ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2006 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)
  88 {
  89         tinfo_key = &mul::tinfo_static;
  90         overall_coeff = oc;
  91         construct_from_epvector(v);
  92         GINAC_ASSERT(is_canonical());
  93 }
  94
  95 mul::mul(std::auto_ptr<epvector> vp, const ex & oc)
  96 {
  97         tinfo_key = &mul::tinfo_static;
  98         GINAC_ASSERT(vp.get()!=0);
  99         overall_coeff = oc;
 100         construct_from_epvector(*vp);
 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                 overall_coeff.print(c, precedence());
 222                 c.s << "*";
 223         }
 224
 225         // Print arguments, separated by "*" or "/"
 226         epvector::const_iterator it = seq.begin(), itend = seq.end();
 227         while (it != itend) {
 228
 229                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 230                 bool needclosingparenthesis = false;
 231                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 232                         if (is_a<print_csrc_cl_N>(c)) {
 233                                 c.s << "recip(";
 234                                 needclosingparenthesis = true;
 235                         } else
 236                                 c.s << "1.0/";
 237                 }
 238
 239                 // If the exponent is 1 or -1, it is left out
 240                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 241                         it->rest.print(c, precedence());
 242                 else if (it->coeff.info(info_flags::negint))
 243                         // Outer parens around ex needed for broken GCC parser:
 244                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 245                 else
 246                         // Outer parens around ex needed for broken GCC parser:
 247                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 248
 249                 if (needclosingparenthesis)
 250                         c.s << ")";
 251
 252                 // Separator is "/" for negative integer powers, "*" otherwise
 253                 ++it;
 254                 if (it != itend) {
 255                         if (it->coeff.info(info_flags::negint))
 256                                 c.s << "/";
 257                         else
 258                                 c.s << "*";
 259                 }
 260         }
 261
 262         if (precedence() <= level)
 263                 c.s << ")";
 264 }
 265
 266 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 267 {
 268         c.s << class_name() << '(';
 269         op(0).print(c);
 270         for (size_t i=1; i<nops(); ++i) {
 271                 c.s << ',';
 272                 op(i).print(c);
 273         }
 274         c.s << ')';
 275 }
 276
 277 bool mul::info(unsigned inf) const
 278 {
 279         switch (inf) {
 280                 case info_flags::polynomial:
 281                 case info_flags::integer_polynomial:
 282                 case info_flags::cinteger_polynomial:
 283                 case info_flags::rational_polynomial:
 284                 case info_flags::crational_polynomial:
 285                 case info_flags::rational_function: {
 286                         epvector::const_iterator i = seq.begin(), end = seq.end();
 287                         while (i != end) {
 288                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 289                                         return false;
 290                                 ++i;
 291                         }
 292                         return overall_coeff.info(inf);
 293                 }
 294                 case info_flags::algebraic: {
 295                         epvector::const_iterator i = seq.begin(), end = seq.end();
 296                         while (i != end) {
 297                                 if ((recombine_pair_to_ex(*i).info(inf)))
 298                                         return true;
 299                                 ++i;
 300                         }
 301                         return false;
 302                 }
 303         }
 304         return inherited::info(inf);
 305 }
 306
 307 int mul::degree(const ex & s) const
 308 {
 309         // Sum up degrees of factors
 310         int deg_sum = 0;
 311         epvector::const_iterator i = seq.begin(), end = seq.end();
 312         while (i != end) {
 313                 if (ex_to<numeric>(i->coeff).is_integer())
 314                         deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
 315                 ++i;
 316         }
 317         return deg_sum;
 318 }
 319
 320 int mul::ldegree(const ex & s) const
 321 {
 322         // Sum up degrees of factors
 323         int deg_sum = 0;
 324         epvector::const_iterator i = seq.begin(), end = seq.end();
 325         while (i != end) {
 326                 if (ex_to<numeric>(i->coeff).is_integer())
 327                         deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
 328                 ++i;
 329         }
 330         return deg_sum;
 331 }
 332
 333 ex mul::coeff(const ex & s, int n) const
 334 {
 335         exvector coeffseq;
 336         coeffseq.reserve(seq.size()+1);
 337
 338         if (n==0) {
 339                 // product of individual coeffs
 340                 // if a non-zero power of s is found, the resulting product will be 0
 341                 epvector::const_iterator i = seq.begin(), end = seq.end();
 342                 while (i != end) {
 343                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 344                         ++i;
 345                 }
 346                 coeffseq.push_back(overall_coeff);
 347                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 348         }
 349
 350         epvector::const_iterator i = seq.begin(), end = seq.end();
 351         bool coeff_found = false;
 352         while (i != end) {
 353                 ex t = recombine_pair_to_ex(*i);
 354                 ex c = t.coeff(s, n);
 355                 if (!c.is_zero()) {
 356                         coeffseq.push_back(c);
 357                         coeff_found = 1;
 358                 } else {
 359                         coeffseq.push_back(t);
 360                 }
 361                 ++i;
 362         }
 363         if (coeff_found) {
 364                 coeffseq.push_back(overall_coeff);
 365                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 366         }
 367
 368         return _ex0;
 369 }
 370
 371 /** Perform automatic term rewriting rules in this class.  In the following
 372  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 373  *  stand for such expressions that contain a plain number.
 374  *  - *(...,x;0) -> 0
 375  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 376  *  - *(x;1) -> x
 377  *  - *(;c) -> c
 378  *
 379  *  @param level cut-off in recursive evaluation */
 380 ex mul::eval(int level) const
 381 {
 382         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 383         if (evaled_seqp.get()) {
 384                 // do more evaluation later
 385                 return (new mul(evaled_seqp, overall_coeff))->
 386                            setflag(status_flags::dynallocated);
 387         }
 388
 389 #ifdef DO_GINAC_ASSERT
 390         epvector::const_iterator i = seq.begin(), end = seq.end();
 391         while (i != end) {
 392                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 393                              (!(ex_to<numeric>(i->coeff).is_integer())));
 394                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 395                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 396                     print(print_tree(std::cerr));
 397                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 398                 /* for paranoia */
 399                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 400                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 401                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 402                 /* end paranoia */
 403                 ++i;
 404         }
 405 #endif // def DO_GINAC_ASSERT
 406
 407         if (flags & status_flags::evaluated) {
 408                 GINAC_ASSERT(seq.size()>0);
 409                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 410                 return *this;
 411         }
 412
 413         int seq_size = seq.size();
 414         if (overall_coeff.is_zero()) {
 415                 // *(...,x;0) -> 0
 416                 return _ex0;
 417         } else if (seq_size==0) {
 418                 // *(;c) -> c
 419                 return overall_coeff;
 420         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 421                 // *(x;1) -> x
 422                 return recombine_pair_to_ex(*(seq.begin()));
 423         } else if ((seq_size==1) &&
 424                    is_exactly_a<add>((*seq.begin()).rest) &&
 425                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 426                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 427                 const add & addref = ex_to<add>((*seq.begin()).rest);
 428                 std::auto_ptr<epvector> distrseq(new epvector);
 429                 distrseq->reserve(addref.seq.size());
 430                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 431                 while (i != end) {
 432                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 433                         ++i;
 434                 }
 435                 return (new add(distrseq,
 436                                 ex_to<numeric>(addref.overall_coeff).
 437                                 mul_dyn(ex_to<numeric>(overall_coeff))))
 438                       ->setflag(status_flags::dynallocated | status_flags::evaluated);
 439         }
 440         return this->hold();
 441 }
 442
 443 ex mul::evalf(int level) const
 444 {
 445         if (level==1)
 446                 return mul(seq,overall_coeff);
 447
 448         if (level==-max_recursion_level)
 449                 throw(std::runtime_error("max recursion level reached"));
 450
 451         std::auto_ptr<epvector> s(new epvector);
 452         s->reserve(seq.size());
 453
 454         --level;
 455         epvector::const_iterator i = seq.begin(), end = seq.end();
 456         while (i != end) {
 457                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 458                                                            i->coeff));
 459                 ++i;
 460         }
 461         return mul(s, overall_coeff.evalf(level));
 462 }
 463
 464 ex mul::evalm() const
 465 {
 466         // numeric*matrix
 467         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 468          && is_a<matrix>(seq[0].rest))
 469                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 470
 471         // Evaluate children first, look whether there are any matrices at all
 472         // (there can be either no matrices or one matrix; if there were more
 473         // than one matrix, it would be a non-commutative product)
 474         std::auto_ptr<epvector> s(new epvector);
 475         s->reserve(seq.size());
 476
 477         bool have_matrix = false;
 478         epvector::iterator the_matrix;
 479
 480         epvector::const_iterator i = seq.begin(), end = seq.end();
 481         while (i != end) {
 482                 const ex &m = recombine_pair_to_ex(*i).evalm();
 483                 s->push_back(split_ex_to_pair(m));
 484                 if (is_a<matrix>(m)) {
 485                         have_matrix = true;
 486                         the_matrix = s->end() - 1;
 487                 }
 488                 ++i;
 489         }
 490
 491         if (have_matrix) {
 492
 493                 // The product contained a matrix. We will multiply all other factors
 494                 // into that matrix.
 495                 matrix m = ex_to<matrix>(the_matrix->rest);
 496                 s->erase(the_matrix);
 497                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 498                 return m.mul_scalar(scalar);
 499
 500         } else
 501                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 502 }
 503
 504 ex mul::eval_ncmul(const exvector & v) const
 505 {
 506         if (seq.empty())
 507                 return inherited::eval_ncmul(v);
 508
 509         // Find first noncommutative element and call its eval_ncmul()
 510         epvector::const_iterator i = seq.begin(), end = seq.end();
 511         while (i != end) {
 512                 if (i->rest.return_type() == return_types::noncommutative)
 513                         return i->rest.eval_ncmul(v);
 514                 ++i;
 515         }
 516         return inherited::eval_ncmul(v);
 517 }
 518
 519 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
 520 {
 521         ex origbase;
 522         int origexponent;
 523         int origexpsign;
 524
 525         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 526                 origbase = origfactor.op(0);
 527                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 528                 origexponent = expon > 0 ? expon : -expon;
 529                 origexpsign = expon > 0 ? 1 : -1;
 530         } else {
 531                 origbase = origfactor;
 532                 origexponent = 1;
 533                 origexpsign = 1;
 534         }
 535
 536         ex patternbase;
 537         int patternexponent;
 538         int patternexpsign;
 539
 540         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 541                 patternbase = patternfactor.op(0);
 542                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 543                 patternexponent = expon > 0 ? expon : -expon;
 544                 patternexpsign = expon > 0 ? 1 : -1;
 545         } else {
 546                 patternbase = patternfactor;
 547                 patternexponent = 1;
 548                 patternexpsign = 1;
 549         }
 550
 551         lst saverepls = repls;
 552         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 553                 return false;
 554         repls = saverepls;
 555
 556         int newnummatches = origexponent / patternexponent;
 557         if (newnummatches < nummatches)
 558                 nummatches = newnummatches;
 559         return true;
 560 }
 561
 562 /** Checks wheter e matches to the pattern pat and the (possibly to be updated
 563   * list of replacements repls. This matching is in the sense of algebraic
 564   * substitutions. Matching starts with pat.op(factor) of the pattern because
 565   * the factors before this one have already been matched. The (possibly
 566   * updated) number of matches is in nummatches. subsed[i] is true for factors
 567   * that already have been replaced by previous substitutions and matched[i]
 568   * is true for factors that have been matched by the current match.
 569   */
 570 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
 571                 int factor, int &nummatches, const std::vector<bool> &subsed,
 572                 std::vector<bool> &matched)
 573 {
 574         if (factor == pat.nops())
 575                 return true;
 576
 577         for (size_t i=0; i<e.nops(); ++i) {
 578                 if(subsed[i] || matched[i])
 579                         continue;
 580                 lst newrepls = repls;
 581                 int newnummatches = nummatches;
 582                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 583                         matched[i] = true;
 584                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 585                                         newnummatches, subsed, matched)) {
 586                                 repls = newrepls;
 587                                 nummatches = newnummatches;
 588                                 return true;
 589                         }
 590                         else
 591                                 matched[i] = false;
 592                 }
 593         }
 594
 595         return false;
 596 }
 597
 598 bool mul::has(const ex & pattern, unsigned options) const
 599 {
 600         if(!(options&has_options::algebraic))
 601                 return basic::has(pattern,options);
 602         if(is_a<mul>(pattern)) {
 603                 lst repls;
 604                 int nummatches = std::numeric_limits<int>::max();
 605                 std::vector<bool> subsed(seq.size(), false);
 606                 std::vector<bool> matched(seq.size(), false);
 607                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 608                                 subsed, matched))
 609                         return true;
 610         }
 611         return basic::has(pattern, options);
 612 }
 613
 614 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 615 {
 616         std::vector<bool> subsed(seq.size(), false);
 617         exvector subsresult(seq.size());
 618
 619         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 620
 621                 if (is_exactly_a<mul>(it->first)) {
 622 retry1:
 623                         int nummatches = std::numeric_limits<int>::max();
 624                         std::vector<bool> currsubsed(seq.size(), false);
 625                         bool succeed = true;
 626                         lst repls;
 627
 628                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 629                                 continue;
 630
 631                         bool foundfirstsubsedfactor = false;
 632                         for (size_t j=0; j<subsed.size(); j++) {
 633                                 if (currsubsed[j]) {
 634                                         if (foundfirstsubsedfactor)
 635                                                 subsresult[j] = op(j);
 636                                         else {
 637                                                 foundfirstsubsedfactor = true;
 638                                                 subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
 639                                         }
 640                                         subsed[j] = true;
 641                                 }
 642                         }
 643                         goto retry1;
 644
 645                 } else {
 646 retry2:
 647                         int nummatches = std::numeric_limits<int>::max();
 648                         lst repls;
 649
 650                         for (size_t j=0; j<this->nops(); j++) {
 651                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)) {
 652                                         subsed[j] = true;
 653                                         subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
 654                                         goto retry2;
 655                                 }
 656                         }
 657                 }
 658         }
 659
 660         bool subsfound = false;
 661         for (size_t i=0; i<subsed.size(); i++) {
 662                 if (subsed[i]) {
 663                         subsfound = true;
 664                         break;
 665                 }
 666         }
 667         if (!subsfound)
 668                 return subs_one_level(m, options | subs_options::algebraic);
 669
 670         exvector ev; ev.reserve(nops());
 671         for (size_t i=0; i<nops(); i++) {
 672                 if (subsed[i])
 673                         ev.push_back(subsresult[i]);
 674                 else
 675                         ev.push_back(op(i));
 676         }
 677
 678         return (new mul(ev))->setflag(status_flags::dynallocated);
 679 }
 680
 681 // protected
 682
 683 /** Implementation of ex::diff() for a product.  It applies the product rule.
 684  *  @see ex::diff */
 685 ex mul::derivative(const symbol & s) const
 686 {
 687         size_t num = seq.size();
 688         exvector addseq;
 689         addseq.reserve(num);
 690
 691         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 692         epvector mulseq = seq;
 693         epvector::const_iterator i = seq.begin(), end = seq.end();
 694         epvector::iterator i2 = mulseq.begin();
 695         while (i != end) {
 696                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 697                                              i->rest.diff(s));
 698                 ep.swap(*i2);
 699                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 700                 ep.swap(*i2);
 701                 ++i; ++i2;
 702         }
 703         return (new add(addseq))->setflag(status_flags::dynallocated);
 704 }
 705
 706 int mul::compare_same_type(const basic & other) const
 707 {
 708         return inherited::compare_same_type(other);
 709 }
 710
 711 unsigned mul::return_type() const
 712 {
 713         if (seq.empty()) {
 714                 // mul without factors: should not happen, but commutates
 715                 return return_types::commutative;
 716         }
 717
 718         bool all_commutative = true;
 719         epvector::const_iterator noncommutative_element; // point to first found nc element
 720
 721         epvector::const_iterator i = seq.begin(), end = seq.end();
 722         while (i != end) {
 723                 unsigned rt = i->rest.return_type();
 724                 if (rt == return_types::noncommutative_composite)
 725                         return rt; // one ncc -> mul also ncc
 726                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 727                         // first nc element found, remember position
 728                         noncommutative_element = i;
 729                         all_commutative = false;
 730                 }
 731                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 732                         // another nc element found, compare type_infos
 733                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 734                                         // different types -> mul is ncc
 735                                         return return_types::noncommutative_composite;
 736                         }
 737                 }
 738                 ++i;
 739         }
 740         // all factors checked
 741         return all_commutative ? return_types::commutative : return_types::noncommutative;
 742 }
 743
 744 tinfo_t mul::return_type_tinfo() const
 745 {
 746         if (seq.empty())
 747                 return this;  // mul without factors: should not happen
 748
 749         // return type_info of first noncommutative element
 750         epvector::const_iterator i = seq.begin(), end = seq.end();
 751         while (i != end) {
 752                 if (i->rest.return_type() == return_types::noncommutative)
 753                         return i->rest.return_type_tinfo();
 754                 ++i;
 755         }
 756         // no noncommutative element found, should not happen
 757         return this;
 758 }
 759
 760 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
 761 {
 762         return (new mul(v, oc))->setflag(status_flags::dynallocated);
 763 }
 764
 765 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc) const
 766 {
 767         return (new mul(vp, oc))->setflag(status_flags::dynallocated);
 768 }
 769
 770 expair mul::split_ex_to_pair(const ex & e) const
 771 {
 772         if (is_exactly_a<power>(e)) {
 773                 const power & powerref = ex_to<power>(e);
 774                 if (is_exactly_a<numeric>(powerref.exponent))
 775                         return expair(powerref.basis,powerref.exponent);
 776         }
 777         return expair(e,_ex1);
 778 }
 779
 780 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 781                                           const ex & c) const
 782 {
 783         // to avoid duplication of power simplification rules,
 784         // we create a temporary power object
 785         // otherwise it would be hard to correctly evaluate
 786         // expression like (4^(1/3))^(3/2)
 787         if (c.is_equal(_ex1))
 788                 return split_ex_to_pair(e);
 789
 790         return split_ex_to_pair(power(e,c));
 791 }
 792
 793 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 794                                             const ex & c) const
 795 {
 796         // to avoid duplication of power simplification rules,
 797         // we create a temporary power object
 798         // otherwise it would be hard to correctly evaluate
 799         // expression like (4^(1/3))^(3/2)
 800         if (c.is_equal(_ex1))
 801                 return p;
 802
 803         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 804 }
 805
 806 ex mul::recombine_pair_to_ex(const expair & p) const
 807 {
 808         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 809                 return p.rest;
 810         else
 811                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 812 }
 813
 814 bool mul::expair_needs_further_processing(epp it)
 815 {
 816         if (is_exactly_a<mul>(it->rest) &&
 817                 ex_to<numeric>(it->coeff).is_integer()) {
 818                 // combined pair is product with integer power -> expand it
 819                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 820                 return true;
 821         }
 822         if (is_exactly_a<numeric>(it->rest)) {
 823                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 824                 if (!ep.is_equal(*it)) {
 825                         // combined pair is a numeric power which can be simplified
 826                         *it = ep;
 827                         return true;
 828                 }
 829                 if (it->coeff.is_equal(_ex1)) {
 830                         // combined pair has coeff 1 and must be moved to the end
 831                         return true;
 832                 }
 833         }
 834         return false;
 835 }
 836
 837 ex mul::default_overall_coeff() const
 838 {
 839         return _ex1;
 840 }
 841
 842 void mul::combine_overall_coeff(const ex & c)
 843 {
 844         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 845         GINAC_ASSERT(is_exactly_a<numeric>(c));
 846         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 847 }
 848
 849 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 850 {
 851         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 852         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 853         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 854         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 855 }
 856
 857 bool mul::can_make_flat(const expair & p) const
 858 {
 859         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 860         // this assertion will probably fail somewhere
 861         // it would require a more careful make_flat, obeying the power laws
 862         // probably should return true only if p.coeff is integer
 863         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 864 }
 865
 866 bool mul::can_be_further_expanded(const ex & e)
 867 {
 868         if (is_exactly_a<mul>(e)) {
 869                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 870                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 871                                 return true;
 872                 }
 873         } else if (is_exactly_a<power>(e)) {
 874                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 875                         return true;
 876         }
 877         return false;
 878 }
 879
 880 ex mul::expand(unsigned options) const
 881 {
 882         // First, expand the children
 883         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
 884         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
 885
 886         // Now, look for all the factors that are sums and multiply each one out
 887         // with the next one that is found while collecting the factors which are
 888         // not sums
 889         ex last_expanded = _ex1;
 890
 891         epvector non_adds;
 892         non_adds.reserve(expanded_seq.size());
 893
 894         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
 895                 if (is_exactly_a<add>(cit->rest) &&
 896                         (cit->coeff.is_equal(_ex1))) {
 897                         if (is_exactly_a<add>(last_expanded)) {
 898
 899                                 // Expand a product of two sums, aggressive version.
 900                                 // Caring for the overall coefficients in separate loops can
 901                                 // sometimes give a performance gain of up to 15%!
 902
 903                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
 904                                 // add2 is for the inner loop and should be the bigger of the two sums
 905                                 // in the presence of asymptotically good sorting:
 906                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
 907                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
 908                                 const epvector::const_iterator add1begin = add1.seq.begin();
 909                                 const epvector::const_iterator add1end   = add1.seq.end();
 910                                 const epvector::const_iterator add2begin = add2.seq.begin();
 911                                 const epvector::const_iterator add2end   = add2.seq.end();
 912                                 epvector distrseq;
 913                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
 914
 915                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
 916                                 if (!add1.overall_coeff.is_zero()) {
 917                                         if (add1.overall_coeff.is_equal(_ex1))
 918                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
 919                                         else
 920                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
 921                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
 922                                 }
 923
 924                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
 925                                 if (!add2.overall_coeff.is_zero()) {
 926                                         if (add2.overall_coeff.is_equal(_ex1))
 927                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
 928                                         else
 929                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
 930                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
 931                                 }
 932
 933                                 // Compute the new overall coefficient and put it together:
 934                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
 935
 936                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
 937
 938                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
 939                                         add_indices = get_all_dummy_indices(i->rest);
 940                                         add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
 941                                 }
 942                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
 943                                         add_indices = get_all_dummy_indices(i->rest);
 944                                         add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
 945                                 }
 946
 947                                 sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
 948                                 sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
 949                                 lst dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
 950
 951                                 // Multiply explicitly all non-numeric terms of add1 and add2:
 952                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
 953                                         // We really have to combine terms here in order to compactify
 954                                         // the result.  Otherwise it would become waayy tooo bigg.
 955                                         numeric oc;
 956                                         distrseq.clear();
 957                                         ex i2_new = (dummy_subs.op(0).nops()>0?
 958                                                                  i2->rest.subs((lst)dummy_subs.op(0), (lst)dummy_subs.op(1), subs_options::no_pattern) : i2->rest);
 959                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
 960                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
 961                                                 // since that would violate an invariant of expairseq:
 962                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
 963                                                 if (is_exactly_a<numeric>(rest)) {
 964                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
 965                                                 } else {
 966                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
 967                                                 }
 968                                         }
 969                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
 970                                 }
 971                                 last_expanded = tmp_accu;
 972
 973                         } else {
 974                                 if (!last_expanded.is_equal(_ex1))
 975                                         non_adds.push_back(split_ex_to_pair(last_expanded));
 976                                 last_expanded = cit->rest;
 977                         }
 978
 979                 } else {
 980                         non_adds.push_back(*cit);
 981                 }
 982         }
 983
 984         // Now the only remaining thing to do is to multiply the factors which
 985         // were not sums into the "last_expanded" sum
 986         if (is_exactly_a<add>(last_expanded)) {
 987                 size_t n = last_expanded.nops();
 988                 exvector distrseq;
 989                 distrseq.reserve(n);
 990                 exvector va = get_all_dummy_indices(mul(non_adds));
 991                 sort(va.begin(), va.end(), ex_is_less());
 992
 993                 for (size_t i=0; i<n; ++i) {
 994                         epvector factors = non_adds;
 995                         factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
 996                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
 997                         if (can_be_further_expanded(term)) {
 998                                 distrseq.push_back(term.expand());
 999                         } else {
1000                                 if (options == 0)
1001                                         ex_to<basic>(term).setflag(status_flags::expanded);
1002                                 distrseq.push_back(term);
1003                         }
1004                 }
1005
1006                 return ((new add(distrseq))->
1007                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1008         }
1009
1010         non_adds.push_back(split_ex_to_pair(last_expanded));
1011         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1012         if (can_be_further_expanded(result)) {
1013                 return result.expand();
1014         } else {
1015                 if (options == 0)
1016                         ex_to<basic>(result).setflag(status_flags::expanded);
1017                 return result;
1018         }
1019 }
1020
1021
1022 //////////
1023 // new virtual functions which can be overridden by derived classes
1024 //////////
1025
1026 // none
1027
1028 //////////
1029 // non-virtual functions in this class
1030 //////////
1031
1032
1033 /** Member-wise expand the expairs representing this sequence.  This must be
1034  *  overridden from expairseq::expandchildren() and done iteratively in order
1035  *  to allow for early cancallations and thus safe memory.
1036  *
1037  *  @see mul::expand()
1038  *  @return pointer to epvector containing expanded representation or zero
1039  *  pointer, if sequence is unchanged. */
1040 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1041 {
1042         const epvector::const_iterator last = seq.end();
1043         epvector::const_iterator cit = seq.begin();
1044         while (cit!=last) {
1045                 const ex & factor = recombine_pair_to_ex(*cit);
1046                 const ex & expanded_factor = factor.expand(options);
1047                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1048
1049                         // something changed, copy seq, eval and return it
1050                         std::auto_ptr<epvector> s(new epvector);
1051                         s->reserve(seq.size());
1052
1053                         // copy parts of seq which are known not to have changed
1054                         epvector::const_iterator cit2 = seq.begin();
1055                         while (cit2!=cit) {
1056                                 s->push_back(*cit2);
1057                                 ++cit2;
1058                         }
1059
1060                         // copy first changed element
1061                         s->push_back(split_ex_to_pair(expanded_factor));
1062                         ++cit2;
1063
1064                         // copy rest
1065                         while (cit2!=last) {
1066                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1067                                 ++cit2;
1068                         }
1069                         return s;
1070                 }
1071                 ++cit;
1072         }
1073
1074         return std::auto_ptr<epvector>(0); // nothing has changed
1075 }
1076
1077 } // namespace GiNaC