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 += recombine_pair_to_ex(*i).degree(s);
 319                 else {
 320                         if (i->rest.has(s))
 321                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
 322                 }
 323                 ++i;
 324         }
 325         return deg_sum;
 326 }
 327
 328 int mul::ldegree(const ex & s) const
 329 {
 330         // Sum up degrees of factors
 331         int deg_sum = 0;
 332         epvector::const_iterator i = seq.begin(), end = seq.end();
 333         while (i != end) {
 334                 if (ex_to<numeric>(i->coeff).is_integer())
 335                         deg_sum += recombine_pair_to_ex(*i).ldegree(s);
 336                 else {
 337                         if (i->rest.has(s))
 338                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
 339                 }
 340                 ++i;
 341         }
 342         return deg_sum;
 343 }
 344
 345 ex mul::coeff(const ex & s, int n) const
 346 {
 347         exvector coeffseq;
 348         coeffseq.reserve(seq.size()+1);
 349
 350         if (n==0) {
 351                 // product of individual coeffs
 352                 // if a non-zero power of s is found, the resulting product will be 0
 353                 epvector::const_iterator i = seq.begin(), end = seq.end();
 354                 while (i != end) {
 355                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 356                         ++i;
 357                 }
 358                 coeffseq.push_back(overall_coeff);
 359                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 360         }
 361
 362         epvector::const_iterator i = seq.begin(), end = seq.end();
 363         bool coeff_found = false;
 364         while (i != end) {
 365                 ex t = recombine_pair_to_ex(*i);
 366                 ex c = t.coeff(s, n);
 367                 if (!c.is_zero()) {
 368                         coeffseq.push_back(c);
 369                         coeff_found = 1;
 370                 } else {
 371                         coeffseq.push_back(t);
 372                 }
 373                 ++i;
 374         }
 375         if (coeff_found) {
 376                 coeffseq.push_back(overall_coeff);
 377                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 378         }
 379
 380         return _ex0;
 381 }
 382
 383 /** Perform automatic term rewriting rules in this class.  In the following
 384  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 385  *  stand for such expressions that contain a plain number.
 386  *  - *(...,x;0) -> 0
 387  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 388  *  - *(x;1) -> x
 389  *  - *(;c) -> c
 390  *
 391  *  @param level cut-off in recursive evaluation */
 392 ex mul::eval(int level) const
 393 {
 394         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 395         if (evaled_seqp.get()) {
 396                 // do more evaluation later
 397                 return (new mul(evaled_seqp, overall_coeff))->
 398                            setflag(status_flags::dynallocated);
 399         }
 400
 401 #ifdef DO_GINAC_ASSERT
 402         epvector::const_iterator i = seq.begin(), end = seq.end();
 403         while (i != end) {
 404                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 405                              (!(ex_to<numeric>(i->coeff).is_integer())));
 406                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 407                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 408                     print(print_tree(std::cerr));
 409                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 410                 /* for paranoia */
 411                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 412                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 413                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 414                 /* end paranoia */
 415                 ++i;
 416         }
 417 #endif // def DO_GINAC_ASSERT
 418
 419         if (flags & status_flags::evaluated) {
 420                 GINAC_ASSERT(seq.size()>0);
 421                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 422                 return *this;
 423         }
 424
 425         int seq_size = seq.size();
 426         if (overall_coeff.is_zero()) {
 427                 // *(...,x;0) -> 0
 428                 return _ex0;
 429         } else if (seq_size==0) {
 430                 // *(;c) -> c
 431                 return overall_coeff;
 432         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 433                 // *(x;1) -> x
 434                 return recombine_pair_to_ex(*(seq.begin()));
 435         } else if ((seq_size==1) &&
 436                    is_exactly_a<add>((*seq.begin()).rest) &&
 437                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 438                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 439                 const add & addref = ex_to<add>((*seq.begin()).rest);
 440                 std::auto_ptr<epvector> distrseq(new epvector);
 441                 distrseq->reserve(addref.seq.size());
 442                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 443                 while (i != end) {
 444                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 445                         ++i;
 446                 }
 447                 return (new add(distrseq,
 448                                 ex_to<numeric>(addref.overall_coeff).
 449                                 mul_dyn(ex_to<numeric>(overall_coeff))))
 450                       ->setflag(status_flags::dynallocated | status_flags::evaluated);
 451         }
 452         return this->hold();
 453 }
 454
 455 ex mul::evalf(int level) const
 456 {
 457         if (level==1)
 458                 return mul(seq,overall_coeff);
 459
 460         if (level==-max_recursion_level)
 461                 throw(std::runtime_error("max recursion level reached"));
 462
 463         std::auto_ptr<epvector> s(new epvector);
 464         s->reserve(seq.size());
 465
 466         --level;
 467         epvector::const_iterator i = seq.begin(), end = seq.end();
 468         while (i != end) {
 469                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 470                                                            i->coeff));
 471                 ++i;
 472         }
 473         return mul(s, overall_coeff.evalf(level));
 474 }
 475
 476 void mul::find_real_imag(ex & rp, ex & ip) const
 477 {
 478         rp = overall_coeff.real_part();
 479         ip = overall_coeff.imag_part();
 480         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 481                 ex factor = recombine_pair_to_ex(*i);
 482                 ex new_rp = factor.real_part();
 483                 ex new_ip = factor.imag_part();
 484                 if(new_ip.is_zero()) {
 485                         rp *= new_rp;
 486                         ip *= new_rp;
 487                 } else {
 488                         ex temp = rp*new_rp - ip*new_ip;
 489                         ip = ip*new_rp + rp*new_ip;
 490                         rp = temp;
 491                 }
 492         }
 493         rp = rp.expand();
 494         ip = ip.expand();
 495 }
 496
 497 ex mul::real_part() const
 498 {
 499         ex rp, ip;
 500         find_real_imag(rp, ip);
 501         return rp;
 502 }
 503
 504 ex mul::imag_part() const
 505 {
 506         ex rp, ip;
 507         find_real_imag(rp, ip);
 508         return ip;
 509 }
 510
 511 ex mul::evalm() const
 512 {
 513         // numeric*matrix
 514         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 515          && is_a<matrix>(seq[0].rest))
 516                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 517
 518         // Evaluate children first, look whether there are any matrices at all
 519         // (there can be either no matrices or one matrix; if there were more
 520         // than one matrix, it would be a non-commutative product)
 521         std::auto_ptr<epvector> s(new epvector);
 522         s->reserve(seq.size());
 523
 524         bool have_matrix = false;
 525         epvector::iterator the_matrix;
 526
 527         epvector::const_iterator i = seq.begin(), end = seq.end();
 528         while (i != end) {
 529                 const ex &m = recombine_pair_to_ex(*i).evalm();
 530                 s->push_back(split_ex_to_pair(m));
 531                 if (is_a<matrix>(m)) {
 532                         have_matrix = true;
 533                         the_matrix = s->end() - 1;
 534                 }
 535                 ++i;
 536         }
 537
 538         if (have_matrix) {
 539
 540                 // The product contained a matrix. We will multiply all other factors
 541                 // into that matrix.
 542                 matrix m = ex_to<matrix>(the_matrix->rest);
 543                 s->erase(the_matrix);
 544                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 545                 return m.mul_scalar(scalar);
 546
 547         } else
 548                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 549 }
 550
 551 ex mul::eval_ncmul(const exvector & v) const
 552 {
 553         if (seq.empty())
 554                 return inherited::eval_ncmul(v);
 555
 556         // Find first noncommutative element and call its eval_ncmul()
 557         epvector::const_iterator i = seq.begin(), end = seq.end();
 558         while (i != end) {
 559                 if (i->rest.return_type() == return_types::noncommutative)
 560                         return i->rest.eval_ncmul(v);
 561                 ++i;
 562         }
 563         return inherited::eval_ncmul(v);
 564 }
 565
 566 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
 567 {
 568         ex origbase;
 569         int origexponent;
 570         int origexpsign;
 571
 572         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 573                 origbase = origfactor.op(0);
 574                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 575                 origexponent = expon > 0 ? expon : -expon;
 576                 origexpsign = expon > 0 ? 1 : -1;
 577         } else {
 578                 origbase = origfactor;
 579                 origexponent = 1;
 580                 origexpsign = 1;
 581         }
 582
 583         ex patternbase;
 584         int patternexponent;
 585         int patternexpsign;
 586
 587         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 588                 patternbase = patternfactor.op(0);
 589                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 590                 patternexponent = expon > 0 ? expon : -expon;
 591                 patternexpsign = expon > 0 ? 1 : -1;
 592         } else {
 593                 patternbase = patternfactor;
 594                 patternexponent = 1;
 595                 patternexpsign = 1;
 596         }
 597
 598         lst saverepls = repls;
 599         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 600                 return false;
 601         repls = saverepls;
 602
 603         int newnummatches = origexponent / patternexponent;
 604         if (newnummatches < nummatches)
 605                 nummatches = newnummatches;
 606         return true;
 607 }
 608
 609 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
 610   * list of replacements repls. This matching is in the sense of algebraic
 611   * substitutions. Matching starts with pat.op(factor) of the pattern because
 612   * the factors before this one have already been matched. The (possibly
 613   * updated) number of matches is in nummatches. subsed[i] is true for factors
 614   * that already have been replaced by previous substitutions and matched[i]
 615   * is true for factors that have been matched by the current match.
 616   */
 617 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
 618                 int factor, int &nummatches, const std::vector<bool> &subsed,
 619                 std::vector<bool> &matched)
 620 {
 621         if (factor == pat.nops())
 622                 return true;
 623
 624         for (size_t i=0; i<e.nops(); ++i) {
 625                 if(subsed[i] || matched[i])
 626                         continue;
 627                 lst newrepls = repls;
 628                 int newnummatches = nummatches;
 629                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 630                         matched[i] = true;
 631                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 632                                         newnummatches, subsed, matched)) {
 633                                 repls = newrepls;
 634                                 nummatches = newnummatches;
 635                                 return true;
 636                         }
 637                         else
 638                                 matched[i] = false;
 639                 }
 640         }
 641
 642         return false;
 643 }
 644
 645 bool mul::has(const ex & pattern, unsigned options) const
 646 {
 647         if(!(options&has_options::algebraic))
 648                 return basic::has(pattern,options);
 649         if(is_a<mul>(pattern)) {
 650                 lst repls;
 651                 int nummatches = std::numeric_limits<int>::max();
 652                 std::vector<bool> subsed(seq.size(), false);
 653                 std::vector<bool> matched(seq.size(), false);
 654                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 655                                 subsed, matched))
 656                         return true;
 657         }
 658         return basic::has(pattern, options);
 659 }
 660
 661 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 662 {
 663         std::vector<bool> subsed(seq.size(), false);
 664         exvector subsresult(seq.size());
 665         ex divide_by = 1;
 666         ex multiply_by = 1;
 667
 668         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 669
 670                 if (is_exactly_a<mul>(it->first)) {
 671 retry1:
 672                         int nummatches = std::numeric_limits<int>::max();
 673                         std::vector<bool> currsubsed(seq.size(), false);
 674                         lst repls;
 675
 676                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 677                                 continue;
 678
 679                         for (size_t j=0; j<subsed.size(); j++)
 680                                 if (currsubsed[j])
 681                                         subsed[j] = true;
 682                         ex subsed_pattern
 683                                 = it->first.subs(ex(repls), subs_options::no_pattern);
 684                         divide_by *= power(subsed_pattern, nummatches);
 685                         ex subsed_result
 686                                 = it->second.subs(ex(repls), subs_options::no_pattern);
 687                         multiply_by *= power(subsed_result, nummatches);
 688                         goto retry1;
 689
 690                 } else {
 691
 692                         for (size_t j=0; j<this->nops(); j++) {
 693                                 int nummatches = std::numeric_limits<int>::max();
 694                                 lst repls;
 695                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
 696                                         subsed[j] = true;
 697                                         ex subsed_pattern
 698                                                 = it->first.subs(ex(repls), subs_options::no_pattern);
 699                                         divide_by *= power(subsed_pattern, nummatches);
 700                                         ex subsed_result
 701                                                 = it->second.subs(ex(repls), subs_options::no_pattern);
 702                                         multiply_by *= power(subsed_result, nummatches);
 703                                 }
 704                         }
 705                 }
 706         }
 707
 708         bool subsfound = false;
 709         for (size_t i=0; i<subsed.size(); i++) {
 710                 if (subsed[i]) {
 711                         subsfound = true;
 712                         break;
 713                 }
 714         }
 715         if (!subsfound)
 716                 return subs_one_level(m, options | subs_options::algebraic);
 717
 718         return ((*this)/divide_by)*multiply_by;
 719 }
 720
 721 // protected
 722
 723 /** Implementation of ex::diff() for a product.  It applies the product rule.
 724  *  @see ex::diff */
 725 ex mul::derivative(const symbol & s) const
 726 {
 727         size_t num = seq.size();
 728         exvector addseq;
 729         addseq.reserve(num);
 730
 731         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 732         epvector mulseq = seq;
 733         epvector::const_iterator i = seq.begin(), end = seq.end();
 734         epvector::iterator i2 = mulseq.begin();
 735         while (i != end) {
 736                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 737                                              i->rest.diff(s));
 738                 ep.swap(*i2);
 739                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 740                 ep.swap(*i2);
 741                 ++i; ++i2;
 742         }
 743         return (new add(addseq))->setflag(status_flags::dynallocated);
 744 }
 745
 746 int mul::compare_same_type(const basic & other) const
 747 {
 748         return inherited::compare_same_type(other);
 749 }
 750
 751 unsigned mul::return_type() const
 752 {
 753         if (seq.empty()) {
 754                 // mul without factors: should not happen, but commutates
 755                 return return_types::commutative;
 756         }
 757
 758         bool all_commutative = true;
 759         epvector::const_iterator noncommutative_element; // point to first found nc element
 760
 761         epvector::const_iterator i = seq.begin(), end = seq.end();
 762         while (i != end) {
 763                 unsigned rt = i->rest.return_type();
 764                 if (rt == return_types::noncommutative_composite)
 765                         return rt; // one ncc -> mul also ncc
 766                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 767                         // first nc element found, remember position
 768                         noncommutative_element = i;
 769                         all_commutative = false;
 770                 }
 771                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 772                         // another nc element found, compare type_infos
 773                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 774                                         // different types -> mul is ncc
 775                                         return return_types::noncommutative_composite;
 776                         }
 777                 }
 778                 ++i;
 779         }
 780         // all factors checked
 781         return all_commutative ? return_types::commutative : return_types::noncommutative;
 782 }
 783
 784 tinfo_t mul::return_type_tinfo() const
 785 {
 786         if (seq.empty())
 787                 return this;  // mul without factors: should not happen
 788
 789         // return type_info of first noncommutative element
 790         epvector::const_iterator i = seq.begin(), end = seq.end();
 791         while (i != end) {
 792                 if (i->rest.return_type() == return_types::noncommutative)
 793                         return i->rest.return_type_tinfo();
 794                 ++i;
 795         }
 796         // no noncommutative element found, should not happen
 797         return this;
 798 }
 799
 800 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 801 {
 802         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 803 }
 804
 805 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
 806 {
 807         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 808 }
 809
 810 expair mul::split_ex_to_pair(const ex & e) const
 811 {
 812         if (is_exactly_a<power>(e)) {
 813                 const power & powerref = ex_to<power>(e);
 814                 if (is_exactly_a<numeric>(powerref.exponent))
 815                         return expair(powerref.basis,powerref.exponent);
 816         }
 817         return expair(e,_ex1);
 818 }
 819
 820 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 821                                           const ex & c) const
 822 {
 823         // to avoid duplication of power simplification rules,
 824         // we create a temporary power object
 825         // otherwise it would be hard to correctly evaluate
 826         // expression like (4^(1/3))^(3/2)
 827         if (c.is_equal(_ex1))
 828                 return split_ex_to_pair(e);
 829
 830         return split_ex_to_pair(power(e,c));
 831 }
 832
 833 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 834                                             const ex & c) const
 835 {
 836         // to avoid duplication of power simplification rules,
 837         // we create a temporary power object
 838         // otherwise it would be hard to correctly evaluate
 839         // expression like (4^(1/3))^(3/2)
 840         if (c.is_equal(_ex1))
 841                 return p;
 842
 843         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 844 }
 845
 846 ex mul::recombine_pair_to_ex(const expair & p) const
 847 {
 848         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 849                 return p.rest;
 850         else
 851                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 852 }
 853
 854 bool mul::expair_needs_further_processing(epp it)
 855 {
 856         if (is_exactly_a<mul>(it->rest) &&
 857                 ex_to<numeric>(it->coeff).is_integer()) {
 858                 // combined pair is product with integer power -> expand it
 859                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 860                 return true;
 861         }
 862         if (is_exactly_a<numeric>(it->rest)) {
 863                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 864                 if (!ep.is_equal(*it)) {
 865                         // combined pair is a numeric power which can be simplified
 866                         *it = ep;
 867                         return true;
 868                 }
 869                 if (it->coeff.is_equal(_ex1)) {
 870                         // combined pair has coeff 1 and must be moved to the end
 871                         return true;
 872                 }
 873         }
 874         return false;
 875 }
 876
 877 ex mul::default_overall_coeff() const
 878 {
 879         return _ex1;
 880 }
 881
 882 void mul::combine_overall_coeff(const ex & c)
 883 {
 884         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 885         GINAC_ASSERT(is_exactly_a<numeric>(c));
 886         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 887 }
 888
 889 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 890 {
 891         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 892         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 893         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 894         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 895 }
 896
 897 bool mul::can_make_flat(const expair & p) const
 898 {
 899         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 900         // this assertion will probably fail somewhere
 901         // it would require a more careful make_flat, obeying the power laws
 902         // probably should return true only if p.coeff is integer
 903         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 904 }
 905
 906 bool mul::can_be_further_expanded(const ex & e)
 907 {
 908         if (is_exactly_a<mul>(e)) {
 909                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 910                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 911                                 return true;
 912                 }
 913         } else if (is_exactly_a<power>(e)) {
 914                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 915                         return true;
 916         }
 917         return false;
 918 }
 919
 920 ex mul::expand(unsigned options) const
 921 {
 922         // First, expand the children
 923         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
 924         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
 925
 926         // Now, look for all the factors that are sums and multiply each one out
 927         // with the next one that is found while collecting the factors which are
 928         // not sums
 929         ex last_expanded = _ex1;
 930
 931         epvector non_adds;
 932         non_adds.reserve(expanded_seq.size());
 933
 934         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
 935                 if (is_exactly_a<add>(cit->rest) &&
 936                         (cit->coeff.is_equal(_ex1))) {
 937                         if (is_exactly_a<add>(last_expanded)) {
 938
 939                                 // Expand a product of two sums, aggressive version.
 940                                 // Caring for the overall coefficients in separate loops can
 941                                 // sometimes give a performance gain of up to 15%!
 942
 943                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
 944                                 // add2 is for the inner loop and should be the bigger of the two sums
 945                                 // in the presence of asymptotically good sorting:
 946                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
 947                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
 948                                 const epvector::const_iterator add1begin = add1.seq.begin();
 949                                 const epvector::const_iterator add1end   = add1.seq.end();
 950                                 const epvector::const_iterator add2begin = add2.seq.begin();
 951                                 const epvector::const_iterator add2end   = add2.seq.end();
 952                                 epvector distrseq;
 953                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
 954
 955                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
 956                                 if (!add1.overall_coeff.is_zero()) {
 957                                         if (add1.overall_coeff.is_equal(_ex1))
 958                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
 959                                         else
 960                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
 961                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
 962                                 }
 963
 964                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
 965                                 if (!add2.overall_coeff.is_zero()) {
 966                                         if (add2.overall_coeff.is_equal(_ex1))
 967                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
 968                                         else
 969                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
 970                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
 971                                 }
 972
 973                                 // Compute the new overall coefficient and put it together:
 974                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
 975
 976                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
 977
 978                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
 979                                         add_indices = get_all_dummy_indices_safely(i->rest);
 980                                         add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
 981                                 }
 982                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
 983                                         add_indices = get_all_dummy_indices_safely(i->rest);
 984                                         add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
 985                                 }
 986
 987                                 sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
 988                                 sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
 989                                 lst dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
 990
 991                                 // Multiply explicitly all non-numeric terms of add1 and add2:
 992                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
 993                                         // We really have to combine terms here in order to compactify
 994                                         // the result.  Otherwise it would become waayy tooo bigg.
 995                                         numeric oc;
 996                                         distrseq.clear();
 997                                         ex i2_new = (dummy_subs.op(0).nops()>0?
 998                                                                  i2->rest.subs((lst)dummy_subs.op(0), (lst)dummy_subs.op(1), subs_options::no_pattern) : i2->rest);
 999                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1000                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1001                                                 // since that would violate an invariant of expairseq:
1002                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
1003                                                 if (is_exactly_a<numeric>(rest)) {
1004                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1005                                                 } else {
1006                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1007                                                 }
1008                                         }
1009                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
1010                                 }
1011                                 last_expanded = tmp_accu;
1012
1013                         } else {
1014                                 if (!last_expanded.is_equal(_ex1))
1015                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1016                                 last_expanded = cit->rest;
1017                         }
1018
1019                 } else {
1020                         non_adds.push_back(*cit);
1021                 }
1022         }
1023
1024         // Now the only remaining thing to do is to multiply the factors which
1025         // were not sums into the "last_expanded" sum
1026         if (is_exactly_a<add>(last_expanded)) {
1027                 size_t n = last_expanded.nops();
1028                 exvector distrseq;
1029                 distrseq.reserve(n);
1030                 exvector va = get_all_dummy_indices_safely(mul(non_adds));
1031                 sort(va.begin(), va.end(), ex_is_less());
1032
1033                 for (size_t i=0; i<n; ++i) {
1034                         epvector factors = non_adds;
1035                         factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1036                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1037                         if (can_be_further_expanded(term)) {
1038                                 distrseq.push_back(term.expand());
1039                         } else {
1040                                 if (options == 0)
1041                                         ex_to<basic>(term).setflag(status_flags::expanded);
1042                                 distrseq.push_back(term);
1043                         }
1044                 }
1045
1046                 return ((new add(distrseq))->
1047                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1048         }
1049
1050         non_adds.push_back(split_ex_to_pair(last_expanded));
1051         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1052         if (can_be_further_expanded(result)) {
1053                 return result.expand();
1054         } else {
1055                 if (options == 0)
1056                         ex_to<basic>(result).setflag(status_flags::expanded);
1057                 return result;
1058         }
1059 }
1060
1061
1062 //////////
1063 // new virtual functions which can be overridden by derived classes
1064 //////////
1065
1066 // none
1067
1068 //////////
1069 // non-virtual functions in this class
1070 //////////
1071
1072
1073 /** Member-wise expand the expairs representing this sequence.  This must be
1074  *  overridden from expairseq::expandchildren() and done iteratively in order
1075  *  to allow for early cancallations and thus safe memory.
1076  *
1077  *  @see mul::expand()
1078  *  @return pointer to epvector containing expanded representation or zero
1079  *  pointer, if sequence is unchanged. */
1080 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1081 {
1082         const epvector::const_iterator last = seq.end();
1083         epvector::const_iterator cit = seq.begin();
1084         while (cit!=last) {
1085                 const ex & factor = recombine_pair_to_ex(*cit);
1086                 const ex & expanded_factor = factor.expand(options);
1087                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1088
1089                         // something changed, copy seq, eval and return it
1090                         std::auto_ptr<epvector> s(new epvector);
1091                         s->reserve(seq.size());
1092
1093                         // copy parts of seq which are known not to have changed
1094                         epvector::const_iterator cit2 = seq.begin();
1095                         while (cit2!=cit) {
1096                                 s->push_back(*cit2);
1097                                 ++cit2;
1098                         }
1099
1100                         // copy first changed element
1101                         s->push_back(split_ex_to_pair(expanded_factor));
1102                         ++cit2;
1103
1104                         // copy rest
1105                         while (cit2!=last) {
1106                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1107                                 ++cit2;
1108                         }
1109                         return s;
1110                 }
1111                 ++cit;
1112         }
1113
1114         return std::auto_ptr<epvector>(0); // nothing has changed
1115 }
1116
1117 } // namespace GiNaC