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 #include "symbol.h"
  38 #include "compiler.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
  43   print_func<print_context>(&mul::do_print).
  44   print_func<print_latex>(&mul::do_print_latex).
  45   print_func<print_csrc>(&mul::do_print_csrc).
  46   print_func<print_tree>(&mul::do_print_tree).
  47   print_func<print_python_repr>(&mul::do_print_python_repr))
  48
  49
  50 //////////
  51 // default constructor
  52 //////////
  53
  54 mul::mul()
  55 {
  56         tinfo_key = TINFO_mul;
  57 }
  58
  59 //////////
  60 // other constructors
  61 //////////
  62
  63 // public
  64
  65 mul::mul(const ex & lh, const ex & rh)
  66 {
  67         tinfo_key = TINFO_mul;
  68         overall_coeff = _ex1;
  69         construct_from_2_ex(lh,rh);
  70         GINAC_ASSERT(is_canonical());
  71 }
  72
  73 mul::mul(const exvector & v)
  74 {
  75         tinfo_key = TINFO_mul;
  76         overall_coeff = _ex1;
  77         construct_from_exvector(v);
  78         GINAC_ASSERT(is_canonical());
  79 }
  80
  81 mul::mul(const epvector & v)
  82 {
  83         tinfo_key = TINFO_mul;
  84         overall_coeff = _ex1;
  85         construct_from_epvector(v);
  86         GINAC_ASSERT(is_canonical());
  87 }
  88
  89 mul::mul(const epvector & v, const ex & oc)
  90 {
  91         tinfo_key = TINFO_mul;
  92         overall_coeff = oc;
  93         construct_from_epvector(v);
  94         GINAC_ASSERT(is_canonical());
  95 }
  96
  97 mul::mul(std::auto_ptr<epvector> vp, const ex & oc)
  98 {
  99         tinfo_key = TINFO_mul;
 100         GINAC_ASSERT(vp.get()!=0);
 101         overall_coeff = oc;
 102         construct_from_epvector(*vp);
 103         GINAC_ASSERT(is_canonical());
 104 }
 105
 106 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 107 {
 108         tinfo_key = TINFO_mul;
 109         exvector factors;
 110         factors.reserve(3);
 111         factors.push_back(lh);
 112         factors.push_back(mh);
 113         factors.push_back(rh);
 114         overall_coeff = _ex1;
 115         construct_from_exvector(factors);
 116         GINAC_ASSERT(is_canonical());
 117 }
 118
 119 //////////
 120 // archiving
 121 //////////
 122
 123 DEFAULT_ARCHIVING(mul)
 124
 125 //////////
 126 // functions overriding virtual functions from base classes
 127 //////////
 128
 129 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
 130 {
 131         const numeric &coeff = ex_to<numeric>(overall_coeff);
 132         if (coeff.csgn() == -1)
 133                 c.s << '-';
 134         if (!coeff.is_equal(*_num1_p) &&
 135                 !coeff.is_equal(*_num_1_p)) {
 136                 if (coeff.is_rational()) {
 137                         if (coeff.is_negative())
 138                                 (-coeff).print(c);
 139                         else
 140                                 coeff.print(c);
 141                 } else {
 142                         if (coeff.csgn() == -1)
 143                                 (-coeff).print(c, precedence());
 144                         else
 145                                 coeff.print(c, precedence());
 146                 }
 147                 c.s << mul_sym;
 148         }
 149 }
 150
 151 void mul::do_print(const print_context & c, unsigned level) const
 152 {
 153         if (precedence() <= level)
 154                 c.s << '(';
 155
 156         print_overall_coeff(c, "*");
 157
 158         epvector::const_iterator it = seq.begin(), itend = seq.end();
 159         bool first = true;
 160         while (it != itend) {
 161                 if (!first)
 162                         c.s << '*';
 163                 else
 164                         first = false;
 165                 recombine_pair_to_ex(*it).print(c, precedence());
 166                 ++it;
 167         }
 168
 169         if (precedence() <= level)
 170                 c.s << ')';
 171 }
 172
 173 void mul::do_print_latex(const print_latex & c, unsigned level) const
 174 {
 175         if (precedence() <= level)
 176                 c.s << "{(";
 177
 178         print_overall_coeff(c, " ");
 179
 180         // Separate factors into those with negative numeric exponent
 181         // and all others
 182         epvector::const_iterator it = seq.begin(), itend = seq.end();
 183         exvector neg_powers, others;
 184         while (it != itend) {
 185                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
 186                 if (ex_to<numeric>(it->coeff).is_negative())
 187                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
 188                 else
 189                         others.push_back(recombine_pair_to_ex(*it));
 190                 ++it;
 191         }
 192
 193         if (!neg_powers.empty()) {
 194
 195                 // Factors with negative exponent are printed as a fraction
 196                 c.s << "\\frac{";
 197                 mul(others).eval().print(c);
 198                 c.s << "}{";
 199                 mul(neg_powers).eval().print(c);
 200                 c.s << "}";
 201
 202         } else {
 203
 204                 // All other factors are printed in the ordinary way
 205                 exvector::const_iterator vit = others.begin(), vitend = others.end();
 206                 while (vit != vitend) {
 207                         c.s << ' ';
 208                         vit->print(c, precedence());
 209                         ++vit;
 210                 }
 211         }
 212
 213         if (precedence() <= level)
 214                 c.s << ")}";
 215 }
 216
 217 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
 218 {
 219         if (precedence() <= level)
 220                 c.s << "(";
 221
 222         if (!overall_coeff.is_equal(_ex1)) {
 223                 if (overall_coeff.is_equal(_ex_1))
 224                         c.s << "-";
 225                 else {
 226                         overall_coeff.print(c, precedence());
 227                         c.s << "*";
 228                 }
 229         }
 230
 231         // Print arguments, separated by "*" or "/"
 232         epvector::const_iterator it = seq.begin(), itend = seq.end();
 233         while (it != itend) {
 234
 235                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 236                 bool needclosingparenthesis = false;
 237                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 238                         if (is_a<print_csrc_cl_N>(c)) {
 239                                 c.s << "recip(";
 240                                 needclosingparenthesis = true;
 241                         } else
 242                                 c.s << "1.0/";
 243                 }
 244
 245                 // If the exponent is 1 or -1, it is left out
 246                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 247                         it->rest.print(c, precedence());
 248                 else if (it->coeff.info(info_flags::negint))
 249                         // Outer parens around ex needed for broken GCC parser:
 250                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 251                 else
 252                         // Outer parens around ex needed for broken GCC parser:
 253                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 254
 255                 if (needclosingparenthesis)
 256                         c.s << ")";
 257
 258                 // Separator is "/" for negative integer powers, "*" otherwise
 259                 ++it;
 260                 if (it != itend) {
 261                         if (it->coeff.info(info_flags::negint))
 262                                 c.s << "/";
 263                         else
 264                                 c.s << "*";
 265                 }
 266         }
 267
 268         if (precedence() <= level)
 269                 c.s << ")";
 270 }
 271
 272 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 273 {
 274         c.s << class_name() << '(';
 275         op(0).print(c);
 276         for (size_t i=1; i<nops(); ++i) {
 277                 c.s << ',';
 278                 op(i).print(c);
 279         }
 280         c.s << ')';
 281 }
 282
 283 bool mul::info(unsigned inf) const
 284 {
 285         switch (inf) {
 286                 case info_flags::polynomial:
 287                 case info_flags::integer_polynomial:
 288                 case info_flags::cinteger_polynomial:
 289                 case info_flags::rational_polynomial:
 290                 case info_flags::crational_polynomial:
 291                 case info_flags::rational_function: {
 292                         epvector::const_iterator i = seq.begin(), end = seq.end();
 293                         while (i != end) {
 294                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 295                                         return false;
 296                                 ++i;
 297                         }
 298                         return overall_coeff.info(inf);
 299                 }
 300                 case info_flags::algebraic: {
 301                         epvector::const_iterator i = seq.begin(), end = seq.end();
 302                         while (i != end) {
 303                                 if ((recombine_pair_to_ex(*i).info(inf)))
 304                                         return true;
 305                                 ++i;
 306                         }
 307                         return false;
 308                 }
 309         }
 310         return inherited::info(inf);
 311 }
 312
 313 int mul::degree(const ex & s) const
 314 {
 315         // Sum up degrees of factors
 316         int deg_sum = 0;
 317         epvector::const_iterator i = seq.begin(), end = seq.end();
 318         while (i != end) {
 319                 if (ex_to<numeric>(i->coeff).is_integer())
 320                         deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
 321                 ++i;
 322         }
 323         return deg_sum;
 324 }
 325
 326 int mul::ldegree(const ex & s) const
 327 {
 328         // Sum up degrees of factors
 329         int deg_sum = 0;
 330         epvector::const_iterator i = seq.begin(), end = seq.end();
 331         while (i != end) {
 332                 if (ex_to<numeric>(i->coeff).is_integer())
 333                         deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
 334                 ++i;
 335         }
 336         return deg_sum;
 337 }
 338
 339 ex mul::coeff(const ex & s, int n) const
 340 {
 341         exvector coeffseq;
 342         coeffseq.reserve(seq.size()+1);
 343
 344         if (n==0) {
 345                 // product of individual coeffs
 346                 // if a non-zero power of s is found, the resulting product will be 0
 347                 epvector::const_iterator i = seq.begin(), end = seq.end();
 348                 while (i != end) {
 349                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 350                         ++i;
 351                 }
 352                 coeffseq.push_back(overall_coeff);
 353                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 354         }
 355
 356         epvector::const_iterator i = seq.begin(), end = seq.end();
 357         bool coeff_found = false;
 358         while (i != end) {
 359                 ex t = recombine_pair_to_ex(*i);
 360                 ex c = t.coeff(s, n);
 361                 if (!c.is_zero()) {
 362                         coeffseq.push_back(c);
 363                         coeff_found = 1;
 364                 } else {
 365                         coeffseq.push_back(t);
 366                 }
 367                 ++i;
 368         }
 369         if (coeff_found) {
 370                 coeffseq.push_back(overall_coeff);
 371                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 372         }
 373
 374         return _ex0;
 375 }
 376
 377 /** Perform automatic term rewriting rules in this class.  In the following
 378  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 379  *  stand for such expressions that contain a plain number.
 380  *  - *(...,x;0) -> 0
 381  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 382  *  - *(x;1) -> x
 383  *  - *(;c) -> c
 384  *
 385  *  @param level cut-off in recursive evaluation */
 386 ex mul::eval(int level) const
 387 {
 388         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 389         if (evaled_seqp.get()) {
 390                 // do more evaluation later
 391                 return (new mul(evaled_seqp, overall_coeff))->
 392                            setflag(status_flags::dynallocated);
 393         }
 394
 395 #ifdef DO_GINAC_ASSERT
 396         epvector::const_iterator i = seq.begin(), end = seq.end();
 397         while (i != end) {
 398                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 399                              (!(ex_to<numeric>(i->coeff).is_integer())));
 400                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 401                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 402                     print(print_tree(std::cerr));
 403                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 404                 /* for paranoia */
 405                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 406                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 407                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 408                 /* end paranoia */
 409                 ++i;
 410         }
 411 #endif // def DO_GINAC_ASSERT
 412
 413         if (flags & status_flags::evaluated) {
 414                 GINAC_ASSERT(seq.size()>0);
 415                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 416                 return *this;
 417         }
 418
 419         size_t seq_size = seq.size();
 420         if (overall_coeff.is_zero()) {
 421                 // *(...,x;0) -> 0
 422                 return _ex0;
 423         } else if (seq_size==0) {
 424                 // *(;c) -> c
 425                 return overall_coeff;
 426         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 427                 // *(x;1) -> x
 428                 return recombine_pair_to_ex(*(seq.begin()));
 429         } else if ((seq_size==1) &&
 430                    is_exactly_a<add>((*seq.begin()).rest) &&
 431                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 432                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 433                 const add & addref = ex_to<add>((*seq.begin()).rest);
 434                 std::auto_ptr<epvector> distrseq(new epvector);
 435                 distrseq->reserve(addref.seq.size());
 436                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 437                 while (i != end) {
 438                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 439                         ++i;
 440                 }
 441                 return (new add(distrseq,
 442                                 ex_to<numeric>(addref.overall_coeff).
 443                                 mul_dyn(ex_to<numeric>(overall_coeff))))
 444                       ->setflag(status_flags::dynallocated | status_flags::evaluated);
 445         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
 446                 // Strip the content and the unit part from each term. Thus
 447                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)2
 448
 449                 epvector::const_iterator last = seq.end();
 450                 epvector::const_iterator i = seq.begin();
 451                 epvector::const_iterator j = seq.begin();
 452                 std::auto_ptr<epvector> s(new epvector);
 453                 numeric oc = *_num1_p;
 454                 bool something_changed = false;
 455                 while (i!=last) {
 456                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
 457                                 // power::eval has such a rule, no need to handle powers here
 458                                 ++i;
 459                                 continue;
 460                         }
 461
 462                         // XXX: What is the best way to check if the polynomial is a primitive?
 463                         numeric c = i->rest.integer_content();
 464                         const numeric& lead_coeff =
 465                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div_dyn(c);
 466                         const bool canonicalizable = lead_coeff.is_integer();
 467
 468                         // XXX: The main variable is chosen in a random way, so this code
 469                         // does NOT transform the term into the canonical form (thus, in some
 470                         // very unlucky event it can even loop forever). Hopefully the main
 471                         // variable will be the same for all terms in *this
 472                         const bool unit_normal = lead_coeff.is_pos_integer();
 473                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
 474                                 ++i;
 475                                 continue;
 476                         }
 477
 478                         if (! something_changed) {
 479                                 s->reserve(seq_size);
 480                                 something_changed = true;
 481                         }
 482
 483                         while ((j!=i) && (j!=last)) {
 484                                 s->push_back(*j);
 485                                 ++j;
 486                         }
 487
 488                         if (! unit_normal)
 489                                 c = c.mul(*_num_1_p);
 490
 491                         oc = oc.mul(c);
 492
 493                         // divide add by the number in place to save at least 2 .eval() calls
 494                         const add& addref = ex_to<add>(i->rest);
 495                         add* primitive = new add(addref);
 496                         primitive->setflag(status_flags::dynallocated);
 497                         primitive->clearflag(status_flags::hash_calculated);
 498                         primitive->overall_coeff = ex_to<numeric>(primitive->overall_coeff).div_dyn(c);
 499                         for (epvector::iterator ai = primitive->seq.begin();
 500                                         ai != primitive->seq.end(); ++ai)
 501                                 ai->coeff = ex_to<numeric>(ai->coeff).div_dyn(c);
 502
 503                         s->push_back(expair(*primitive, _ex1));
 504
 505                         ++i;
 506                         ++j;
 507                 }
 508                 if (something_changed) {
 509                         while (j!=last) {
 510                                 s->push_back(*j);
 511                                 ++j;
 512                         }
 513                         return (new mul(s, ex_to<numeric>(overall_coeff).mul_dyn(oc))
 514                                )->setflag(status_flags::dynallocated);
 515                 }
 516         }
 517
 518         return this->hold();
 519 }
 520
 521 ex mul::evalf(int level) const
 522 {
 523         if (level==1)
 524                 return mul(seq,overall_coeff);
 525
 526         if (level==-max_recursion_level)
 527                 throw(std::runtime_error("max recursion level reached"));
 528
 529         std::auto_ptr<epvector> s(new epvector);
 530         s->reserve(seq.size());
 531
 532         --level;
 533         epvector::const_iterator i = seq.begin(), end = seq.end();
 534         while (i != end) {
 535                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 536                                                            i->coeff));
 537                 ++i;
 538         }
 539         return mul(s, overall_coeff.evalf(level));
 540 }
 541
 542 ex mul::evalm() const
 543 {
 544         // numeric*matrix
 545         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 546          && is_a<matrix>(seq[0].rest))
 547                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 548
 549         // Evaluate children first, look whether there are any matrices at all
 550         // (there can be either no matrices or one matrix; if there were more
 551         // than one matrix, it would be a non-commutative product)
 552         std::auto_ptr<epvector> s(new epvector);
 553         s->reserve(seq.size());
 554
 555         bool have_matrix = false;
 556         epvector::iterator the_matrix;
 557
 558         epvector::const_iterator i = seq.begin(), end = seq.end();
 559         while (i != end) {
 560                 const ex &m = recombine_pair_to_ex(*i).evalm();
 561                 s->push_back(split_ex_to_pair(m));
 562                 if (is_a<matrix>(m)) {
 563                         have_matrix = true;
 564                         the_matrix = s->end() - 1;
 565                 }
 566                 ++i;
 567         }
 568
 569         if (have_matrix) {
 570
 571                 // The product contained a matrix. We will multiply all other factors
 572                 // into that matrix.
 573                 matrix m = ex_to<matrix>(the_matrix->rest);
 574                 s->erase(the_matrix);
 575                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 576                 return m.mul_scalar(scalar);
 577
 578         } else
 579                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 580 }
 581
 582 ex mul::eval_ncmul(const exvector & v) const
 583 {
 584         if (seq.empty())
 585                 return inherited::eval_ncmul(v);
 586
 587         // Find first noncommutative element and call its eval_ncmul()
 588         epvector::const_iterator i = seq.begin(), end = seq.end();
 589         while (i != end) {
 590                 if (i->rest.return_type() == return_types::noncommutative)
 591                         return i->rest.eval_ncmul(v);
 592                 ++i;
 593         }
 594         return inherited::eval_ncmul(v);
 595 }
 596
 597 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
 598 {
 599         ex origbase;
 600         int origexponent;
 601         int origexpsign;
 602
 603         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 604                 origbase = origfactor.op(0);
 605                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 606                 origexponent = expon > 0 ? expon : -expon;
 607                 origexpsign = expon > 0 ? 1 : -1;
 608         } else {
 609                 origbase = origfactor;
 610                 origexponent = 1;
 611                 origexpsign = 1;
 612         }
 613
 614         ex patternbase;
 615         int patternexponent;
 616         int patternexpsign;
 617
 618         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 619                 patternbase = patternfactor.op(0);
 620                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 621                 patternexponent = expon > 0 ? expon : -expon;
 622                 patternexpsign = expon > 0 ? 1 : -1;
 623         } else {
 624                 patternbase = patternfactor;
 625                 patternexponent = 1;
 626                 patternexpsign = 1;
 627         }
 628
 629         lst saverepls = repls;
 630         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 631                 return false;
 632         repls = saverepls;
 633
 634         int newnummatches = origexponent / patternexponent;
 635         if (newnummatches < nummatches)
 636                 nummatches = newnummatches;
 637         return true;
 638 }
 639
 640 /** Checks wheter e matches to the pattern pat and the (possibly to be updated
 641   * list of replacements repls. This matching is in the sense of algebraic
 642   * substitutions. Matching starts with pat.op(factor) of the pattern because
 643   * the factors before this one have already been matched. The (possibly
 644   * updated) number of matches is in nummatches. subsed[i] is true for factors
 645   * that already have been replaced by previous substitutions and matched[i]
 646   * is true for factors that have been matched by the current match.
 647   */
 648 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
 649                 int factor, int &nummatches, const std::vector<bool> &subsed,
 650                 std::vector<bool> &matched)
 651 {
 652         if (factor == pat.nops())
 653                 return true;
 654
 655         for (size_t i=0; i<e.nops(); ++i) {
 656                 if(subsed[i] || matched[i])
 657                         continue;
 658                 lst newrepls = repls;
 659                 int newnummatches = nummatches;
 660                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 661                         matched[i] = true;
 662                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 663                                         newnummatches, subsed, matched)) {
 664                                 repls = newrepls;
 665                                 nummatches = newnummatches;
 666                                 return true;
 667                         }
 668                         else
 669                                 matched[i] = false;
 670                 }
 671         }
 672
 673         return false;
 674 }
 675
 676 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 677 {
 678         std::vector<bool> subsed(seq.size(), false);
 679         exvector subsresult(seq.size());
 680
 681         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 682
 683                 if (is_exactly_a<mul>(it->first)) {
 684 retry1:
 685                         int nummatches = std::numeric_limits<int>::max();
 686                         std::vector<bool> currsubsed(seq.size(), false);
 687                         bool succeed = true;
 688                         lst repls;
 689
 690                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 691                                 continue;
 692
 693                         bool foundfirstsubsedfactor = false;
 694                         for (size_t j=0; j<subsed.size(); j++) {
 695                                 if (currsubsed[j]) {
 696                                         if (foundfirstsubsedfactor)
 697                                                 subsresult[j] = op(j);
 698                                         else {
 699                                                 foundfirstsubsedfactor = true;
 700                                                 subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
 701                                         }
 702                                         subsed[j] = true;
 703                                 }
 704                         }
 705                         goto retry1;
 706
 707                 } else {
 708 retry2:
 709                         int nummatches = std::numeric_limits<int>::max();
 710                         lst repls;
 711
 712                         for (size_t j=0; j<this->nops(); j++) {
 713                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)) {
 714                                         subsed[j] = true;
 715                                         subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
 716                                         goto retry2;
 717                                 }
 718                         }
 719                 }
 720         }
 721
 722         bool subsfound = false;
 723         for (size_t i=0; i<subsed.size(); i++) {
 724                 if (subsed[i]) {
 725                         subsfound = true;
 726                         break;
 727                 }
 728         }
 729         if (!subsfound)
 730                 return subs_one_level(m, options | subs_options::algebraic);
 731
 732         exvector ev; ev.reserve(nops());
 733         for (size_t i=0; i<nops(); i++) {
 734                 if (subsed[i])
 735                         ev.push_back(subsresult[i]);
 736                 else
 737                         ev.push_back(op(i));
 738         }
 739
 740         return (new mul(ev))->setflag(status_flags::dynallocated);
 741 }
 742
 743 // protected
 744
 745 /** Implementation of ex::diff() for a product.  It applies the product rule.
 746  *  @see ex::diff */
 747 ex mul::derivative(const symbol & s) const
 748 {
 749         size_t num = seq.size();
 750         exvector addseq;
 751         addseq.reserve(num);
 752
 753         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 754         epvector mulseq = seq;
 755         epvector::const_iterator i = seq.begin(), end = seq.end();
 756         epvector::iterator i2 = mulseq.begin();
 757         while (i != end) {
 758                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 759                                              i->rest.diff(s));
 760                 ep.swap(*i2);
 761                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 762                 ep.swap(*i2);
 763                 ++i; ++i2;
 764         }
 765         return (new add(addseq))->setflag(status_flags::dynallocated);
 766 }
 767
 768 int mul::compare_same_type(const basic & other) const
 769 {
 770         return inherited::compare_same_type(other);
 771 }
 772
 773 unsigned mul::return_type() const
 774 {
 775         if (seq.empty()) {
 776                 // mul without factors: should not happen, but commutates
 777                 return return_types::commutative;
 778         }
 779
 780         bool all_commutative = true;
 781         epvector::const_iterator noncommutative_element; // point to first found nc element
 782
 783         epvector::const_iterator i = seq.begin(), end = seq.end();
 784         while (i != end) {
 785                 unsigned rt = i->rest.return_type();
 786                 if (rt == return_types::noncommutative_composite)
 787                         return rt; // one ncc -> mul also ncc
 788                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 789                         // first nc element found, remember position
 790                         noncommutative_element = i;
 791                         all_commutative = false;
 792                 }
 793                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 794                         // another nc element found, compare type_infos
 795                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 796                                 // diffent types -> mul is ncc
 797                                 return return_types::noncommutative_composite;
 798                         }
 799                 }
 800                 ++i;
 801         }
 802         // all factors checked
 803         return all_commutative ? return_types::commutative : return_types::noncommutative;
 804 }
 805
 806 unsigned mul::return_type_tinfo() const
 807 {
 808         if (seq.empty())
 809                 return tinfo_key;  // mul without factors: should not happen
 810
 811         // return type_info of first noncommutative element
 812         epvector::const_iterator i = seq.begin(), end = seq.end();
 813         while (i != end) {
 814                 if (i->rest.return_type() == return_types::noncommutative)
 815                         return i->rest.return_type_tinfo();
 816                 ++i;
 817         }
 818         // no noncommutative element found, should not happen
 819         return tinfo_key;
 820 }
 821
 822 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
 823 {
 824         return (new mul(v, oc))->setflag(status_flags::dynallocated);
 825 }
 826
 827 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc) const
 828 {
 829         return (new mul(vp, oc))->setflag(status_flags::dynallocated);
 830 }
 831
 832 expair mul::split_ex_to_pair(const ex & e) const
 833 {
 834         if (is_exactly_a<power>(e)) {
 835                 const power & powerref = ex_to<power>(e);
 836                 if (is_exactly_a<numeric>(powerref.exponent))
 837                         return expair(powerref.basis,powerref.exponent);
 838         }
 839         return expair(e,_ex1);
 840 }
 841
 842 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 843                                           const ex & c) const
 844 {
 845         // to avoid duplication of power simplification rules,
 846         // we create a temporary power object
 847         // otherwise it would be hard to correctly evaluate
 848         // expression like (4^(1/3))^(3/2)
 849         if (c.is_equal(_ex1))
 850                 return split_ex_to_pair(e);
 851
 852         return split_ex_to_pair(power(e,c));
 853 }
 854
 855 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 856                                             const ex & c) const
 857 {
 858         // to avoid duplication of power simplification rules,
 859         // we create a temporary power object
 860         // otherwise it would be hard to correctly evaluate
 861         // expression like (4^(1/3))^(3/2)
 862         if (c.is_equal(_ex1))
 863                 return p;
 864
 865         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 866 }
 867
 868 ex mul::recombine_pair_to_ex(const expair & p) const
 869 {
 870         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 871                 return p.rest;
 872         else
 873                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 874 }
 875
 876 bool mul::expair_needs_further_processing(epp it)
 877 {
 878         if (is_exactly_a<mul>(it->rest) &&
 879                 ex_to<numeric>(it->coeff).is_integer()) {
 880                 // combined pair is product with integer power -> expand it
 881                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 882                 return true;
 883         }
 884         if (is_exactly_a<numeric>(it->rest)) {
 885                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 886                 if (!ep.is_equal(*it)) {
 887                         // combined pair is a numeric power which can be simplified
 888                         *it = ep;
 889                         return true;
 890                 }
 891                 if (it->coeff.is_equal(_ex1)) {
 892                         // combined pair has coeff 1 and must be moved to the end
 893                         return true;
 894                 }
 895         }
 896         return false;
 897 }
 898
 899 ex mul::default_overall_coeff() const
 900 {
 901         return _ex1;
 902 }
 903
 904 void mul::combine_overall_coeff(const ex & c)
 905 {
 906         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 907         GINAC_ASSERT(is_exactly_a<numeric>(c));
 908         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 909 }
 910
 911 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 912 {
 913         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 914         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 915         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 916         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 917 }
 918
 919 bool mul::can_make_flat(const expair & p) const
 920 {
 921         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 922         // this assertion will probably fail somewhere
 923         // it would require a more careful make_flat, obeying the power laws
 924         // probably should return true only if p.coeff is integer
 925         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 926 }
 927
 928 bool mul::can_be_further_expanded(const ex & e)
 929 {
 930         if (is_exactly_a<mul>(e)) {
 931                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 932                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 933                                 return true;
 934                 }
 935         } else if (is_exactly_a<power>(e)) {
 936                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 937                         return true;
 938         }
 939         return false;
 940 }
 941
 942 ex mul::expand(unsigned options) const
 943 {
 944         {
 945         // trivial case: expanding the monomial (~ 30% of all calls)
 946                 epvector::const_iterator i = seq.begin(), seq_end = seq.end();
 947                 while ((i != seq.end()) &&  is_a<symbol>(i->rest) && i->coeff.info(info_flags::integer))
 948                         ++i;
 949                 if (i == seq_end) {
 950                         setflag(status_flags::expanded);
 951                         return *this;
 952                 }
 953         }
 954
 955         // do not rename indices if the object has no indices at all
 956         if ((!(options & expand_options::expand_rename_idx)) &&
 957                         this->info(info_flags::has_indices))
 958                 options |= expand_options::expand_rename_idx;
 959
 960         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
 961
 962         // First, expand the children
 963         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
 964         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
 965
 966         // Now, look for all the factors that are sums and multiply each one out
 967         // with the next one that is found while collecting the factors which are
 968         // not sums
 969         ex last_expanded = _ex1;
 970
 971         epvector non_adds;
 972         non_adds.reserve(expanded_seq.size());
 973
 974         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
 975                 if (is_exactly_a<add>(cit->rest) &&
 976                         (cit->coeff.is_equal(_ex1))) {
 977                         if (is_exactly_a<add>(last_expanded)) {
 978
 979                                 // Expand a product of two sums, aggressive version.
 980                                 // Caring for the overall coefficients in separate loops can
 981                                 // sometimes give a performance gain of up to 15%!
 982
 983                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
 984                                 // add2 is for the inner loop and should be the bigger of the two sums
 985                                 // in the presence of asymptotically good sorting:
 986                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
 987                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
 988                                 const epvector::const_iterator add1begin = add1.seq.begin();
 989                                 const epvector::const_iterator add1end   = add1.seq.end();
 990                                 const epvector::const_iterator add2begin = add2.seq.begin();
 991                                 const epvector::const_iterator add2end   = add2.seq.end();
 992                                 epvector distrseq;
 993                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
 994
 995                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
 996                                 if (!add1.overall_coeff.is_zero()) {
 997                                         if (add1.overall_coeff.is_equal(_ex1))
 998                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
 999                                         else
1000                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
1001                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1002                                 }
1003
1004                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1005                                 if (!add2.overall_coeff.is_zero()) {
1006                                         if (add2.overall_coeff.is_equal(_ex1))
1007                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
1008                                         else
1009                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
1010                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1011                                 }
1012
1013                                 // Compute the new overall coefficient and put it together:
1014                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
1015
1016                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1017                                 for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1018                                         // We really have to combine terms here in order to compactify
1019                                         // the result.  Otherwise it would become waayy tooo bigg.
1020                                         numeric oc;
1021                                         distrseq.clear();
1022                                         for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
1023                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1024                                                 // since that would violate an invariant of expairseq:
1025                                                 const ex rest = (new mul(i1->rest, skip_idx_rename ? i2->rest : rename_dummy_indices_uniquely(i1->rest, i2->rest)))->setflag(status_flags::dynallocated);
1026                                                 if (is_exactly_a<numeric>(rest)) {
1027                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1028                                                 } else {
1029                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1030                                                 }
1031                                         }
1032                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
1033                                 }
1034                                 last_expanded = tmp_accu;
1035
1036                         } else {
1037                                 if (!last_expanded.is_equal(_ex1))
1038                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1039                                 last_expanded = cit->rest;
1040                         }
1041
1042                 } else {
1043                         non_adds.push_back(*cit);
1044                 }
1045         }
1046
1047         // Now the only remaining thing to do is to multiply the factors which
1048         // were not sums into the "last_expanded" sum
1049         if (is_exactly_a<add>(last_expanded)) {
1050                 size_t n = last_expanded.nops();
1051                 exvector distrseq;
1052                 distrseq.reserve(n);
1053
1054                 for (size_t i=0; i<n; ++i) {
1055                         epvector factors = non_adds;
1056                         if (skip_idx_rename)
1057                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1058                         else
1059                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(mul(non_adds), last_expanded.op(i))));
1060                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1061                         if (can_be_further_expanded(term)) {
1062                                 distrseq.push_back(term.expand());
1063                         } else {
1064                                 if (options == 0)
1065                                         ex_to<basic>(term).setflag(status_flags::expanded);
1066                                 distrseq.push_back(term);
1067                         }
1068                 }
1069
1070                 return ((new add(distrseq))->
1071                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1072         }
1073
1074         non_adds.push_back(split_ex_to_pair(last_expanded));
1075         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1076         if (can_be_further_expanded(result)) {
1077                 return result.expand();
1078         } else {
1079                 if (options == 0)
1080                         ex_to<basic>(result).setflag(status_flags::expanded);
1081                 return result;
1082         }
1083 }
1084
1085
1086 //////////
1087 // new virtual functions which can be overridden by derived classes
1088 //////////
1089
1090 // none
1091
1092 //////////
1093 // non-virtual functions in this class
1094 //////////
1095
1096
1097 /** Member-wise expand the expairs representing this sequence.  This must be
1098  *  overridden from expairseq::expandchildren() and done iteratively in order
1099  *  to allow for early cancallations and thus safe memory.
1100  *
1101  *  @see mul::expand()
1102  *  @return pointer to epvector containing expanded representation or zero
1103  *  pointer, if sequence is unchanged. */
1104 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1105 {
1106         const epvector::const_iterator last = seq.end();
1107         epvector::const_iterator cit = seq.begin();
1108         while (cit!=last) {
1109                 const ex & factor = recombine_pair_to_ex(*cit);
1110                 const ex & expanded_factor = factor.expand(options);
1111                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1112
1113                         // something changed, copy seq, eval and return it
1114                         std::auto_ptr<epvector> s(new epvector);
1115                         s->reserve(seq.size());
1116
1117                         // copy parts of seq which are known not to have changed
1118                         epvector::const_iterator cit2 = seq.begin();
1119                         while (cit2!=cit) {
1120                                 s->push_back(*cit2);
1121                                 ++cit2;
1122                         }
1123
1124                         // copy first changed element
1125                         s->push_back(split_ex_to_pair(expanded_factor));
1126                         ++cit2;
1127
1128                         // copy rest
1129                         while (cit2!=last) {
1130                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1131                                 ++cit2;
1132                         }
1133                         return s;
1134                 }
1135                 ++cit;
1136         }
1137
1138         return std::auto_ptr<epvector>(0); // nothing has changed
1139 }
1140
1141 } // namespace GiNaC