq.begin(), end = seq.end();
 316         while (i != end) {
 317                 if (ex_to<numeric>(i->coeff).is_integer())
 318                         deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
 319                 ++i;
 320         }
 321         return deg_sum;
 322 }
 323
 324 int mul::ldegree(const ex & s) const
 325 {
 326         // Sum up degrees of factors
 327         int deg_sum = 0;
 328         epvector::const_iterator i = seq.begin(), end = seq.end();
 329         while (i != end) {
 330                 if (ex_to<numeric>(i->coeff).is_integer())
 331                         deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
 332                 ++i;
 333         }
 334         return deg_sum;
 335 }
 336
 337 ex mul::coeff(const ex & s, int n) const
 338 {
 339         exvector coeffseq;
 340         coeffseq.reserve(seq.size()+1);
 341
 342         if (n==0) {
 343                 // product of individual coeffs
 344                 // if a non-zero power of s is found, the resulting product will be 0
 345                 epvector::const_iterator i = seq.begin(), end = seq.end();
 346                 while (i != end) {
 347                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 348                         ++i;
 349                 }
 350                 coeffseq.push_back(overall_coeff);
 351                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 352         }
 353
 354         epvector::const_iterator i = seq.begin(), end = seq.end();
 355         bool coeff_found = false;
 356         while (i != end) {
 357                 ex t = recombine_pair_to_ex(*i);
 358                 ex c = t.coeff(s, n);
 359                 if (!c.is_zero()) {
 360                         coeffseq.push_back(c);
 361                         coeff_found = 1;
 362                 } else {
 363                         coeffseq.push_back(t);
 364                 }
 365                 ++i;
 366         }
 367         if (coeff_found) {
 368                 coeffseq.push_back(overall_coeff);
 369                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 370         }
 371
 372         return _ex0;
 373 }
 374
 375 /** Perform automatic term rewriting rules in this class.  In the following
 376  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 377  *  stand for such expressions that contain a plain number.
 378  *  - *(...,x;0) -> 0
 379  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 380  *  - *(x;1) -> x
 381  *  - *(;c) -> c
 382  *
 383  *  @param level cut-off in recursive evaluation */
 384 ex mul::eval(int level) const
 385 {
 386         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 387         if (evaled_seqp.get()) {
 388                 // do more evaluation later
 389                 return (new mul(evaled_seqp, overall_coeff))->
 390                            setflag(status_flags::dynallocated);
 391         }
 392
 393 #ifdef DO_GINAC_ASSERT
 394         epvector::const_iterator i = seq.begin(), end = seq.end();
 395         while (i != end) {
 396                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 397                              (!(ex_to<numeric>(i->coeff).is_integer())));
 398                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 399                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 400                     print(print_tree(std::cerr));
 401                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 402                 /* for paranoia */
 403                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 404                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 405                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 406                 /* end paranoia */
 407                 ++i;
 408         }
 409 #endif // def DO_GINAC_ASSERT
 410
 411         if (flags & status_flags::evaluated) {
 412                 GINAC_ASSERT(seq.size()>0);
 413                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 414                 return *this;
 415         }
 416
 417         int seq_size = seq.size();
 418         if (overall_coeff.is_zero()) {
 419                 // *(...,x;0) -> 0
 420                 return _ex0;
 421         } else if (seq_size==0) {
 422                 // *(;c) -> c
 423                 return overall_coeff;
 424         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 425                 // *(x;1) -> x
 426                 return recombine_pair_to_ex(*(seq.begin()));
 427         } else if ((seq_size==1) &&
 428                    is_exactly_a<add>((*seq.begin()).rest) &&
 429                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 430                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 431                 const add & addref = ex_to<add>((*seq.begin()).rest);
 432                 std::auto_ptr<epvector> distrseq(new epvector);
 433                 distrseq->reserve(addref.seq.size());
 434                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 435                 while (i != end) {
 436                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 437                         ++i;
 438                 }
 439                 return (new add(distrseq,
 440                                 ex_to<numeric>(addref.overall_coeff).
 441                                 mul_dyn(ex_to<numeric>(overall_coeff))))
 442                       ->setflag(status_flags::dynallocated | status_flags::evaluated);
 443         }
 444         return this->hold();
 445 }
 446
 447 ex mul::evalf(int level) const
 448 {
 449         if (level==1)
 450                 return mul(seq,overall_coeff);
 451
 452         if (level==-max_recursion_level)
 453                 throw(std::runtime_error("max recursion level reached"));
 454
 455         std::auto_ptr<epvector> s(new epvector);
 456         s->reserve(seq.size());
 457
 458         --level;
 459         epvector::const_iterator i = seq.begin(), end = seq.end();
 460         while (i != end) {
 461                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 462                                                            i->coeff));
 463                 ++i;
 464         }
 465         return mul(s, overall_coeff.evalf(level));
 466 }
 467
 468 void mul::find_real_imag(ex & rp, ex & ip) const
 469 {
 470         rp = overall_coeff.real_part();
 471         ip = overall_coeff.imag_part();
 472         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 473                 ex factor = recombine_pair_to_ex(*i);
 474                 ex new_rp = factor.real_part();
 475                 ex new_ip = factor.imag_part();
 476                 if(new_ip.is_zero()) {
 477                         rp *= new_rp;
 478                         ip *= new_rp;
 479                 } else {
 480                         ex temp = rp*new_rp - ip*new_ip;
 481                         ip = ip*new_rp + rp*new_ip;
 482                         rp = temp;
 483                 }
 484         }
 485         rp = rp.expand();
 486         ip = ip.expand();
 487 }
 488
 489 ex mul::real_part() const
 490 {
 491         ex rp, ip;
 492         find_real_imag(rp, ip);
 493         return rp;
 494 }
 495
 496 ex mul::imag_part() const
 497 {
 498         ex rp, ip;
 499         find_real_imag(rp, ip);
 500         return ip;
 501 }
 502
 503 ex mul::evalm() const
 504 {
 505         // numeric*matrix
 506         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 507          && is_a<matrix>(seq[0].rest))
 508                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 509
 510         // Evaluate children first, look whether there are any matrices at all
 511         // (there can be either no matrices or one matrix; if there were more
 512         // than one matrix, it would be a non-commutative product)
 513         std::auto_ptr<epvector> s(new epvector);
 514         s->reserve(seq.size());
 515
 516         bool have_matrix = false;
 517         epvector::iterator the_matrix;
 518
 519         epvector::const_iterator i = seq.begin(), end = seq.end();
 520         while (i != end) {
 521                 const ex &m = recombine_pair_to_ex(*i).evalm();
 522                 s->push_back(split_ex_to_pair(m));
 523                 if (is_a<matrix>(m)) {
 524                         have_matrix = true;
 525                         the_matrix = s->end() - 1;
 526                 }
 527                 ++i;
 528         }
 529
 530         if (have_matrix) {
 531
 532                 // The product contained a matrix. We will multiply all other factors
 533                 // into that matrix.
 534                 matrix m = ex_to<matrix>(the_matrix->rest);
 535                 s->erase(the_matrix);
 536                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 537                 return m.mul_scalar(scalar);
 538
 539         } else
 540                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 541 }
 542
 543 ex mul::eval_ncmul(const exvector & v) const
 544 {
 545         if (seq.empty())
 546                 return inherited::eval_ncmul(v);
 547
 548         // Find first noncommutative element and call its eval_ncmul()
 549         epvector::const_iterator i = seq.begin(), end = seq.end();
 550         while (i != end) {
 551                 if (i->rest.return_type() == return_types::noncommutative)
 552                         return i->rest.eval_ncmul(v);
 553                 ++i;
 554         }
 555         return inherited::eval_ncmul(v);
 556 }
 557
 558 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
 559 {
 560         ex origbase;
 561         int origexponent;
 562         int origexpsign;
 563
 564         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 565                 origbase = origfactor.op(0);
 566                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 567                 origexponent = expon > 0 ? expon : -expon;
 568                 origexpsign = expon > 0 ? 1 : -1;
 569         } else {
 570                 origbase = origfactor;
 571                 origexponent = 1;
 572                 origexpsign = 1;
 573         }
 574
 575         ex patternbase;
 576         int patternexponent;
 577         int patternexpsign;
 578
 579         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 580                 patternbase = patternfactor.op(0);
 581                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 582                 patternexponent = expon > 0 ? expon : -expon;
 583                 patternexpsign = expon > 0 ? 1 : -1;
 584         } else {
 585                 patternbase = patternfactor;
 586                 patternexponent = 1;
 587                 patternexpsign = 1;
 588         }
 589
 590         lst saverepls = repls;
 591         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 592                 return false;
 593         repls = saverepls;
 594
 595         int newnummatches = origexponent / patternexponent;
 596         if (newnummatches < nummatches)
 597                 nummatches = newnummatches;
 598         return true;
 599 }
 600
 601 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
 602   * list of replacements repls. This matching is in the sense of algebraic
 603   * substitutions. Matching starts with pat.op(factor) of the pattern because
 604   * the factors before this one have already been matched. The (possibly
 605   * updated) number of matches is in nummatches. subsed[i] is true for factors
 606   * that already have been replaced by previous substitutions and matched[i]
 607   * is true for factors that have been matched by the current match.
 608   */
 609 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
 610                 int factor, int &nummatches, const std::vector<bool> &subsed,
 611                 std::vector<bool> &matched)
 612 {
 613         if (factor == pat.nops())
 614                 return true;
 615
 616         for (size_t i=0; i<e.nops(); ++i) {
 617                 if(subsed[i] || matched[i])
 618                         continue;
 619                 lst newrepls = repls;
 620                 int newnummatches = nummatches;
 621                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 622                         matched[i] = true;
 623                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 624                                         newnummatches, subsed, matched)) {
 625                                 repls = newrepls;
 626                                 nummatches = newnummatches;
 627                                 return true;
 628                         }
 629                         else
 630                                 matched[i] = false;
 631                 }
 632         }
 633
 634         return false;
 635 }
 636
 637 bool mul::has(const ex & pattern, unsigned options) const
 638 {
 639         if(!(options&has_options::algebraic))
 640                 return basic::has(pattern,options);
 641         if(is_a<mul>(pattern)) {
 642                 lst repls;
 643                 int nummatches = std::numeric_limits<int>::max();
 644                 std::vector<bool> subsed(seq.size(), false);
 645                 std::vector<bool> matched(seq.size(), false);
 646                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 647                                 subsed, matched))
 648                         return true;
 649         }
 650         return basic::has(pattern, options);
 651 }
 652
 653 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 654 {
 655         std::vector<bool> subsed(seq.size(), false);
 656         exvector subsresult(seq.size());
 657         ex divide_by = 1;
 658         ex multiply_by = 1;
 659
 660         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 661
 662                 if (is_exactly_a<mul>(it->first)) {
 663 retry1:
 664                         int nummatches = std::numeric_limits<int>::max();
 665                         std::vector<bool> currsubsed(seq.size(), false);
 666                         lst repls;
 667
 668                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 669                                 continue;
 670
 671                         for (size_t j=0; j<subsed.size(); j++)
 672                                 if (currsubsed[j])
 673                                         subsed[j] = true;
 674                         ex subsed_pattern
 675                                 = it->first.subs(ex(repls), subs_options::no_pattern);
 676                         divide_by *= power(subsed_pattern, nummatches);
 677                         ex subsed_result
 678                                 = it->second.subs(ex(repls), subs_options::no_pattern);
 679                         multiply_by *= power(subsed_result, nummatches);
 680                         goto retry1;
 681
 682                 } else {
 683
 684                         for (size_t j=0; j<this->nops(); j++) {
 685                                 int nummatches = std::numeric_limits<int>::max();
 686                                 lst repls;
 687                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
 688                                         subsed[j] = true;
 689                                         ex subsed_pattern
 690                                                 = it->first.subs(ex(repls), subs_options::no_pattern);
 691                                         divide_by *= power(subsed_pattern, nummatches);
 692                                         ex subsed_result
 693                                                 = it->second.subs(ex(repls), subs_options::no_pattern);
 694                                         multiply_by *= power(subsed_result, nummatches);
 695                                 }
 696                         }
 697                 }
 698         }
 699
 700         bool subsfound = false;
 701         for (size_t i=0; i<subsed.size(); i++) {
 702                 if (subsed[i]) {
 703                         subsfound = true;
 704                         break;
 705                 }
 706         }
 707         if (!subsfound)
 708                 return subs_one_level(m, options | subs_options::algebraic);
 709
 710         return ((*this)/divide_by)*multiply_by;
 711 }
 712
 713 // protected
 714
 715 /** Implementation of ex::diff() for a product.  It applies the product rule.
 716  *  @see ex::diff */
 717 ex mul::derivative(const symbol & s) const
 718 {
 719         size_t num = seq.size();
 720         exvector addseq;
 721         addseq.reserve(num);
 722
 723         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 724         epvector mulseq = seq;
 725         epvector::const_iterator i = seq.begin(), end = seq.end();
 726         epvector::iterator i2 = mulseq.begin();
 727         while (i != end) {
 728                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 729                                              i->rest.diff(s));
 730                 ep.swap(*i2);
 731                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 732                 ep.swap(*i2);
 733                 ++i; ++i2;
 734         }
 735         return (new add(addseq))->setflag(status_flags::dynallocated);
 736 }
 737
 738 int mul::compare_same_type(const basic & other) const
 739 {
 740         return inherited::compare_same_type(other);
 741 }
 742
 743 unsigned mul::return_type() const
 744 {
 745         if (seq.empty()) {
 746                 // mul without factors: should not happen, but commutates
 747                 return return_types::commutative;
 748         }
 749
 750         bool all_commutative = true;
 751         epvector::const_iterator noncommutative_element; // point to first found nc element
 752
 753         epvector::const_iterator i = seq.begin(), end = seq.end();
 754         while (i != end) {
 755                 unsigned rt = i->rest.return_type();
 756                 if (rt == return_types::noncommutative_composite)
 757                         return rt; // one ncc -> mul also ncc
 758                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 759                         // first nc element found, remember position
 760                         noncommutative_element = i;
 761                         all_commutative = false;
 762                 }
 763                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 764                         // another nc element found, compare type_infos
 765                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 766                                         // different types -> mul is ncc
 767                                         return return_types::noncommutative_composite;
 768                         }
 769                 }
 770                 ++i;
 771         }
 772         // all factors checked
 773         return all_commutative ? return_types::commutative : return_types::noncommutative;
 774 }
 775
 776 tinfo_t mul::return_type_tinfo() const
 777 {
 778         if (seq.empty())
 779                 return this;  // mul without factors: should not happen
 780
 781         // return type_info of first noncommutative element
 782         epvector::const_iterator i = seq.begin(), end = seq.end();
 783         while (i != end) {
 784                 if (i->rest.return_type() == return_types::noncommutative)
 785                         return i->rest.return_type_tinfo();
 786                 ++i;
 787         }
 788         // no noncommutative element found, should not happen
 789         return this;
 790 }
 791
 792 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 793 {
 794         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 795 }
 796
 797 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
 798 {
 799         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 800 }
 801
 802 expair mul::split_ex_to_pair(const ex & e) const
 803 {
 804         if (is_exactly_a<power>(e)) {
 805                 const power & powerref = ex_to<power>(e);
 806                 if (is_exactly_a<numeric>(powerref.exponent))
 807                         return expair(powerref.basis,powerref.exponent);
 808         }
 809         return expair(e,_ex1);
 810 }
 811
 812 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 813                                           const ex & c) const
 814 {
 815         // to avoid duplication of power simplification rules,
 816         // we create a temporary power object
 817         // otherwise it would be hard to correctly evaluate
 818         // expression like (4^(1/3))^(3/2)
 819         if (c.is_equal(_ex1))
 820                 return split_ex_to_pair(e);
 821
 822         return split_ex_to_pair(power(e,c));
 823 }
 824
 825 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 826                                             const ex & c) const
 827 {
 828         // to avoid duplication of power simplification rules,
 829         // we create a temporary power object
 830         // otherwise it would be hard to correctly evaluate
 831         // expression like (4^(1/3))^(3/2)
 832         if (c.is_equal(_ex1))
 833                 return p;
 834
 835         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 836 }
 837
 838 ex mul::recombine_pair_to_ex(const expair & p) const
 839 {
 840         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 841                 return p.rest;
 842         else
 843                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 844 }
 845
 846 bool mul::expair_needs_further_processing(epp it)
 847 {
 848         if (is_exactly_a<mul>(it->rest) &&
 849                 ex_to<numeric>(it->coeff).is_integer()) {
 850                 // combined pair is product with integer power -> expand it
 851                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 852                 return true;
 853         }
 854         if (is_exactly_a<numeric>(it->rest)) {
 855                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 856                 if (!ep.is_equal(*it)) {
 857                         // combined pair is a numeric power which can be simplified
 858                         *it = ep;
 859                         return true;
 860                 }
 861                 if (it->coeff.is_equal(_ex1)) {
 862                         // combined pair has coeff 1 and must be moved to the end
 863                         return true;
 864                 }
 865         }
 866         return false;
 867 }
 868
 869 ex mul::default_overall_coeff() const
 870 {
 871         return _ex1;
 872 }
 873
 874 void mul::combine_overall_coeff(const ex & c)
 875 {
 876         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 877         GINAC_ASSERT(is_exactly_a<numeric>(c));
 878         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 879 }
 880
 881 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 882 {
 883         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 884         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 885         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 886         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 887 }
 888
 889 bool mul::can_make_flat(const expair & p) const
 890 {
 891         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 892         // this assertion will probably fail somewhere
 893         // it would require a more careful make_flat, obeying the power laws
 894         // probably should return true only if p.coeff is integer
 895         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 896 }
 897
 898 bool mul::can_be_further_expanded(const ex & e)
 899 {
 900         if (is_exactly_a<mul>(e)) {
 901                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 902                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 903                                 return true;
 904                 }
 905         } else if (is_exactly_a<power>(e)) {
 906                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 907                         return true;
 908         }
 909         return false;
 910 }
 911
 912 ex mul::expand(unsigned options) const
 913 {
 914         // First, expand the children
 915         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
 916         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
 917
 918         // Now, look for all the factors that are sums and multiply each one out
 919         // with the next one that is found while collecting the factors which are
 920         // not sums
 921         ex last_expanded = _ex1;
 922
 923         epvector non_adds;
 924         non_adds.reserve(expanded_seq.size());
 925
 926         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
 927                 if (is_exactly_a<add>(cit->rest) &&
 928                         (cit->coeff.is_equal(_ex1))) {
 929                         if (is_exactly_a<add>(last_expanded)) {
 930
 931                                 // Expand a product of two sums, aggressive version.
 932                                 // Caring for the overall coefficients in separate loops can
 933                                 // sometimes give a performance gain of up to 15%!
 934
 935                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
 936                                 // add2 is for the inner loop and should be the bigger of the two sums
 937                                 // in the presence of asymptotically good sorting:
 938                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
 939                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
 940                                 const epvector::const_iterator add1begin = add1.seq.begin();
 941                                 const epvector::const_iterator add1end   = add1.seq.end();
 942                                 const epvector::const_iterator add2begin = add2.seq.begin();
 943                                 const epvector::const_iterator add2end   = add2.seq.end();
 944                                 epvector distrseq;
 945                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
 946
 947                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
 948                                 if (!add1.overall_coeff.is_zero()) {
 949                                         if (add1.overall_coeff.is_equal(_ex1))
 950                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
 951                                         else
 952                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
 953                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
 954                                 }
 955
 956                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
 957                                 if (!add2.overall_coeff.is_zero()) {
 958                                         if (add2.overall_coeff.is_equal(_ex1))
 959                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
 960                                         else
 961                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
 962                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
 963                                 }
 964
 965                                 // Compute the new overall coefficient and put it together:
 966                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
 967
 968                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
 969
 970                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
 971                                         add_indices = get_all_dummy_indices_safely(i->rest);
 972                                         add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
 973                                 }
 974                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
 975                                         add_indices = get_all_dummy_indices_safely(i->rest);
 976                                         add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
 977                                 }
 978
 979                                 sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
 980                                 sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
 981                                 lst dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
 982
 983                                 // Multiply explicitly all non-numeric terms of add1 and add2:
 984                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
 985                                         // We really have to combine terms here in order to compactify
 986                                         // the result.  Otherwise it would become waayy tooo bigg.
 987                                         numeric oc;
 988                                         distrseq.clear();
 989                                         ex i2_new = (dummy_subs.op(0).nops()>0?
 990                                                                  i2->rest.subs((lst)dummy_subs.op(0), (lst)dummy_subs.op(1), subs_options::no_pattern) : i2->rest);
 991                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
 992                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
 993                                                 // since that would violate an invariant of expairseq:
 994                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
 995                                                 if (is_exactly_a<numeric>(rest)) {
 996                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
 997                                                 } else {
 998                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
 999                                                 }
1000                                         }
1001                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
1002                                 }
1003                                 last_expanded = tmp_accu;
1004
1005                         } else {
1006                                 if (!last_expanded.is_equal(_ex1))
1007                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1008                                 last_expanded = cit->rest;
1009                         }
1010
1011                 } else {
1012                         non_adds.push_back(*cit);
1013                 }
1014         }
1015
1016         // Now the only remaining thing to do is to multiply the factors which
1017         // were not sums into the "last_expanded" sum
1018         if (is_exactly_a<add>(last_expanded)) {
1019                 size_t n = last_expanded.nops();
1020                 exvector distrseq;
1021                 distrseq.reserve(n);
1022                 exvector va = get_all_dummy_indices_safely(mul(non_adds));
1023                 sort(va.begin(), va.end(), ex_is_less());
1024
1025                 for (size_t i=0; i<n; ++i) {
1026                         epvector factors = non_adds;
1027                         factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1028                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1029                         if (can_be_further_expanded(term)) {
1030                                 distrseq.push_back(term.expand());
1031                         } else {
1032                                 if (options == 0)
1033                                         ex_to<basic>(term).setflag(status_flags::expanded);
1034                                 distrseq.push_back(term);
1035                         }
1036                 }
1037
1038                 return ((new add(distrseq))->
1039                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1040         }
1041
1042         non_adds.push_back(split_ex_to_pair(last_expanded));
1043         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1044         if (can_be_further_expanded(result)) {
1045                 return result.expand();
1046         } else {
1047                 if (options == 0)
1048                         ex_to<basic>(result).setflag(status_flags::expanded);
1049                 return result;
1050         }
1051 }
1052
1053
1054 //////////
1055 // new virtual functions which can be overridden by derived classes
1056 //////////
1057
1058 // none
1059
1060 //////////
1061 // non-virtual functions in this class
1062 //////////
1063
1064
1065 /** Member-wise expand the expairs representing this sequence.  This must be
1066  *  overridden from expairseq::expandchildren() and done iteratively in order
1067  *  to allow for early cancallations and thus safe memory.
1068  *
1069  *  @see mul::expand()
1070  *  @return pointer to epvector containing expanded representation or zero
1071  *  pointer, if sequence is unchanged. */
1072 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1073 {
1074         const epvector::const_iterator last = seq.end();
1075         epvector::const_iterator cit = seq.begin();
1076         while (cit!=last) {
1077                 const ex & factor = recombine_pair_to_ex(*cit);
1078                 const ex & expanded_factor = factor.expand(options);
1079                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1080
1081                         // something changed, copy seq, eval and return it
1082                         std::auto_ptr<epvector> s(new epvector);
1083                         s->reserve(seq.size());
1084
1085                         // copy parts of seq which are known not to have changed
1086                         epvector::const_iterator cit2 = seq.begin();
1087                         while (cit2!=cit) {
1088                                 s->push_back(*cit2);
1089                                 ++cit2;
1090                         }
1091
1092                         // copy first changed element
1093                         s->push_back(split_ex_to_pair(expanded_factor));
1094                         ++cit2;
1095
1096                         // copy rest
1097                         while (cit2!=last) {
1098                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1099                                 ++cit2;
1100                         }
1101                         return s;
1102                 }
1103                 ++cit;
1104         }
1105
1106         return std::auto_ptr<epvector>(0); // nothing has changed
1107 }
1108
1109 } // namespace GiNaC