ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2008 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 }
  57
  58 //////////
  59 // other constructors
  60 //////////
  61
  62 // public
  63
  64 mul::mul(const ex & lh, const ex & rh)
  65 {
  66         overall_coeff = _ex1;
  67         construct_from_2_ex(lh,rh);
  68         GINAC_ASSERT(is_canonical());
  69 }
  70
  71 mul::mul(const exvector & v)
  72 {
  73         overall_coeff = _ex1;
  74         construct_from_exvector(v);
  75         GINAC_ASSERT(is_canonical());
  76 }
  77
  78 mul::mul(const epvector & v)
  79 {
  80         overall_coeff = _ex1;
  81         construct_from_epvector(v);
  82         GINAC_ASSERT(is_canonical());
  83 }
  84
  85 mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
  86 {
  87         overall_coeff = oc;
  88         construct_from_epvector(v, do_index_renaming);
  89         GINAC_ASSERT(is_canonical());
  90 }
  91
  92 mul::mul(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming)
  93 {
  94         GINAC_ASSERT(vp.get()!=0);
  95         overall_coeff = oc;
  96         construct_from_epvector(*vp, do_index_renaming);
  97         GINAC_ASSERT(is_canonical());
  98 }
  99
 100 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 101 {
 102         exvector factors;
 103         factors.reserve(3);
 104         factors.push_back(lh);
 105         factors.push_back(mh);
 106         factors.push_back(rh);
 107         overall_coeff = _ex1;
 108         construct_from_exvector(factors);
 109         GINAC_ASSERT(is_canonical());
 110 }
 111
 112 //////////
 113 // archiving
 114 //////////
 115
 116 DEFAULT_ARCHIVING(mul)
 117
 118 //////////
 119 // functions overriding virtual functions from base classes
 120 //////////
 121
 122 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
 123 {
 124         const numeric &coeff = ex_to<numeric>(overall_coeff);
 125         if (coeff.csgn() == -1)
 126                 c.s << '-';
 127         if (!coeff.is_equal(*_num1_p) &&
 128                 !coeff.is_equal(*_num_1_p)) {
 129                 if (coeff.is_rational()) {
 130                         if (coeff.is_negative())
 131                                 (-coeff).print(c);
 132                         else
 133                                 coeff.print(c);
 134                 } else {
 135                         if (coeff.csgn() == -1)
 136                                 (-coeff).print(c, precedence());
 137                         else
 138                                 coeff.print(c, precedence());
 139                 }
 140                 c.s << mul_sym;
 141         }
 142 }
 143
 144 void mul::do_print(const print_context & c, unsigned level) const
 145 {
 146         if (precedence() <= level)
 147                 c.s << '(';
 148
 149         print_overall_coeff(c, "*");
 150
 151         epvector::const_iterator it = seq.begin(), itend = seq.end();
 152         bool first = true;
 153         while (it != itend) {
 154                 if (!first)
 155                         c.s << '*';
 156                 else
 157                         first = false;
 158                 recombine_pair_to_ex(*it).print(c, precedence());
 159                 ++it;
 160         }
 161
 162         if (precedence() <= level)
 163                 c.s << ')';
 164 }
 165
 166 void mul::do_print_latex(const print_latex & c, unsigned level) const
 167 {
 168         if (precedence() <= level)
 169                 c.s << "{(";
 170
 171         print_overall_coeff(c, " ");
 172
 173         // Separate factors into those with negative numeric exponent
 174         // and all others
 175         epvector::const_iterator it = seq.begin(), itend = seq.end();
 176         exvector neg_powers, others;
 177         while (it != itend) {
 178                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
 179                 if (ex_to<numeric>(it->coeff).is_negative())
 180                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
 181                 else
 182                         others.push_back(recombine_pair_to_ex(*it));
 183                 ++it;
 184         }
 185
 186         if (!neg_powers.empty()) {
 187
 188                 // Factors with negative exponent are printed as a fraction
 189                 c.s << "\\frac{";
 190                 mul(others).eval().print(c);
 191                 c.s << "}{";
 192                 mul(neg_powers).eval().print(c);
 193                 c.s << "}";
 194
 195         } else {
 196
 197                 // All other factors are printed in the ordinary way
 198                 exvector::const_iterator vit = others.begin(), vitend = others.end();
 199                 while (vit != vitend) {
 200                         c.s << ' ';
 201                         vit->print(c, precedence());
 202                         ++vit;
 203                 }
 204         }
 205
 206         if (precedence() <= level)
 207                 c.s << ")}";
 208 }
 209
 210 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
 211 {
 212         if (precedence() <= level)
 213                 c.s << "(";
 214
 215         if (!overall_coeff.is_equal(_ex1)) {
 216                 if (overall_coeff.is_equal(_ex_1))
 217                         c.s << "-";
 218                 else {
 219                         overall_coeff.print(c, precedence());
 220                         c.s << "*";
 221                 }
 222         }
 223
 224         // Print arguments, separated by "*" or "/"
 225         epvector::const_iterator it = seq.begin(), itend = seq.end();
 226         while (it != itend) {
 227
 228                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 229                 bool needclosingparenthesis = false;
 230                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 231                         if (is_a<print_csrc_cl_N>(c)) {
 232                                 c.s << "recip(";
 233                                 needclosingparenthesis = true;
 234                         } else
 235                                 c.s << "1.0/";
 236                 }
 237
 238                 // If the exponent is 1 or -1, it is left out
 239                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 240                         it->rest.print(c, precedence());
 241                 else if (it->coeff.info(info_flags::negint))
 242                         // Outer parens around ex needed for broken GCC parser:
 243                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 244                 else
 245                         // Outer parens around ex needed for broken GCC parser:
 246                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 247
 248                 if (needclosingparenthesis)
 249                         c.s << ")";
 250
 251                 // Separator is "/" for negative integer powers, "*" otherwise
 252                 ++it;
 253                 if (it != itend) {
 254                         if (it->coeff.info(info_flags::negint))
 255                                 c.s << "/";
 256                         else
 257                                 c.s << "*";
 258                 }
 259         }
 260
 261         if (precedence() <= level)
 262                 c.s << ")";
 263 }
 264
 265 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 266 {
 267         c.s << class_name() << '(';
 268         op(0).print(c);
 269         for (size_t i=1; i<nops(); ++i) {
 270                 c.s << ',';
 271                 op(i).print(c);
 272         }
 273         c.s << ')';
 274 }
 275
 276 bool mul::info(unsigned inf) const
 277 {
 278         switch (inf) {
 279                 case info_flags::polynomial:
 280                 case info_flags::integer_polynomial:
 281                 case info_flags::cinteger_polynomial:
 282                 case info_flags::rational_polynomial:
 283                 case info_flags::crational_polynomial:
 284                 case info_flags::rational_function: {
 285                         epvector::const_iterator i = seq.begin(), end = seq.end();
 286                         while (i != end) {
 287                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 288                                         return false;
 289                                 ++i;
 290                         }
 291                         return overall_coeff.info(inf);
 292                 }
 293                 case info_flags::algebraic: {
 294                         epvector::const_iterator i = seq.begin(), end = seq.end();
 295                         while (i != end) {
 296                                 if ((recombine_pair_to_ex(*i).info(inf)))
 297                                         return true;
 298                                 ++i;
 299                         }
 300                         return false;
 301                 }
 302         }
 303         return inherited::info(inf);
 304 }
 305
 306 int mul::degree(const ex & s) const
 307 {
 308         // Sum up degrees of factors
 309         int deg_sum = 0;
 310         epvector::const_iterator i = seq.begin(), end = seq.end();
 311         while (i != end) {
 312                 if (ex_to<numeric>(i->coeff).is_integer())
 313                         deg_sum += recombine_pair_to_ex(*i).degree(s);
 314                 else {
 315                         if (i->rest.has(s))
 316                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
 317                 }
 318                 ++i;
 319         }
 320         return deg_sum;
 321 }
 322
 323 int mul::ldegree(const ex & s) const
 324 {
 325         // Sum up degrees of factors
 326         int deg_sum = 0;
 327         epvector::const_iterator i = seq.begin(), end = seq.end();
 328         while (i != end) {
 329                 if (ex_to<numeric>(i->coeff).is_integer())
 330                         deg_sum += recombine_pair_to_ex(*i).ldegree(s);
 331                 else {
 332                         if (i->rest.has(s))
 333                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
 334                 }
 335                 ++i;
 336         }
 337         return deg_sum;
 338 }
 339
 340 ex mul::coeff(const ex & s, int n) const
 341 {
 342         exvector coeffseq;
 343         coeffseq.reserve(seq.size()+1);
 344
 345         if (n==0) {
 346                 // product of individual coeffs
 347                 // if a non-zero power of s is found, the resulting product will be 0
 348                 epvector::const_iterator i = seq.begin(), end = seq.end();
 349                 while (i != end) {
 350                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 351                         ++i;
 352                 }
 353                 coeffseq.push_back(overall_coeff);
 354                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 355         }
 356
 357         epvector::const_iterator i = seq.begin(), end = seq.end();
 358         bool coeff_found = false;
 359         while (i != end) {
 360                 ex t = recombine_pair_to_ex(*i);
 361                 ex c = t.coeff(s, n);
 362                 if (!c.is_zero()) {
 363                         coeffseq.push_back(c);
 364                         coeff_found = 1;
 365                 } else {
 366                         coeffseq.push_back(t);
 367                 }
 368                 ++i;
 369         }
 370         if (coeff_found) {
 371                 coeffseq.push_back(overall_coeff);
 372                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 373         }
 374
 375         return _ex0;
 376 }
 377
 378 /** Perform automatic term rewriting rules in this class.  In the following
 379  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 380  *  stand for such expressions that contain a plain number.
 381  *  - *(...,x;0) -> 0
 382  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 383  *  - *(x;1) -> x
 384  *  - *(;c) -> c
 385  *
 386  *  @param level cut-off in recursive evaluation */
 387 ex mul::eval(int level) const
 388 {
 389         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 390         if (evaled_seqp.get()) {
 391                 // do more evaluation later
 392                 return (new mul(evaled_seqp, overall_coeff))->
 393                            setflag(status_flags::dynallocated);
 394         }
 395
 396 #ifdef DO_GINAC_ASSERT
 397         epvector::const_iterator i = seq.begin(), end = seq.end();
 398         while (i != end) {
 399                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 400                              (!(ex_to<numeric>(i->coeff).is_integer())));
 401                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 402                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 403                     print(print_tree(std::cerr));
 404                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 405                 /* for paranoia */
 406                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 407                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 408                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 409                 /* end paranoia */
 410                 ++i;
 411         }
 412 #endif // def DO_GINAC_ASSERT
 413
 414         if (flags & status_flags::evaluated) {
 415                 GINAC_ASSERT(seq.size()>0);
 416                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 417                 return *this;
 418         }
 419
 420         size_t seq_size = seq.size();
 421         if (overall_coeff.is_zero()) {
 422                 // *(...,x;0) -> 0
 423                 return _ex0;
 424         } else if (seq_size==0) {
 425                 // *(;c) -> c
 426                 return overall_coeff;
 427         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 428                 // *(x;1) -> x
 429                 return recombine_pair_to_ex(*(seq.begin()));
 430         } else if ((seq_size==1) &&
 431                    is_exactly_a<add>((*seq.begin()).rest) &&
 432                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 433                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 434                 const add & addref = ex_to<add>((*seq.begin()).rest);
 435                 std::auto_ptr<epvector> distrseq(new epvector);
 436                 distrseq->reserve(addref.seq.size());
 437                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 438                 while (i != end) {
 439                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 440                         ++i;
 441                 }
 442                 return (new add(distrseq,
 443                                 ex_to<numeric>(addref.overall_coeff).
 444                                 mul_dyn(ex_to<numeric>(overall_coeff)))
 445                        )->setflag(status_flags::dynallocated | status_flags::evaluated);
 446         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
 447                 // Strip the content and the unit part from each term. Thus
 448                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)2
 449
 450                 epvector::const_iterator last = seq.end();
 451                 epvector::const_iterator i = seq.begin();
 452                 epvector::const_iterator j = seq.begin();
 453                 std::auto_ptr<epvector> s(new epvector);
 454                 numeric oc = *_num1_p;
 455                 bool something_changed = false;
 456                 while (i!=last) {
 457                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
 458                                 // power::eval has such a rule, no need to handle powers here
 459                                 ++i;
 460                                 continue;
 461                         }
 462
 463                         // XXX: What is the best way to check if the polynomial is a primitive?
 464                         numeric c = i->rest.integer_content();
 465                         const numeric lead_coeff =
 466                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
 467                         const bool canonicalizable = lead_coeff.is_integer();
 468
 469                         // XXX: The main variable is chosen in a random way, so this code
 470                         // does NOT transform the term into the canonical form (thus, in some
 471                         // very unlucky event it can even loop forever). Hopefully the main
 472                         // variable will be the same for all terms in *this
 473                         const bool unit_normal = lead_coeff.is_pos_integer();
 474                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
 475                                 ++i;
 476                                 continue;
 477                         }
 478
 479                         if (! something_changed) {
 480                                 s->reserve(seq_size);
 481                                 something_changed = true;
 482                         }
 483
 484                         while ((j!=i) && (j!=last)) {
 485                                 s->push_back(*j);
 486                                 ++j;
 487                         }
 488
 489                         if (! unit_normal)
 490                                 c = c.mul(*_num_1_p);
 491
 492                         oc = oc.mul(c);
 493
 494                         // divide add by the number in place to save at least 2 .eval() calls
 495                         const add& addref = ex_to<add>(i->rest);
 496                         add* primitive = new add(addref);
 497                         primitive->setflag(status_flags::dynallocated);
 498                         primitive->clearflag(status_flags::hash_calculated);
 499                         primitive->overall_coeff = ex_to<numeric>(primitive->overall_coeff).div_dyn(c);
 500                         for (epvector::iterator ai = primitive->seq.begin();
 501                                         ai != primitive->seq.end(); ++ai)
 502                                 ai->coeff = ex_to<numeric>(ai->coeff).div_dyn(c);
 503
 504                         s->push_back(expair(*primitive, _ex1));
 505
 506                         ++i;
 507                         ++j;
 508                 }
 509                 if (something_changed) {
 510                         while (j!=last) {
 511                                 s->push_back(*j);
 512                                 ++j;
 513                         }
 514                         return (new mul(s, ex_to<numeric>(overall_coeff).mul_dyn(oc))
 515                                )->setflag(status_flags::dynallocated);
 516                 }
 517         }
 518
 519         return this->hold();
 520 }
 521
 522 ex mul::evalf(int level) const
 523 {
 524         if (level==1)
 525                 return mul(seq,overall_coeff);
 526
 527         if (level==-max_recursion_level)
 528                 throw(std::runtime_error("max recursion level reached"));
 529
 530         std::auto_ptr<epvector> s(new epvector);
 531         s->reserve(seq.size());
 532
 533         --level;
 534         epvector::const_iterator i = seq.begin(), end = seq.end();
 535         while (i != end) {
 536                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 537                                                            i->coeff));
 538                 ++i;
 539         }
 540         return mul(s, overall_coeff.evalf(level));
 541 }
 542
 543 void mul::find_real_imag(ex & rp, ex & ip) const
 544 {
 545         rp = overall_coeff.real_part();
 546         ip = overall_coeff.imag_part();
 547         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 548                 ex factor = recombine_pair_to_ex(*i);
 549                 ex new_rp = factor.real_part();
 550                 ex new_ip = factor.imag_part();
 551                 if(new_ip.is_zero()) {
 552                         rp *= new_rp;
 553                         ip *= new_rp;
 554                 } else {
 555                         ex temp = rp*new_rp - ip*new_ip;
 556                         ip = ip*new_rp + rp*new_ip;
 557                         rp = temp;
 558                 }
 559         }
 560         rp = rp.expand();
 561         ip = ip.expand();
 562 }
 563
 564 ex mul::real_part() const
 565 {
 566         ex rp, ip;
 567         find_real_imag(rp, ip);
 568         return rp;
 569 }
 570
 571 ex mul::imag_part() const
 572 {
 573         ex rp, ip;
 574         find_real_imag(rp, ip);
 575         return ip;
 576 }
 577
 578 ex mul::evalm() const
 579 {
 580         // numeric*matrix
 581         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 582          && is_a<matrix>(seq[0].rest))
 583                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 584
 585         // Evaluate children first, look whether there are any matrices at all
 586         // (there can be either no matrices or one matrix; if there were more
 587         // than one matrix, it would be a non-commutative product)
 588         std::auto_ptr<epvector> s(new epvector);
 589         s->reserve(seq.size());
 590
 591         bool have_matrix = false;
 592         epvector::iterator the_matrix;
 593
 594         epvector::const_iterator i = seq.begin(), end = seq.end();
 595         while (i != end) {
 596                 const ex &m = recombine_pair_to_ex(*i).evalm();
 597                 s->push_back(split_ex_to_pair(m));
 598                 if (is_a<matrix>(m)) {
 599                         have_matrix = true;
 600                         the_matrix = s->end() - 1;
 601                 }
 602                 ++i;
 603         }
 604
 605         if (have_matrix) {
 606
 607                 // The product contained a matrix. We will multiply all other factors
 608                 // into that matrix.
 609                 matrix m = ex_to<matrix>(the_matrix->rest);
 610                 s->erase(the_matrix);
 611                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 612                 return m.mul_scalar(scalar);
 613
 614         } else
 615                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 616 }
 617
 618 ex mul::eval_ncmul(const exvector & v) const
 619 {
 620         if (seq.empty())
 621                 return inherited::eval_ncmul(v);
 622
 623         // Find first noncommutative element and call its eval_ncmul()
 624         epvector::const_iterator i = seq.begin(), end = seq.end();
 625         while (i != end) {
 626                 if (i->rest.return_type() == return_types::noncommutative)
 627                         return i->rest.eval_ncmul(v);
 628                 ++i;
 629         }
 630         return inherited::eval_ncmul(v);
 631 }
 632
 633 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
 634 {
 635         ex origbase;
 636         int origexponent;
 637         int origexpsign;
 638
 639         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 640                 origbase = origfactor.op(0);
 641                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 642                 origexponent = expon > 0 ? expon : -expon;
 643                 origexpsign = expon > 0 ? 1 : -1;
 644         } else {
 645                 origbase = origfactor;
 646                 origexponent = 1;
 647                 origexpsign = 1;
 648         }
 649
 650         ex patternbase;
 651         int patternexponent;
 652         int patternexpsign;
 653
 654         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 655                 patternbase = patternfactor.op(0);
 656                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 657                 patternexponent = expon > 0 ? expon : -expon;
 658                 patternexpsign = expon > 0 ? 1 : -1;
 659         } else {
 660                 patternbase = patternfactor;
 661                 patternexponent = 1;
 662                 patternexpsign = 1;
 663         }
 664
 665         exmap saverepls = repls;
 666         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 667                 return false;
 668         repls = saverepls;
 669
 670         int newnummatches = origexponent / patternexponent;
 671         if (newnummatches < nummatches)
 672                 nummatches = newnummatches;
 673         return true;
 674 }
 675
 676 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
 677   * list of replacements repls. This matching is in the sense of algebraic
 678   * substitutions. Matching starts with pat.op(factor) of the pattern because
 679   * the factors before this one have already been matched. The (possibly
 680   * updated) number of matches is in nummatches. subsed[i] is true for factors
 681   * that already have been replaced by previous substitutions and matched[i]
 682   * is true for factors that have been matched by the current match.
 683   */
 684 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
 685                 int factor, int &nummatches, const std::vector<bool> &subsed,
 686                 std::vector<bool> &matched)
 687 {
 688         if (factor == pat.nops())
 689                 return true;
 690
 691         for (size_t i=0; i<e.nops(); ++i) {
 692                 if(subsed[i] || matched[i])
 693                         continue;
 694                 exmap newrepls = repls;
 695                 int newnummatches = nummatches;
 696                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 697                         matched[i] = true;
 698                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 699                                         newnummatches, subsed, matched)) {
 700                                 repls = newrepls;
 701                                 nummatches = newnummatches;
 702                                 return true;
 703                         }
 704                         else
 705                                 matched[i] = false;
 706                 }
 707         }
 708
 709         return false;
 710 }
 711
 712 bool mul::has(const ex & pattern, unsigned options) const
 713 {
 714         if(!(options&has_options::algebraic))
 715                 return basic::has(pattern,options);
 716         if(is_a<mul>(pattern)) {
 717                 exmap repls;
 718                 int nummatches = std::numeric_limits<int>::max();
 719                 std::vector<bool> subsed(seq.size(), false);
 720                 std::vector<bool> matched(seq.size(), false);
 721                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 722                                 subsed, matched))
 723                         return true;
 724         }
 725         return basic::has(pattern, options);
 726 }
 727
 728 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 729 {
 730         std::vector<bool> subsed(seq.size(), false);
 731         exvector subsresult(seq.size());
 732         ex divide_by = 1;
 733         ex multiply_by = 1;
 734
 735         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 736
 737                 if (is_exactly_a<mul>(it->first)) {
 738 retry1:
 739                         int nummatches = std::numeric_limits<int>::max();
 740                         std::vector<bool> currsubsed(seq.size(), false);
 741                         exmap repls;
 742
 743                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 744                                 continue;
 745
 746                         for (size_t j=0; j<subsed.size(); j++)
 747                                 if (currsubsed[j])
 748                                         subsed[j] = true;
 749                         ex subsed_pattern
 750                                 = it->first.subs(repls, subs_options::no_pattern);
 751                         divide_by *= power(subsed_pattern, nummatches);
 752                         ex subsed_result
 753                                 = it->second.subs(repls, subs_options::no_pattern);
 754                         multiply_by *= power(subsed_result, nummatches);
 755                         goto retry1;
 756
 757                 } else {
 758
 759                         for (size_t j=0; j<this->nops(); j++) {
 760                                 int nummatches = std::numeric_limits<int>::max();
 761                                 exmap repls;
 762                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
 763                                         subsed[j] = true;
 764                                         ex subsed_pattern
 765                                                 = it->first.subs(repls, subs_options::no_pattern);
 766                                         divide_by *= power(subsed_pattern, nummatches);
 767                                         ex subsed_result
 768                                                 = it->second.subs(repls, subs_options::no_pattern);
 769                                         multiply_by *= power(subsed_result, nummatches);
 770                                 }
 771                         }
 772                 }
 773         }
 774
 775         bool subsfound = false;
 776         for (size_t i=0; i<subsed.size(); i++) {
 777                 if (subsed[i]) {
 778                         subsfound = true;
 779                         break;
 780                 }
 781         }
 782         if (!subsfound)
 783                 return subs_one_level(m, options | subs_options::algebraic);
 784
 785         return ((*this)/divide_by)*multiply_by;
 786 }
 787
 788 // protected
 789
 790 /** Implementation of ex::diff() for a product.  It applies the product rule.
 791  *  @see ex::diff */
 792 ex mul::derivative(const symbol & s) const
 793 {
 794         size_t num = seq.size();
 795         exvector addseq;
 796         addseq.reserve(num);
 797
 798         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 799         epvector mulseq = seq;
 800         epvector::const_iterator i = seq.begin(), end = seq.end();
 801         epvector::iterator i2 = mulseq.begin();
 802         while (i != end) {
 803                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 804                                              i->rest.diff(s));
 805                 ep.swap(*i2);
 806                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 807                 ep.swap(*i2);
 808                 ++i; ++i2;
 809         }
 810         return (new add(addseq))->setflag(status_flags::dynallocated);
 811 }
 812
 813 int mul::compare_same_type(const basic & other) const
 814 {
 815         return inherited::compare_same_type(other);
 816 }
 817
 818 unsigned mul::return_type() const
 819 {
 820         if (seq.empty()) {
 821                 // mul without factors: should not happen, but commutates
 822                 return return_types::commutative;
 823         }
 824
 825         bool all_commutative = true;
 826         epvector::const_iterator noncommutative_element; // point to first found nc element
 827
 828         epvector::const_iterator i = seq.begin(), end = seq.end();
 829         while (i != end) {
 830                 unsigned rt = i->rest.return_type();
 831                 if (rt == return_types::noncommutative_composite)
 832                         return rt; // one ncc -> mul also ncc
 833                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 834                         // first nc element found, remember position
 835                         noncommutative_element = i;
 836                         all_commutative = false;
 837                 }
 838                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 839                         // another nc element found, compare type_infos
 840                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 841                                         // different types -> mul is ncc
 842                                         return return_types::noncommutative_composite;
 843                         }
 844                 }
 845                 ++i;
 846         }
 847         // all factors checked
 848         return all_commutative ? return_types::commutative : return_types::noncommutative;
 849 }
 850
 851 return_type_t mul::return_type_tinfo() const
 852 {
 853         if (seq.empty())
 854                 return make_return_type_t<mul>(); // mul without factors: should not happen
 855
 856         // return type_info of first noncommutative element
 857         epvector::const_iterator i = seq.begin(), end = seq.end();
 858         while (i != end) {
 859                 if (i->rest.return_type() == return_types::noncommutative)
 860                         return i->rest.return_type_tinfo();
 861                 ++i;
 862         }
 863         // no noncommutative element found, should not happen
 864         return make_return_type_t<mul>();
 865 }
 866
 867 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 868 {
 869         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 870 }
 871
 872 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
 873 {
 874         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 875 }
 876
 877 expair mul::split_ex_to_pair(const ex & e) const
 878 {
 879         if (is_exactly_a<power>(e)) {
 880                 const power & powerref = ex_to<power>(e);
 881                 if (is_exactly_a<numeric>(powerref.exponent))
 882                         return expair(powerref.basis,powerref.exponent);
 883         }
 884         return expair(e,_ex1);
 885 }
 886
 887 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 888                                           const ex & c) const
 889 {
 890         // to avoid duplication of power simplification rules,
 891         // we create a temporary power object
 892         // otherwise it would be hard to correctly evaluate
 893         // expression like (4^(1/3))^(3/2)
 894         if (c.is_equal(_ex1))
 895                 return split_ex_to_pair(e);
 896
 897         return split_ex_to_pair(power(e,c));
 898 }
 899
 900 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 901                                             const ex & c) const
 902 {
 903         // to avoid duplication of power simplification rules,
 904         // we create a temporary power object
 905         // otherwise it would be hard to correctly evaluate
 906         // expression like (4^(1/3))^(3/2)
 907         if (c.is_equal(_ex1))
 908                 return p;
 909
 910         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 911 }
 912
 913 ex mul::recombine_pair_to_ex(const expair & p) const
 914 {
 915         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 916                 return p.rest;
 917         else
 918                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 919 }
 920
 921 bool mul::expair_needs_further_processing(epp it)
 922 {
 923         if (is_exactly_a<mul>(it->rest) &&
 924                 ex_to<numeric>(it->coeff).is_integer()) {
 925                 // combined pair is product with integer power -> expand it
 926                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 927                 return true;
 928         }
 929         if (is_exactly_a<numeric>(it->rest)) {
 930                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 931                 if (!ep.is_equal(*it)) {
 932                         // combined pair is a numeric power which can be simplified
 933                         *it = ep;
 934                         return true;
 935                 }
 936                 if (it->coeff.is_equal(_ex1)) {
 937                         // combined pair has coeff 1 and must be moved to the end
 938                         return true;
 939                 }
 940         }
 941         return false;
 942 }
 943
 944 ex mul::default_overall_coeff() const
 945 {
 946         return _ex1;
 947 }
 948
 949 void mul::combine_overall_coeff(const ex & c)
 950 {
 951         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 952         GINAC_ASSERT(is_exactly_a<numeric>(c));
 953         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 954 }
 955
 956 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 957 {
 958         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 959         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 960         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 961         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 962 }
 963
 964 bool mul::can_make_flat(const expair & p) const
 965 {
 966         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 967         // this assertion will probably fail somewhere
 968         // it would require a more careful make_flat, obeying the power laws
 969         // probably should return true only if p.coeff is integer
 970         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 971 }
 972
 973 bool mul::can_be_further_expanded(const ex & e)
 974 {
 975         if (is_exactly_a<mul>(e)) {
 976                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 977                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 978                                 return true;
 979                 }
 980         } else if (is_exactly_a<power>(e)) {
 981                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 982                         return true;
 983         }
 984         return false;
 985 }
 986
 987 ex mul::expand(unsigned options) const
 988 {
 989         {
 990         // trivial case: expanding the monomial (~ 30% of all calls)
 991                 epvector::const_iterator i = seq.begin(), seq_end = seq.end();
 992                 while ((i != seq.end()) &&  is_a<symbol>(i->rest) && i->coeff.info(info_flags::integer))
 993                         ++i;
 994                 if (i == seq_end) {
 995                         setflag(status_flags::expanded);
 996                         return *this;
 997                 }
 998         }
 999
1000         // do not rename indices if the object has no indices at all
1001         if ((!(options & expand_options::expand_rename_idx)) &&
1002                         this->info(info_flags::has_indices))
1003                 options |= expand_options::expand_rename_idx;
1004
1005         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1006
1007         // First, expand the children
1008         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
1009         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
1010
1011         // Now, look for all the factors that are sums and multiply each one out
1012         // with the next one that is found while collecting the factors which are
1013         // not sums
1014         ex last_expanded = _ex1;
1015
1016         epvector non_adds;
1017         non_adds.reserve(expanded_seq.size());
1018
1019         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
1020                 if (is_exactly_a<add>(cit->rest) &&
1021                         (cit->coeff.is_equal(_ex1))) {
1022                         if (is_exactly_a<add>(last_expanded)) {
1023
1024                                 // Expand a product of two sums, aggressive version.
1025                                 // Caring for the overall coefficients in separate loops can
1026                                 // sometimes give a performance gain of up to 15%!
1027
1028                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
1029                                 // add2 is for the inner loop and should be the bigger of the two sums
1030                                 // in the presence of asymptotically good sorting:
1031                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
1032                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
1033                                 const epvector::const_iterator add1begin = add1.seq.begin();
1034                                 const epvector::const_iterator add1end   = add1.seq.end();
1035                                 const epvector::const_iterator add2begin = add2.seq.begin();
1036                                 const epvector::const_iterator add2end   = add2.seq.end();
1037                                 epvector distrseq;
1038                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
1039
1040                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1041                                 if (!add1.overall_coeff.is_zero()) {
1042                                         if (add1.overall_coeff.is_equal(_ex1))
1043                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
1044                                         else
1045                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
1046                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1047                                 }
1048
1049                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1050                                 if (!add2.overall_coeff.is_zero()) {
1051                                         if (add2.overall_coeff.is_equal(_ex1))
1052                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
1053                                         else
1054                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
1055                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1056                                 }
1057
1058                                 // Compute the new overall coefficient and put it together:
1059                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
1060
1061                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1062                                 lst dummy_subs;
1063
1064                                 if (!skip_idx_rename) {
1065                                         for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
1066                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1067                                                 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1068                                         }
1069                                         for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
1070                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1071                                                 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1072                                         }
1073
1074                                         sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1075                                         sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1076                                         dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1077                                 }
1078
1079                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1080                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
1081                                         // We really have to combine terms here in order to compactify
1082                                         // the result.  Otherwise it would become waayy tooo bigg.
1083                                         numeric oc(*_num0_p);
1084                                         epvector distrseq2;
1085                                         distrseq2.reserve(add1.seq.size());
1086                                         const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1087                                                         i2->rest :
1088                                                         i2->rest.subs(ex_to<lst>(dummy_subs.op(0)),
1089                                                                 ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1090                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1091                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1092                                                 // since that would violate an invariant of expairseq:
1093                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
1094                                                 if (is_exactly_a<numeric>(rest)) {
1095                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1096                                                 } else {
1097                                                         distrseq2.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1098                                                 }
1099                                         }
1100                                         tmp_accu += (new add(distrseq2, oc))->setflag(status_flags::dynallocated);
1101                                 }
1102                                 last_expanded = tmp_accu;
1103                         } else {
1104                                 if (!last_expanded.is_equal(_ex1))
1105                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1106                                 last_expanded = cit->rest;
1107                         }
1108
1109                 } else {
1110                         non_adds.push_back(*cit);
1111                 }
1112         }
1113
1114         // Now the only remaining thing to do is to multiply the factors which
1115         // were not sums into the "last_expanded" sum
1116         if (is_exactly_a<add>(last_expanded)) {
1117                 size_t n = last_expanded.nops();
1118                 exvector distrseq;
1119                 distrseq.reserve(n);
1120                 exvector va;
1121                 if (! skip_idx_rename) {
1122                         va = get_all_dummy_indices_safely(mul(non_adds));
1123                         sort(va.begin(), va.end(), ex_is_less());
1124                 }
1125
1126                 for (size_t i=0; i<n; ++i) {
1127                         epvector factors = non_adds;
1128                         if (skip_idx_rename)
1129                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1130                         else
1131                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1132                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1133                         if (can_be_further_expanded(term)) {
1134                                 distrseq.push_back(term.expand());
1135                         } else {
1136                                 if (options == 0)
1137                                         ex_to<basic>(term).setflag(status_flags::expanded);
1138                                 distrseq.push_back(term);
1139                         }
1140                 }
1141
1142                 return ((new add(distrseq))->
1143                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1144         }
1145
1146         non_adds.push_back(split_ex_to_pair(last_expanded));
1147         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1148         if (can_be_further_expanded(result)) {
1149                 return result.expand();
1150         } else {
1151                 if (options == 0)
1152                         ex_to<basic>(result).setflag(status_flags::expanded);
1153                 return result;
1154         }
1155 }
1156
1157
1158 //////////
1159 // new virtual functions which can be overridden by derived classes
1160 //////////
1161
1162 // none
1163
1164 //////////
1165 // non-virtual functions in this class
1166 //////////
1167
1168
1169 /** Member-wise expand the expairs representing this sequence.  This must be
1170  *  overridden from expairseq::expandchildren() and done iteratively in order
1171  *  to allow for early cancallations and thus safe memory.
1172  *
1173  *  @see mul::expand()
1174  *  @return pointer to epvector containing expanded representation or zero
1175  *  pointer, if sequence is unchanged. */
1176 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1177 {
1178         const epvector::const_iterator last = seq.end();
1179         epvector::const_iterator cit = seq.begin();
1180         while (cit!=last) {
1181                 const ex & factor = recombine_pair_to_ex(*cit);
1182                 const ex & expanded_factor = factor.expand(options);
1183                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1184
1185                         // something changed, copy seq, eval and return it
1186                         std::auto_ptr<epvector> s(new epvector);
1187                         s->reserve(seq.size());
1188
1189                         // copy parts of seq which are known not to have changed
1190                         epvector::const_iterator cit2 = seq.begin();
1191                         while (cit2!=cit) {
1192                                 s->push_back(*cit2);
1193                                 ++cit2;
1194                         }
1195
1196                         // copy first changed element
1197                         s->push_back(split_ex_to_pair(expanded_factor));
1198                         ++cit2;
1199
1200                         // copy rest
1201                         while (cit2!=last) {
1202                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1203                                 ++cit2;
1204                         }
1205                         return s;
1206                 }
1207                 ++cit;
1208         }
1209
1210         return std::auto_ptr<epvector>(0); // nothing has changed
1211 }
1212
1213 } // namespace GiNaC