ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2009 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 "mul.h"
  24 #include "add.h"
  25 #include "power.h"
  26 #include "operators.h"
  27 #include "matrix.h"
  28 #include "indexed.h"
  29 #include "lst.h"
  30 #include "archive.h"
  31 #include "utils.h"
  32 #include "symbol.h"
  33 #include "compiler.h"
  34
  35 #include <iostream>
  36 #include <limits>
  37 #include <stdexcept>
  38 #include <vector>
  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 //////////
 117 // functions overriding virtual functions from base classes
 118 //////////
 119
 120 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
 121 {
 122         const numeric &coeff = ex_to<numeric>(overall_coeff);
 123         if (coeff.csgn() == -1)
 124                 c.s << '-';
 125         if (!coeff.is_equal(*_num1_p) &&
 126                 !coeff.is_equal(*_num_1_p)) {
 127                 if (coeff.is_rational()) {
 128                         if (coeff.is_negative())
 129                                 (-coeff).print(c);
 130                         else
 131                                 coeff.print(c);
 132                 } else {
 133                         if (coeff.csgn() == -1)
 134                                 (-coeff).print(c, precedence());
 135                         else
 136                                 coeff.print(c, precedence());
 137                 }
 138                 c.s << mul_sym;
 139         }
 140 }
 141
 142 void mul::do_print(const print_context & c, unsigned level) const
 143 {
 144         if (precedence() <= level)
 145                 c.s << '(';
 146
 147         print_overall_coeff(c, "*");
 148
 149         epvector::const_iterator it = seq.begin(), itend = seq.end();
 150         bool first = true;
 151         while (it != itend) {
 152                 if (!first)
 153                         c.s << '*';
 154                 else
 155                         first = false;
 156                 recombine_pair_to_ex(*it).print(c, precedence());
 157                 ++it;
 158         }
 159
 160         if (precedence() <= level)
 161                 c.s << ')';
 162 }
 163
 164 void mul::do_print_latex(const print_latex & c, unsigned level) const
 165 {
 166         if (precedence() <= level)
 167                 c.s << "{(";
 168
 169         print_overall_coeff(c, " ");
 170
 171         // Separate factors into those with negative numeric exponent
 172         // and all others
 173         epvector::const_iterator it = seq.begin(), itend = seq.end();
 174         exvector neg_powers, others;
 175         while (it != itend) {
 176                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
 177                 if (ex_to<numeric>(it->coeff).is_negative())
 178                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
 179                 else
 180                         others.push_back(recombine_pair_to_ex(*it));
 181                 ++it;
 182         }
 183
 184         if (!neg_powers.empty()) {
 185
 186                 // Factors with negative exponent are printed as a fraction
 187                 c.s << "\\frac{";
 188                 mul(others).eval().print(c);
 189                 c.s << "}{";
 190                 mul(neg_powers).eval().print(c);
 191                 c.s << "}";
 192
 193         } else {
 194
 195                 // All other factors are printed in the ordinary way
 196                 exvector::const_iterator vit = others.begin(), vitend = others.end();
 197                 while (vit != vitend) {
 198                         c.s << ' ';
 199                         vit->print(c, precedence());
 200                         ++vit;
 201                 }
 202         }
 203
 204         if (precedence() <= level)
 205                 c.s << ")}";
 206 }
 207
 208 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
 209 {
 210         if (precedence() <= level)
 211                 c.s << "(";
 212
 213         if (!overall_coeff.is_equal(_ex1)) {
 214                 if (overall_coeff.is_equal(_ex_1))
 215                         c.s << "-";
 216                 else {
 217                         overall_coeff.print(c, precedence());
 218                         c.s << "*";
 219                 }
 220         }
 221
 222         // Print arguments, separated by "*" or "/"
 223         epvector::const_iterator it = seq.begin(), itend = seq.end();
 224         while (it != itend) {
 225
 226                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 227                 bool needclosingparenthesis = false;
 228                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 229                         if (is_a<print_csrc_cl_N>(c)) {
 230                                 c.s << "recip(";
 231                                 needclosingparenthesis = true;
 232                         } else
 233                                 c.s << "1.0/";
 234                 }
 235
 236                 // If the exponent is 1 or -1, it is left out
 237                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 238                         it->rest.print(c, precedence());
 239                 else if (it->coeff.info(info_flags::negint))
 240                         // Outer parens around ex needed for broken GCC parser:
 241                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 242                 else
 243                         // Outer parens around ex needed for broken GCC parser:
 244                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 245
 246                 if (needclosingparenthesis)
 247                         c.s << ")";
 248
 249                 // Separator is "/" for negative integer powers, "*" otherwise
 250                 ++it;
 251                 if (it != itend) {
 252                         if (it->coeff.info(info_flags::negint))
 253                                 c.s << "/";
 254                         else
 255                                 c.s << "*";
 256                 }
 257         }
 258
 259         if (precedence() <= level)
 260                 c.s << ")";
 261 }
 262
 263 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 264 {
 265         c.s << class_name() << '(';
 266         op(0).print(c);
 267         for (size_t i=1; i<nops(); ++i) {
 268                 c.s << ',';
 269                 op(i).print(c);
 270         }
 271         c.s << ')';
 272 }
 273
 274 bool mul::info(unsigned inf) const
 275 {
 276         switch (inf) {
 277                 case info_flags::polynomial:
 278                 case info_flags::integer_polynomial:
 279                 case info_flags::cinteger_polynomial:
 280                 case info_flags::rational_polynomial:
 281                 case info_flags::real:
 282                 case info_flags::rational:
 283                 case info_flags::integer:
 284                 case info_flags::crational:
 285                 case info_flags::cinteger:
 286                 case info_flags::positive:
 287                 case info_flags::nonnegative:
 288                 case info_flags::posint:
 289                 case info_flags::nonnegint:
 290                 case info_flags::even:
 291                 case info_flags::crational_polynomial:
 292                 case info_flags::rational_function: {
 293                         epvector::const_iterator i = seq.begin(), end = seq.end();
 294                         while (i != end) {
 295                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 296                                         return false;
 297                                 ++i;
 298                         }
 299                         if (overall_coeff.is_equal(*_num1_p) && inf == info_flags::even)
 300                                 return true;
 301                         return overall_coeff.info(inf);
 302                 }
 303                 case info_flags::algebraic: {
 304                         epvector::const_iterator i = seq.begin(), end = seq.end();
 305                         while (i != end) {
 306                                 if ((recombine_pair_to_ex(*i).info(inf)))
 307                                         return true;
 308                                 ++i;
 309                         }
 310                         return false;
 311                 }
 312         }
 313         return inherited::info(inf);
 314 }
 315
 316 int mul::degree(const ex & s) const
 317 {
 318         // Sum up degrees of factors
 319         int deg_sum = 0;
 320         epvector::const_iterator i = seq.begin(), end = seq.end();
 321         while (i != end) {
 322                 if (ex_to<numeric>(i->coeff).is_integer())
 323                         deg_sum += recombine_pair_to_ex(*i).degree(s);
 324                 else {
 325                         if (i->rest.has(s))
 326                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
 327                 }
 328                 ++i;
 329         }
 330         return deg_sum;
 331 }
 332
 333 int mul::ldegree(const ex & s) const
 334 {
 335         // Sum up degrees of factors
 336         int deg_sum = 0;
 337         epvector::const_iterator i = seq.begin(), end = seq.end();
 338         while (i != end) {
 339                 if (ex_to<numeric>(i->coeff).is_integer())
 340                         deg_sum += recombine_pair_to_ex(*i).ldegree(s);
 341                 else {
 342                         if (i->rest.has(s))
 343                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
 344                 }
 345                 ++i;
 346         }
 347         return deg_sum;
 348 }
 349
 350 ex mul::coeff(const ex & s, int n) const
 351 {
 352         exvector coeffseq;
 353         coeffseq.reserve(seq.size()+1);
 354
 355         if (n==0) {
 356                 // product of individual coeffs
 357                 // if a non-zero power of s is found, the resulting product will be 0
 358                 epvector::const_iterator i = seq.begin(), end = seq.end();
 359                 while (i != end) {
 360                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 361                         ++i;
 362                 }
 363                 coeffseq.push_back(overall_coeff);
 364                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 365         }
 366
 367         epvector::const_iterator i = seq.begin(), end = seq.end();
 368         bool coeff_found = false;
 369         while (i != end) {
 370                 ex t = recombine_pair_to_ex(*i);
 371                 ex c = t.coeff(s, n);
 372                 if (!c.is_zero()) {
 373                         coeffseq.push_back(c);
 374                         coeff_found = 1;
 375                 } else {
 376                         coeffseq.push_back(t);
 377                 }
 378                 ++i;
 379         }
 380         if (coeff_found) {
 381                 coeffseq.push_back(overall_coeff);
 382                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 383         }
 384
 385         return _ex0;
 386 }
 387
 388 /** Perform automatic term rewriting rules in this class.  In the following
 389  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 390  *  stand for such expressions that contain a plain number.
 391  *  - *(...,x;0) -> 0
 392  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 393  *  - *(x;1) -> x
 394  *  - *(;c) -> c
 395  *
 396  *  @param level cut-off in recursive evaluation */
 397 ex mul::eval(int level) const
 398 {
 399         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 400         if (evaled_seqp.get()) {
 401                 // do more evaluation later
 402                 return (new mul(evaled_seqp, overall_coeff))->
 403                            setflag(status_flags::dynallocated);
 404         }
 405
 406 #ifdef DO_GINAC_ASSERT
 407         epvector::const_iterator i = seq.begin(), end = seq.end();
 408         while (i != end) {
 409                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 410                              (!(ex_to<numeric>(i->coeff).is_integer())));
 411                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 412                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 413                     print(print_tree(std::cerr));
 414                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 415                 /* for paranoia */
 416                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 417                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 418                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 419                 /* end paranoia */
 420                 ++i;
 421         }
 422 #endif // def DO_GINAC_ASSERT
 423
 424         if (flags & status_flags::evaluated) {
 425                 GINAC_ASSERT(seq.size()>0);
 426                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 427                 return *this;
 428         }
 429
 430         size_t seq_size = seq.size();
 431         if (overall_coeff.is_zero()) {
 432                 // *(...,x;0) -> 0
 433                 return _ex0;
 434         } else if (seq_size==0) {
 435                 // *(;c) -> c
 436                 return overall_coeff;
 437         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 438                 // *(x;1) -> x
 439                 return recombine_pair_to_ex(*(seq.begin()));
 440         } else if ((seq_size==1) &&
 441                    is_exactly_a<add>((*seq.begin()).rest) &&
 442                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 443                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 444                 const add & addref = ex_to<add>((*seq.begin()).rest);
 445                 std::auto_ptr<epvector> distrseq(new epvector);
 446                 distrseq->reserve(addref.seq.size());
 447                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 448                 while (i != end) {
 449                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 450                         ++i;
 451                 }
 452                 return (new add(distrseq,
 453                                 ex_to<numeric>(addref.overall_coeff).
 454                                 mul_dyn(ex_to<numeric>(overall_coeff)))
 455                        )->setflag(status_flags::dynallocated | status_flags::evaluated);
 456         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
 457                 // Strip the content and the unit part from each term. Thus
 458                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)2
 459
 460                 epvector::const_iterator last = seq.end();
 461                 epvector::const_iterator i = seq.begin();
 462                 epvector::const_iterator j = seq.begin();
 463                 std::auto_ptr<epvector> s(new epvector);
 464                 numeric oc = *_num1_p;
 465                 bool something_changed = false;
 466                 while (i!=last) {
 467                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
 468                                 // power::eval has such a rule, no need to handle powers here
 469                                 ++i;
 470                                 continue;
 471                         }
 472
 473                         // XXX: What is the best way to check if the polynomial is a primitive?
 474                         numeric c = i->rest.integer_content();
 475                         const numeric lead_coeff =
 476                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
 477                         const bool canonicalizable = lead_coeff.is_integer();
 478
 479                         // XXX: The main variable is chosen in a random way, so this code
 480                         // does NOT transform the term into the canonical form (thus, in some
 481                         // very unlucky event it can even loop forever). Hopefully the main
 482                         // variable will be the same for all terms in *this
 483                         const bool unit_normal = lead_coeff.is_pos_integer();
 484                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
 485                                 ++i;
 486                                 continue;
 487                         }
 488
 489                         if (! something_changed) {
 490                                 s->reserve(seq_size);
 491                                 something_changed = true;
 492                         }
 493
 494                         while ((j!=i) && (j!=last)) {
 495                                 s->push_back(*j);
 496                                 ++j;
 497                         }
 498
 499                         if (! unit_normal)
 500                                 c = c.mul(*_num_1_p);
 501
 502                         oc = oc.mul(c);
 503
 504                         // divide add by the number in place to save at least 2 .eval() calls
 505                         const add& addref = ex_to<add>(i->rest);
 506                         add* primitive = new add(addref);
 507                         primitive->setflag(status_flags::dynallocated);
 508                         primitive->clearflag(status_flags::hash_calculated);
 509                         primitive->overall_coeff = ex_to<numeric>(primitive->overall_coeff).div_dyn(c);
 510                         for (epvector::iterator ai = primitive->seq.begin();
 511                                         ai != primitive->seq.end(); ++ai)
 512                                 ai->coeff = ex_to<numeric>(ai->coeff).div_dyn(c);
 513
 514                         s->push_back(expair(*primitive, _ex1));
 515
 516                         ++i;
 517                         ++j;
 518                 }
 519                 if (something_changed) {
 520                         while (j!=last) {
 521                                 s->push_back(*j);
 522                                 ++j;
 523                         }
 524                         return (new mul(s, ex_to<numeric>(overall_coeff).mul_dyn(oc))
 525                                )->setflag(status_flags::dynallocated);
 526                 }
 527         }
 528
 529         return this->hold();
 530 }
 531
 532 ex mul::evalf(int level) const
 533 {
 534         if (level==1)
 535                 return mul(seq,overall_coeff);
 536
 537         if (level==-max_recursion_level)
 538                 throw(std::runtime_error("max recursion level reached"));
 539
 540         std::auto_ptr<epvector> s(new epvector);
 541         s->reserve(seq.size());
 542
 543         --level;
 544         epvector::const_iterator i = seq.begin(), end = seq.end();
 545         while (i != end) {
 546                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 547                                                            i->coeff));
 548                 ++i;
 549         }
 550         return mul(s, overall_coeff.evalf(level));
 551 }
 552
 553 void mul::find_real_imag(ex & rp, ex & ip) const
 554 {
 555         rp = overall_coeff.real_part();
 556         ip = overall_coeff.imag_part();
 557         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 558                 ex factor = recombine_pair_to_ex(*i);
 559                 ex new_rp = factor.real_part();
 560                 ex new_ip = factor.imag_part();
 561                 if(new_ip.is_zero()) {
 562                         rp *= new_rp;
 563                         ip *= new_rp;
 564                 } else {
 565                         ex temp = rp*new_rp - ip*new_ip;
 566                         ip = ip*new_rp + rp*new_ip;
 567                         rp = temp;
 568                 }
 569         }
 570         rp = rp.expand();
 571         ip = ip.expand();
 572 }
 573
 574 ex mul::real_part() const
 575 {
 576         ex rp, ip;
 577         find_real_imag(rp, ip);
 578         return rp;
 579 }
 580
 581 ex mul::imag_part() const
 582 {
 583         ex rp, ip;
 584         find_real_imag(rp, ip);
 585         return ip;
 586 }
 587
 588 ex mul::evalm() const
 589 {
 590         // numeric*matrix
 591         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 592          && is_a<matrix>(seq[0].rest))
 593                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 594
 595         // Evaluate children first, look whether there are any matrices at all
 596         // (there can be either no matrices or one matrix; if there were more
 597         // than one matrix, it would be a non-commutative product)
 598         std::auto_ptr<epvector> s(new epvector);
 599         s->reserve(seq.size());
 600
 601         bool have_matrix = false;
 602         epvector::iterator the_matrix;
 603
 604         epvector::const_iterator i = seq.begin(), end = seq.end();
 605         while (i != end) {
 606                 const ex &m = recombine_pair_to_ex(*i).evalm();
 607                 s->push_back(split_ex_to_pair(m));
 608                 if (is_a<matrix>(m)) {
 609                         have_matrix = true;
 610                         the_matrix = s->end() - 1;
 611                 }
 612                 ++i;
 613         }
 614
 615         if (have_matrix) {
 616
 617                 // The product contained a matrix. We will multiply all other factors
 618                 // into that matrix.
 619                 matrix m = ex_to<matrix>(the_matrix->rest);
 620                 s->erase(the_matrix);
 621                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 622                 return m.mul_scalar(scalar);
 623
 624         } else
 625                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 626 }
 627
 628 ex mul::eval_ncmul(const exvector & v) const
 629 {
 630         if (seq.empty())
 631                 return inherited::eval_ncmul(v);
 632
 633         // Find first noncommutative element and call its eval_ncmul()
 634         epvector::const_iterator i = seq.begin(), end = seq.end();
 635         while (i != end) {
 636                 if (i->rest.return_type() == return_types::noncommutative)
 637                         return i->rest.eval_ncmul(v);
 638                 ++i;
 639         }
 640         return inherited::eval_ncmul(v);
 641 }
 642
 643 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
 644 {
 645         ex origbase;
 646         int origexponent;
 647         int origexpsign;
 648
 649         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 650                 origbase = origfactor.op(0);
 651                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 652                 origexponent = expon > 0 ? expon : -expon;
 653                 origexpsign = expon > 0 ? 1 : -1;
 654         } else {
 655                 origbase = origfactor;
 656                 origexponent = 1;
 657                 origexpsign = 1;
 658         }
 659
 660         ex patternbase;
 661         int patternexponent;
 662         int patternexpsign;
 663
 664         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 665                 patternbase = patternfactor.op(0);
 666                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 667                 patternexponent = expon > 0 ? expon : -expon;
 668                 patternexpsign = expon > 0 ? 1 : -1;
 669         } else {
 670                 patternbase = patternfactor;
 671                 patternexponent = 1;
 672                 patternexpsign = 1;
 673         }
 674
 675         exmap saverepls = repls;
 676         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 677                 return false;
 678         repls = saverepls;
 679
 680         int newnummatches = origexponent / patternexponent;
 681         if (newnummatches < nummatches)
 682                 nummatches = newnummatches;
 683         return true;
 684 }
 685
 686 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
 687   * list of replacements repls. This matching is in the sense of algebraic
 688   * substitutions. Matching starts with pat.op(factor) of the pattern because
 689   * the factors before this one have already been matched. The (possibly
 690   * updated) number of matches is in nummatches. subsed[i] is true for factors
 691   * that already have been replaced by previous substitutions and matched[i]
 692   * is true for factors that have been matched by the current match.
 693   */
 694 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
 695                 int factor, int &nummatches, const std::vector<bool> &subsed,
 696                 std::vector<bool> &matched)
 697 {
 698         if (factor == (int)pat.nops())
 699                 return true;
 700
 701         for (size_t i=0; i<e.nops(); ++i) {
 702                 if(subsed[i] || matched[i])
 703                         continue;
 704                 exmap newrepls = repls;
 705                 int newnummatches = nummatches;
 706                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 707                         matched[i] = true;
 708                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 709                                         newnummatches, subsed, matched)) {
 710                                 repls = newrepls;
 711                                 nummatches = newnummatches;
 712                                 return true;
 713                         }
 714                         else
 715                                 matched[i] = false;
 716                 }
 717         }
 718
 719         return false;
 720 }
 721
 722 bool mul::has(const ex & pattern, unsigned options) const
 723 {
 724         if(!(options&has_options::algebraic))
 725                 return basic::has(pattern,options);
 726         if(is_a<mul>(pattern)) {
 727                 exmap repls;
 728                 int nummatches = std::numeric_limits<int>::max();
 729                 std::vector<bool> subsed(seq.size(), false);
 730                 std::vector<bool> matched(seq.size(), false);
 731                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 732                                 subsed, matched))
 733                         return true;
 734         }
 735         return basic::has(pattern, options);
 736 }
 737
 738 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 739 {
 740         std::vector<bool> subsed(seq.size(), false);
 741         exvector subsresult(seq.size());
 742         ex divide_by = 1;
 743         ex multiply_by = 1;
 744
 745         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 746
 747                 if (is_exactly_a<mul>(it->first)) {
 748 retry1:
 749                         int nummatches = std::numeric_limits<int>::max();
 750                         std::vector<bool> currsubsed(seq.size(), false);
 751                         exmap repls;
 752
 753                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 754                                 continue;
 755
 756                         for (size_t j=0; j<subsed.size(); j++)
 757                                 if (currsubsed[j])
 758                                         subsed[j] = true;
 759                         ex subsed_pattern
 760                                 = it->first.subs(repls, subs_options::no_pattern);
 761                         divide_by *= power(subsed_pattern, nummatches);
 762                         ex subsed_result
 763                                 = it->second.subs(repls, subs_options::no_pattern);
 764                         multiply_by *= power(subsed_result, nummatches);
 765                         goto retry1;
 766
 767                 } else {
 768
 769                         for (size_t j=0; j<this->nops(); j++) {
 770                                 int nummatches = std::numeric_limits<int>::max();
 771                                 exmap repls;
 772                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
 773                                         subsed[j] = true;
 774                                         ex subsed_pattern
 775                                                 = it->first.subs(repls, subs_options::no_pattern);
 776                                         divide_by *= power(subsed_pattern, nummatches);
 777                                         ex subsed_result
 778                                                 = it->second.subs(repls, subs_options::no_pattern);
 779                                         multiply_by *= power(subsed_result, nummatches);
 780                                 }
 781                         }
 782                 }
 783         }
 784
 785         bool subsfound = false;
 786         for (size_t i=0; i<subsed.size(); i++) {
 787                 if (subsed[i]) {
 788                         subsfound = true;
 789                         break;
 790                 }
 791         }
 792         if (!subsfound)
 793                 return subs_one_level(m, options | subs_options::algebraic);
 794
 795         return ((*this)/divide_by)*multiply_by;
 796 }
 797
 798 // protected
 799
 800 /** Implementation of ex::diff() for a product.  It applies the product rule.
 801  *  @see ex::diff */
 802 ex mul::derivative(const symbol & s) const
 803 {
 804         size_t num = seq.size();
 805         exvector addseq;
 806         addseq.reserve(num);
 807
 808         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 809         epvector mulseq = seq;
 810         epvector::const_iterator i = seq.begin(), end = seq.end();
 811         epvector::iterator i2 = mulseq.begin();
 812         while (i != end) {
 813                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 814                                              i->rest.diff(s));
 815                 ep.swap(*i2);
 816                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 817                 ep.swap(*i2);
 818                 ++i; ++i2;
 819         }
 820         return (new add(addseq))->setflag(status_flags::dynallocated);
 821 }
 822
 823 int mul::compare_same_type(const basic & other) const
 824 {
 825         return inherited::compare_same_type(other);
 826 }
 827
 828 unsigned mul::return_type() const
 829 {
 830         if (seq.empty()) {
 831                 // mul without factors: should not happen, but commutates
 832                 return return_types::commutative;
 833         }
 834
 835         bool all_commutative = true;
 836         epvector::const_iterator noncommutative_element; // point to first found nc element
 837
 838         epvector::const_iterator i = seq.begin(), end = seq.end();
 839         while (i != end) {
 840                 unsigned rt = i->rest.return_type();
 841                 if (rt == return_types::noncommutative_composite)
 842                         return rt; // one ncc -> mul also ncc
 843                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 844                         // first nc element found, remember position
 845                         noncommutative_element = i;
 846                         all_commutative = false;
 847                 }
 848                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 849                         // another nc element found, compare type_infos
 850                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 851                                         // different types -> mul is ncc
 852                                         return return_types::noncommutative_composite;
 853                         }
 854                 }
 855                 ++i;
 856         }
 857         // all factors checked
 858         return all_commutative ? return_types::commutative : return_types::noncommutative;
 859 }
 860
 861 return_type_t mul::return_type_tinfo() const
 862 {
 863         if (seq.empty())
 864                 return make_return_type_t<mul>(); // mul without factors: should not happen
 865
 866         // return type_info of first noncommutative element
 867         epvector::const_iterator i = seq.begin(), end = seq.end();
 868         while (i != end) {
 869                 if (i->rest.return_type() == return_types::noncommutative)
 870                         return i->rest.return_type_tinfo();
 871                 ++i;
 872         }
 873         // no noncommutative element found, should not happen
 874         return make_return_type_t<mul>();
 875 }
 876
 877 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 878 {
 879         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 880 }
 881
 882 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
 883 {
 884         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 885 }
 886
 887 expair mul::split_ex_to_pair(const ex & e) const
 888 {
 889         if (is_exactly_a<power>(e)) {
 890                 const power & powerref = ex_to<power>(e);
 891                 if (is_exactly_a<numeric>(powerref.exponent))
 892                         return expair(powerref.basis,powerref.exponent);
 893         }
 894         return expair(e,_ex1);
 895 }
 896
 897 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 898                                           const ex & c) const
 899 {
 900         // to avoid duplication of power simplification rules,
 901         // we create a temporary power object
 902         // otherwise it would be hard to correctly evaluate
 903         // expression like (4^(1/3))^(3/2)
 904         if (c.is_equal(_ex1))
 905                 return split_ex_to_pair(e);
 906
 907         return split_ex_to_pair(power(e,c));
 908 }
 909
 910 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 911                                             const ex & c) const
 912 {
 913         // to avoid duplication of power simplification rules,
 914         // we create a temporary power object
 915         // otherwise it would be hard to correctly evaluate
 916         // expression like (4^(1/3))^(3/2)
 917         if (c.is_equal(_ex1))
 918                 return p;
 919
 920         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 921 }
 922
 923 ex mul::recombine_pair_to_ex(const expair & p) const
 924 {
 925         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 926                 return p.rest;
 927         else
 928                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 929 }
 930
 931 bool mul::expair_needs_further_processing(epp it)
 932 {
 933         if (is_exactly_a<mul>(it->rest) &&
 934                 ex_to<numeric>(it->coeff).is_integer()) {
 935                 // combined pair is product with integer power -> expand it
 936                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 937                 return true;
 938         }
 939         if (is_exactly_a<numeric>(it->rest)) {
 940                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 941                 if (!ep.is_equal(*it)) {
 942                         // combined pair is a numeric power which can be simplified
 943                         *it = ep;
 944                         return true;
 945                 }
 946                 if (it->coeff.is_equal(_ex1)) {
 947                         // combined pair has coeff 1 and must be moved to the end
 948                         return true;
 949                 }
 950         }
 951         return false;
 952 }
 953
 954 ex mul::default_overall_coeff() const
 955 {
 956         return _ex1;
 957 }
 958
 959 void mul::combine_overall_coeff(const ex & c)
 960 {
 961         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 962         GINAC_ASSERT(is_exactly_a<numeric>(c));
 963         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 964 }
 965
 966 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 967 {
 968         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 969         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 970         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 971         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 972 }
 973
 974 bool mul::can_make_flat(const expair & p) const
 975 {
 976         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 977         // this assertion will probably fail somewhere
 978         // it would require a more careful make_flat, obeying the power laws
 979         // probably should return true only if p.coeff is integer
 980         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 981 }
 982
 983 bool mul::can_be_further_expanded(const ex & e)
 984 {
 985         if (is_exactly_a<mul>(e)) {
 986                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 987                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 988                                 return true;
 989                 }
 990         } else if (is_exactly_a<power>(e)) {
 991                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 992                         return true;
 993         }
 994         return false;
 995 }
 996
 997 ex mul::expand(unsigned options) const
 998 {
 999         {
1000         // trivial case: expanding the monomial (~ 30% of all calls)
1001                 epvector::const_iterator i = seq.begin(), seq_end = seq.end();
1002                 while ((i != seq.end()) &&  is_a<symbol>(i->rest) && i->coeff.info(info_flags::integer))
1003                         ++i;
1004                 if (i == seq_end) {
1005                         setflag(status_flags::expanded);
1006                         return *this;
1007                 }
1008         }
1009
1010         // do not rename indices if the object has no indices at all
1011         if ((!(options & expand_options::expand_rename_idx)) &&
1012                         this->info(info_flags::has_indices))
1013                 options |= expand_options::expand_rename_idx;
1014
1015         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1016
1017         // First, expand the children
1018         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
1019         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
1020
1021         // Now, look for all the factors that are sums and multiply each one out
1022         // with the next one that is found while collecting the factors which are
1023         // not sums
1024         ex last_expanded = _ex1;
1025
1026         epvector non_adds;
1027         non_adds.reserve(expanded_seq.size());
1028
1029         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
1030                 if (is_exactly_a<add>(cit->rest) &&
1031                         (cit->coeff.is_equal(_ex1))) {
1032                         if (is_exactly_a<add>(last_expanded)) {
1033
1034                                 // Expand a product of two sums, aggressive version.
1035                                 // Caring for the overall coefficients in separate loops can
1036                                 // sometimes give a performance gain of up to 15%!
1037
1038                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
1039                                 // add2 is for the inner loop and should be the bigger of the two sums
1040                                 // in the presence of asymptotically good sorting:
1041                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
1042                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
1043                                 const epvector::const_iterator add1begin = add1.seq.begin();
1044                                 const epvector::const_iterator add1end   = add1.seq.end();
1045                                 const epvector::const_iterator add2begin = add2.seq.begin();
1046                                 const epvector::const_iterator add2end   = add2.seq.end();
1047                                 epvector distrseq;
1048                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
1049
1050                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1051                                 if (!add1.overall_coeff.is_zero()) {
1052                                         if (add1.overall_coeff.is_equal(_ex1))
1053                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
1054                                         else
1055                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
1056                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1057                                 }
1058
1059                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1060                                 if (!add2.overall_coeff.is_zero()) {
1061                                         if (add2.overall_coeff.is_equal(_ex1))
1062                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
1063                                         else
1064                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
1065                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1066                                 }
1067
1068                                 // Compute the new overall coefficient and put it together:
1069                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
1070
1071                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1072                                 lst dummy_subs;
1073
1074                                 if (!skip_idx_rename) {
1075                                         for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
1076                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1077                                                 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1078                                         }
1079                                         for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
1080                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1081                                                 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1082                                         }
1083
1084                                         sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1085                                         sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1086                                         dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1087                                 }
1088
1089                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1090                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
1091                                         // We really have to combine terms here in order to compactify
1092                                         // the result.  Otherwise it would become waayy tooo bigg.
1093                                         numeric oc(*_num0_p);
1094                                         epvector distrseq2;
1095                                         distrseq2.reserve(add1.seq.size());
1096                                         const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1097                                                         i2->rest :
1098                                                         i2->rest.subs(ex_to<lst>(dummy_subs.op(0)),
1099                                                                 ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1100                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1101                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1102                                                 // since that would violate an invariant of expairseq:
1103                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
1104                                                 if (is_exactly_a<numeric>(rest)) {
1105                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1106                                                 } else {
1107                                                         distrseq2.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1108                                                 }
1109                                         }
1110                                         tmp_accu += (new add(distrseq2, oc))->setflag(status_flags::dynallocated);
1111                                 }
1112                                 last_expanded = tmp_accu;
1113                         } else {
1114                                 if (!last_expanded.is_equal(_ex1))
1115                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1116                                 last_expanded = cit->rest;
1117                         }
1118
1119                 } else {
1120                         non_adds.push_back(*cit);
1121                 }
1122         }
1123
1124         // Now the only remaining thing to do is to multiply the factors which
1125         // were not sums into the "last_expanded" sum
1126         if (is_exactly_a<add>(last_expanded)) {
1127                 size_t n = last_expanded.nops();
1128                 exvector distrseq;
1129                 distrseq.reserve(n);
1130                 exvector va;
1131                 if (! skip_idx_rename) {
1132                         va = get_all_dummy_indices_safely(mul(non_adds));
1133                         sort(va.begin(), va.end(), ex_is_less());
1134                 }
1135
1136                 for (size_t i=0; i<n; ++i) {
1137                         epvector factors = non_adds;
1138                         if (skip_idx_rename)
1139                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1140                         else
1141                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1142                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1143                         if (can_be_further_expanded(term)) {
1144                                 distrseq.push_back(term.expand());
1145                         } else {
1146                                 if (options == 0)
1147                                         ex_to<basic>(term).setflag(status_flags::expanded);
1148                                 distrseq.push_back(term);
1149                         }
1150                 }
1151
1152                 return ((new add(distrseq))->
1153                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1154         }
1155
1156         non_adds.push_back(split_ex_to_pair(last_expanded));
1157         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1158         if (can_be_further_expanded(result)) {
1159                 return result.expand();
1160         } else {
1161                 if (options == 0)
1162                         ex_to<basic>(result).setflag(status_flags::expanded);
1163                 return result;
1164         }
1165 }
1166
1167
1168 //////////
1169 // new virtual functions which can be overridden by derived classes
1170 //////////
1171
1172 // none
1173
1174 //////////
1175 // non-virtual functions in this class
1176 //////////
1177
1178
1179 /** Member-wise expand the expairs representing this sequence.  This must be
1180  *  overridden from expairseq::expandchildren() and done iteratively in order
1181  *  to allow for early cancallations and thus safe memory.
1182  *
1183  *  @see mul::expand()
1184  *  @return pointer to epvector containing expanded representation or zero
1185  *  pointer, if sequence is unchanged. */
1186 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1187 {
1188         const epvector::const_iterator last = seq.end();
1189         epvector::const_iterator cit = seq.begin();
1190         while (cit!=last) {
1191                 const ex & factor = recombine_pair_to_ex(*cit);
1192                 const ex & expanded_factor = factor.expand(options);
1193                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1194
1195                         // something changed, copy seq, eval and return it
1196                         std::auto_ptr<epvector> s(new epvector);
1197                         s->reserve(seq.size());
1198
1199                         // copy parts of seq which are known not to have changed
1200                         epvector::const_iterator cit2 = seq.begin();
1201                         while (cit2!=cit) {
1202                                 s->push_back(*cit2);
1203                                 ++cit2;
1204                         }
1205
1206                         // copy first changed element
1207                         s->push_back(split_ex_to_pair(expanded_factor));
1208                         ++cit2;
1209
1210                         // copy rest
1211                         while (cit2!=last) {
1212                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1213                                 ++cit2;
1214                         }
1215                         return s;
1216                 }
1217                 ++cit;
1218         }
1219
1220         return std::auto_ptr<epvector>(0); // nothing has changed
1221 }
1222
1223 GINAC_BIND_UNARCHIVER(mul);
1224
1225 } // namespace GiNaC