ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <iostream>
  24 #include <vector>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "mul.h"
  29 #include "add.h"
  30 #include "power.h"
  31 #include "operators.h"
  32 #include "matrix.h"
  33 #include "indexed.h"
  34 #include "lst.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37 #include "symbol.h"
  38 #include "compiler.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
  43   print_func<print_context>(&mul::do_print).
  44   print_func<print_latex>(&mul::do_print_latex).
  45   print_func<print_csrc>(&mul::do_print_csrc).
  46   print_func<print_tree>(&mul::do_print_tree).
  47   print_func<print_python_repr>(&mul::do_print_python_repr))
  48
  49
  50 //////////
  51 // default constructor
  52 //////////
  53
  54 mul::mul()
  55 {
  56         tinfo_key = &mul::tinfo_static;
  57 }
  58
  59 //////////
  60 // other constructors
  61 //////////
  62
  63 // public
  64
  65 mul::mul(const ex & lh, const ex & rh)
  66 {
  67         tinfo_key = &mul::tinfo_static;
  68         overall_coeff = _ex1;
  69         construct_from_2_ex(lh,rh);
  70         GINAC_ASSERT(is_canonical());
  71 }
  72
  73 mul::mul(const exvector & v)
  74 {
  75         tinfo_key = &mul::tinfo_static;
  76         overall_coeff = _ex1;
  77         construct_from_exvector(v);
  78         GINAC_ASSERT(is_canonical());
  79 }
  80
  81 mul::mul(const epvector & v)
  82 {
  83         tinfo_key = &mul::tinfo_static;
  84         overall_coeff = _ex1;
  85         construct_from_epvector(v);
  86         GINAC_ASSERT(is_canonical());
  87 }
  88
  89 mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
  90 {
  91         tinfo_key = &mul::tinfo_static;
  92         overall_coeff = oc;
  93         construct_from_epvector(v, do_index_renaming);
  94         GINAC_ASSERT(is_canonical());
  95 }
  96
  97 mul::mul(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming)
  98 {
  99         tinfo_key = &mul::tinfo_static;
 100         GINAC_ASSERT(vp.get()!=0);
 101         overall_coeff = oc;
 102         construct_from_epvector(*vp, do_index_renaming);
 103         GINAC_ASSERT(is_canonical());
 104 }
 105
 106 mul::mul(const ex & lh, const ex & mh, const ex & rh)
 107 {
 108         tinfo_key = &mul::tinfo_static;
 109         exvector factors;
 110         factors.reserve(3);
 111         factors.push_back(lh);
 112         factors.push_back(mh);
 113         factors.push_back(rh);
 114         overall_coeff = _ex1;
 115         construct_from_exvector(factors);
 116         GINAC_ASSERT(is_canonical());
 117 }
 118
 119 //////////
 120 // archiving
 121 //////////
 122
 123 DEFAULT_ARCHIVING(mul)
 124
 125 //////////
 126 // functions overriding virtual functions from base classes
 127 //////////
 128
 129 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
 130 {
 131         const numeric &coeff = ex_to<numeric>(overall_coeff);
 132         if (coeff.csgn() == -1)
 133                 c.s << '-';
 134         if (!coeff.is_equal(*_num1_p) &&
 135                 !coeff.is_equal(*_num_1_p)) {
 136                 if (coeff.is_rational()) {
 137                         if (coeff.is_negative())
 138                                 (-coeff).print(c);
 139                         else
 140                                 coeff.print(c);
 141                 } else {
 142                         if (coeff.csgn() == -1)
 143                                 (-coeff).print(c, precedence());
 144                         else
 145                                 coeff.print(c, precedence());
 146                 }
 147                 c.s << mul_sym;
 148         }
 149 }
 150
 151 void mul::do_print(const print_context & c, unsigned level) const
 152 {
 153         if (precedence() <= level)
 154                 c.s << '(';
 155
 156         print_overall_coeff(c, "*");
 157
 158         epvector::const_iterator it = seq.begin(), itend = seq.end();
 159         bool first = true;
 160         while (it != itend) {
 161                 if (!first)
 162                         c.s << '*';
 163                 else
 164                         first = false;
 165                 recombine_pair_to_ex(*it).print(c, precedence());
 166                 ++it;
 167         }
 168
 169         if (precedence() <= level)
 170                 c.s << ')';
 171 }
 172
 173 void mul::do_print_latex(const print_latex & c, unsigned level) const
 174 {
 175         if (precedence() <= level)
 176                 c.s << "{(";
 177
 178         print_overall_coeff(c, " ");
 179
 180         // Separate factors into those with negative numeric exponent
 181         // and all others
 182         epvector::const_iterator it = seq.begin(), itend = seq.end();
 183         exvector neg_powers, others;
 184         while (it != itend) {
 185                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
 186                 if (ex_to<numeric>(it->coeff).is_negative())
 187                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
 188                 else
 189                         others.push_back(recombine_pair_to_ex(*it));
 190                 ++it;
 191         }
 192
 193         if (!neg_powers.empty()) {
 194
 195                 // Factors with negative exponent are printed as a fraction
 196                 c.s << "\\frac{";
 197                 mul(others).eval().print(c);
 198                 c.s << "}{";
 199                 mul(neg_powers).eval().print(c);
 200                 c.s << "}";
 201
 202         } else {
 203
 204                 // All other factors are printed in the ordinary way
 205                 exvector::const_iterator vit = others.begin(), vitend = others.end();
 206                 while (vit != vitend) {
 207                         c.s << ' ';
 208                         vit->print(c, precedence());
 209                         ++vit;
 210                 }
 211         }
 212
 213         if (precedence() <= level)
 214                 c.s << ")}";
 215 }
 216
 217 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
 218 {
 219         if (precedence() <= level)
 220                 c.s << "(";
 221
 222         if (!overall_coeff.is_equal(_ex1)) {
 223                 if (overall_coeff.is_equal(_ex_1))
 224                         c.s << "-";
 225                 else {
 226                         overall_coeff.print(c, precedence());
 227                         c.s << "*";
 228                 }
 229         }
 230
 231         // Print arguments, separated by "*" or "/"
 232         epvector::const_iterator it = seq.begin(), itend = seq.end();
 233         while (it != itend) {
 234
 235                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
 236                 bool needclosingparenthesis = false;
 237                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
 238                         if (is_a<print_csrc_cl_N>(c)) {
 239                                 c.s << "recip(";
 240                                 needclosingparenthesis = true;
 241                         } else
 242                                 c.s << "1.0/";
 243                 }
 244
 245                 // If the exponent is 1 or -1, it is left out
 246                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
 247                         it->rest.print(c, precedence());
 248                 else if (it->coeff.info(info_flags::negint))
 249                         // Outer parens around ex needed for broken GCC parser:
 250                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
 251                 else
 252                         // Outer parens around ex needed for broken GCC parser:
 253                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
 254
 255                 if (needclosingparenthesis)
 256                         c.s << ")";
 257
 258                 // Separator is "/" for negative integer powers, "*" otherwise
 259                 ++it;
 260                 if (it != itend) {
 261                         if (it->coeff.info(info_flags::negint))
 262                                 c.s << "/";
 263                         else
 264                                 c.s << "*";
 265                 }
 266         }
 267
 268         if (precedence() <= level)
 269                 c.s << ")";
 270 }
 271
 272 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
 273 {
 274         c.s << class_name() << '(';
 275         op(0).print(c);
 276         for (size_t i=1; i<nops(); ++i) {
 277                 c.s << ',';
 278                 op(i).print(c);
 279         }
 280         c.s << ')';
 281 }
 282
 283 bool mul::info(unsigned inf) const
 284 {
 285         switch (inf) {
 286                 case info_flags::polynomial:
 287                 case info_flags::integer_polynomial:
 288                 case info_flags::cinteger_polynomial:
 289                 case info_flags::rational_polynomial:
 290                 case info_flags::crational_polynomial:
 291                 case info_flags::rational_function: {
 292                         epvector::const_iterator i = seq.begin(), end = seq.end();
 293                         while (i != end) {
 294                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 295                                         return false;
 296                                 ++i;
 297                         }
 298                         return overall_coeff.info(inf);
 299                 }
 300                 case info_flags::algebraic: {
 301                         epvector::const_iterator i = seq.begin(), end = seq.end();
 302                         while (i != end) {
 303                                 if ((recombine_pair_to_ex(*i).info(inf)))
 304                                         return true;
 305                                 ++i;
 306                         }
 307                         return false;
 308                 }
 309         }
 310         return inherited::info(inf);
 311 }
 312
 313 int mul::degree(const ex & s) const
 314 {
 315         // Sum up degrees of factors
 316         int deg_sum = 0;
 317         epvector::const_iterator i = seq.begin(), end = seq.end();
 318         while (i != end) {
 319                 if (ex_to<numeric>(i->coeff).is_integer())
 320                         deg_sum += recombine_pair_to_ex(*i).degree(s);
 321                 else {
 322                         if (i->rest.has(s))
 323                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
 324                 }
 325                 ++i;
 326         }
 327         return deg_sum;
 328 }
 329
 330 int mul::ldegree(const ex & s) const
 331 {
 332         // Sum up degrees of factors
 333         int deg_sum = 0;
 334         epvector::const_iterator i = seq.begin(), end = seq.end();
 335         while (i != end) {
 336                 if (ex_to<numeric>(i->coeff).is_integer())
 337                         deg_sum += recombine_pair_to_ex(*i).ldegree(s);
 338                 else {
 339                         if (i->rest.has(s))
 340                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
 341                 }
 342                 ++i;
 343         }
 344         return deg_sum;
 345 }
 346
 347 ex mul::coeff(const ex & s, int n) const
 348 {
 349         exvector coeffseq;
 350         coeffseq.reserve(seq.size()+1);
 351
 352         if (n==0) {
 353                 // product of individual coeffs
 354                 // if a non-zero power of s is found, the resulting product will be 0
 355                 epvector::const_iterator i = seq.begin(), end = seq.end();
 356                 while (i != end) {
 357                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 358                         ++i;
 359                 }
 360                 coeffseq.push_back(overall_coeff);
 361                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 362         }
 363
 364         epvector::const_iterator i = seq.begin(), end = seq.end();
 365         bool coeff_found = false;
 366         while (i != end) {
 367                 ex t = recombine_pair_to_ex(*i);
 368                 ex c = t.coeff(s, n);
 369                 if (!c.is_zero()) {
 370                         coeffseq.push_back(c);
 371                         coeff_found = 1;
 372                 } else {
 373                         coeffseq.push_back(t);
 374                 }
 375                 ++i;
 376         }
 377         if (coeff_found) {
 378                 coeffseq.push_back(overall_coeff);
 379                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 380         }
 381
 382         return _ex0;
 383 }
 384
 385 /** Perform automatic term rewriting rules in this class.  In the following
 386  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 387  *  stand for such expressions that contain a plain number.
 388  *  - *(...,x;0) -> 0
 389  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 390  *  - *(x;1) -> x
 391  *  - *(;c) -> c
 392  *
 393  *  @param level cut-off in recursive evaluation */
 394 ex mul::eval(int level) const
 395 {
 396         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 397         if (evaled_seqp.get()) {
 398                 // do more evaluation later
 399                 return (new mul(evaled_seqp, overall_coeff))->
 400                            setflag(status_flags::dynallocated);
 401         }
 402
 403 #ifdef DO_GINAC_ASSERT
 404         epvector::const_iterator i = seq.begin(), end = seq.end();
 405         while (i != end) {
 406                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
 407                              (!(ex_to<numeric>(i->coeff).is_integer())));
 408                 GINAC_ASSERT(!(i->is_canonical_numeric()));
 409                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
 410                     print(print_tree(std::cerr));
 411                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
 412                 /* for paranoia */
 413                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
 414                 GINAC_ASSERT(p.rest.is_equal(i->rest));
 415                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
 416                 /* end paranoia */
 417                 ++i;
 418         }
 419 #endif // def DO_GINAC_ASSERT
 420
 421         if (flags & status_flags::evaluated) {
 422                 GINAC_ASSERT(seq.size()>0);
 423                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 424                 return *this;
 425         }
 426
 427         size_t seq_size = seq.size();
 428         if (overall_coeff.is_zero()) {
 429                 // *(...,x;0) -> 0
 430                 return _ex0;
 431         } else if (seq_size==0) {
 432                 // *(;c) -> c
 433                 return overall_coeff;
 434         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 435                 // *(x;1) -> x
 436                 return recombine_pair_to_ex(*(seq.begin()));
 437         } else if ((seq_size==1) &&
 438                    is_exactly_a<add>((*seq.begin()).rest) &&
 439                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 440                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 441                 const add & addref = ex_to<add>((*seq.begin()).rest);
 442                 std::auto_ptr<epvector> distrseq(new epvector);
 443                 distrseq->reserve(addref.seq.size());
 444                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 445                 while (i != end) {
 446                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 447                         ++i;
 448                 }
 449                 return (new add(distrseq,
 450                                 ex_to<numeric>(addref.overall_coeff).
 451                                 mul_dyn(ex_to<numeric>(overall_coeff)))
 452                        )->setflag(status_flags::dynallocated | status_flags::evaluated);
 453         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
 454                 // Strip the content and the unit part from each term. Thus
 455                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)2
 456
 457                 epvector::const_iterator last = seq.end();
 458                 epvector::const_iterator i = seq.begin();
 459                 epvector::const_iterator j = seq.begin();
 460                 std::auto_ptr<epvector> s(new epvector);
 461                 numeric oc = *_num1_p;
 462                 bool something_changed = false;
 463                 while (i!=last) {
 464                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
 465                                 // power::eval has such a rule, no need to handle powers here
 466                                 ++i;
 467                                 continue;
 468                         }
 469
 470                         // XXX: What is the best way to check if the polynomial is a primitive?
 471                         numeric c = i->rest.integer_content();
 472                         const numeric& lead_coeff =
 473                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div_dyn(c);
 474                         const bool canonicalizable = lead_coeff.is_integer();
 475
 476                         // XXX: The main variable is chosen in a random way, so this code
 477                         // does NOT transform the term into the canonical form (thus, in some
 478                         // very unlucky event it can even loop forever). Hopefully the main
 479                         // variable will be the same for all terms in *this
 480                         const bool unit_normal = lead_coeff.is_pos_integer();
 481                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
 482                                 ++i;
 483                                 continue;
 484                         }
 485
 486                         if (! something_changed) {
 487                                 s->reserve(seq_size);
 488                                 something_changed = true;
 489                         }
 490
 491                         while ((j!=i) && (j!=last)) {
 492                                 s->push_back(*j);
 493                                 ++j;
 494                         }
 495
 496                         if (! unit_normal)
 497                                 c = c.mul(*_num_1_p);
 498
 499                         oc = oc.mul(c);
 500
 501                         // divide add by the number in place to save at least 2 .eval() calls
 502                         const add& addref = ex_to<add>(i->rest);
 503                         add* primitive = new add(addref);
 504                         primitive->setflag(status_flags::dynallocated);
 505                         primitive->clearflag(status_flags::hash_calculated);
 506                         primitive->overall_coeff = ex_to<numeric>(primitive->overall_coeff).div_dyn(c);
 507                         for (epvector::iterator ai = primitive->seq.begin();
 508                                         ai != primitive->seq.end(); ++ai)
 509                                 ai->coeff = ex_to<numeric>(ai->coeff).div_dyn(c);
 510
 511                         s->push_back(expair(*primitive, _ex1));
 512
 513                         ++i;
 514                         ++j;
 515                 }
 516                 if (something_changed) {
 517                         while (j!=last) {
 518                                 s->push_back(*j);
 519                                 ++j;
 520                         }
 521                         return (new mul(s, ex_to<numeric>(overall_coeff).mul_dyn(oc))
 522                                )->setflag(status_flags::dynallocated);
 523                 }
 524         }
 525
 526         return this->hold();
 527 }
 528
 529 ex mul::evalf(int level) const
 530 {
 531         if (level==1)
 532                 return mul(seq,overall_coeff);
 533
 534         if (level==-max_recursion_level)
 535                 throw(std::runtime_error("max recursion level reached"));
 536
 537         std::auto_ptr<epvector> s(new epvector);
 538         s->reserve(seq.size());
 539
 540         --level;
 541         epvector::const_iterator i = seq.begin(), end = seq.end();
 542         while (i != end) {
 543                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 544                                                            i->coeff));
 545                 ++i;
 546         }
 547         return mul(s, overall_coeff.evalf(level));
 548 }
 549
 550 void mul::find_real_imag(ex & rp, ex & ip) const
 551 {
 552         rp = overall_coeff.real_part();
 553         ip = overall_coeff.imag_part();
 554         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 555                 ex factor = recombine_pair_to_ex(*i);
 556                 ex new_rp = factor.real_part();
 557                 ex new_ip = factor.imag_part();
 558                 if(new_ip.is_zero()) {
 559                         rp *= new_rp;
 560                         ip *= new_rp;
 561                 } else {
 562                         ex temp = rp*new_rp - ip*new_ip;
 563                         ip = ip*new_rp + rp*new_ip;
 564                         rp = temp;
 565                 }
 566         }
 567         rp = rp.expand();
 568         ip = ip.expand();
 569 }
 570
 571 ex mul::real_part() const
 572 {
 573         ex rp, ip;
 574         find_real_imag(rp, ip);
 575         return rp;
 576 }
 577
 578 ex mul::imag_part() const
 579 {
 580         ex rp, ip;
 581         find_real_imag(rp, ip);
 582         return ip;
 583 }
 584
 585 ex mul::evalm() const
 586 {
 587         // numeric*matrix
 588         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 589          && is_a<matrix>(seq[0].rest))
 590                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 591
 592         // Evaluate children first, look whether there are any matrices at all
 593         // (there can be either no matrices or one matrix; if there were more
 594         // than one matrix, it would be a non-commutative product)
 595         std::auto_ptr<epvector> s(new epvector);
 596         s->reserve(seq.size());
 597
 598         bool have_matrix = false;
 599         epvector::iterator the_matrix;
 600
 601         epvector::const_iterator i = seq.begin(), end = seq.end();
 602         while (i != end) {
 603                 const ex &m = recombine_pair_to_ex(*i).evalm();
 604                 s->push_back(split_ex_to_pair(m));
 605                 if (is_a<matrix>(m)) {
 606                         have_matrix = true;
 607                         the_matrix = s->end() - 1;
 608                 }
 609                 ++i;
 610         }
 611
 612         if (have_matrix) {
 613
 614                 // The product contained a matrix. We will multiply all other factors
 615                 // into that matrix.
 616                 matrix m = ex_to<matrix>(the_matrix->rest);
 617                 s->erase(the_matrix);
 618                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 619                 return m.mul_scalar(scalar);
 620
 621         } else
 622                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 623 }
 624
 625 ex mul::eval_ncmul(const exvector & v) const
 626 {
 627         if (seq.empty())
 628                 return inherited::eval_ncmul(v);
 629
 630         // Find first noncommutative element and call its eval_ncmul()
 631         epvector::const_iterator i = seq.begin(), end = seq.end();
 632         while (i != end) {
 633                 if (i->rest.return_type() == return_types::noncommutative)
 634                         return i->rest.eval_ncmul(v);
 635                 ++i;
 636         }
 637         return inherited::eval_ncmul(v);
 638 }
 639
 640 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
 641 {
 642         ex origbase;
 643         int origexponent;
 644         int origexpsign;
 645
 646         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 647                 origbase = origfactor.op(0);
 648                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 649                 origexponent = expon > 0 ? expon : -expon;
 650                 origexpsign = expon > 0 ? 1 : -1;
 651         } else {
 652                 origbase = origfactor;
 653                 origexponent = 1;
 654                 origexpsign = 1;
 655         }
 656
 657         ex patternbase;
 658         int patternexponent;
 659         int patternexpsign;
 660
 661         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 662                 patternbase = patternfactor.op(0);
 663                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 664                 patternexponent = expon > 0 ? expon : -expon;
 665                 patternexpsign = expon > 0 ? 1 : -1;
 666         } else {
 667                 patternbase = patternfactor;
 668                 patternexponent = 1;
 669                 patternexpsign = 1;
 670         }
 671
 672         lst saverepls = repls;
 673         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 674                 return false;
 675         repls = saverepls;
 676
 677         int newnummatches = origexponent / patternexponent;
 678         if (newnummatches < nummatches)
 679                 nummatches = newnummatches;
 680         return true;
 681 }
 682
 683 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
 684   * list of replacements repls. This matching is in the sense of algebraic
 685   * substitutions. Matching starts with pat.op(factor) of the pattern because
 686   * the factors before this one have already been matched. The (possibly
 687   * updated) number of matches is in nummatches. subsed[i] is true for factors
 688   * that already have been replaced by previous substitutions and matched[i]
 689   * is true for factors that have been matched by the current match.
 690   */
 691 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
 692                 int factor, int &nummatches, const std::vector<bool> &subsed,
 693                 std::vector<bool> &matched)
 694 {
 695         if (factor == pat.nops())
 696                 return true;
 697
 698         for (size_t i=0; i<e.nops(); ++i) {
 699                 if(subsed[i] || matched[i])
 700                         continue;
 701                 lst newrepls = repls;
 702                 int newnummatches = nummatches;
 703                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 704                         matched[i] = true;
 705                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 706                                         newnummatches, subsed, matched)) {
 707                                 repls = newrepls;
 708                                 nummatches = newnummatches;
 709                                 return true;
 710                         }
 711                         else
 712                                 matched[i] = false;
 713                 }
 714         }
 715
 716         return false;
 717 }
 718
 719 bool mul::has(const ex & pattern, unsigned options) const
 720 {
 721         if(!(options&has_options::algebraic))
 722                 return basic::has(pattern,options);
 723         if(is_a<mul>(pattern)) {
 724                 lst repls;
 725                 int nummatches = std::numeric_limits<int>::max();
 726                 std::vector<bool> subsed(seq.size(), false);
 727                 std::vector<bool> matched(seq.size(), false);
 728                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 729                                 subsed, matched))
 730                         return true;
 731         }
 732         return basic::has(pattern, options);
 733 }
 734
 735 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 736 {
 737         std::vector<bool> subsed(seq.size(), false);
 738         exvector subsresult(seq.size());
 739         ex divide_by = 1;
 740         ex multiply_by = 1;
 741
 742         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 743
 744                 if (is_exactly_a<mul>(it->first)) {
 745 retry1:
 746                         int nummatches = std::numeric_limits<int>::max();
 747                         std::vector<bool> currsubsed(seq.size(), false);
 748                         lst repls;
 749
 750                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 751                                 continue;
 752
 753                         for (size_t j=0; j<subsed.size(); j++)
 754                                 if (currsubsed[j])
 755                                         subsed[j] = true;
 756                         ex subsed_pattern
 757                                 = it->first.subs(ex(repls), subs_options::no_pattern);
 758                         divide_by *= power(subsed_pattern, nummatches);
 759                         ex subsed_result
 760                                 = it->second.subs(ex(repls), subs_options::no_pattern);
 761                         multiply_by *= power(subsed_result, nummatches);
 762                         goto retry1;
 763
 764                 } else {
 765
 766                         for (size_t j=0; j<this->nops(); j++) {
 767                                 int nummatches = std::numeric_limits<int>::max();
 768                                 lst repls;
 769                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
 770                                         subsed[j] = true;
 771                                         ex subsed_pattern
 772                                                 = it->first.subs(ex(repls), subs_options::no_pattern);
 773                                         divide_by *= power(subsed_pattern, nummatches);
 774                                         ex subsed_result
 775                                                 = it->second.subs(ex(repls), subs_options::no_pattern);
 776                                         multiply_by *= power(subsed_result, nummatches);
 777                                 }
 778                         }
 779                 }
 780         }
 781
 782         bool subsfound = false;
 783         for (size_t i=0; i<subsed.size(); i++) {
 784                 if (subsed[i]) {
 785                         subsfound = true;
 786                         break;
 787                 }
 788         }
 789         if (!subsfound)
 790                 return subs_one_level(m, options | subs_options::algebraic);
 791
 792         return ((*this)/divide_by)*multiply_by;
 793 }
 794
 795 // protected
 796
 797 /** Implementation of ex::diff() for a product.  It applies the product rule.
 798  *  @see ex::diff */
 799 ex mul::derivative(const symbol & s) const
 800 {
 801         size_t num = seq.size();
 802         exvector addseq;
 803         addseq.reserve(num);
 804
 805         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 806         epvector mulseq = seq;
 807         epvector::const_iterator i = seq.begin(), end = seq.end();
 808         epvector::iterator i2 = mulseq.begin();
 809         while (i != end) {
 810                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 811                                              i->rest.diff(s));
 812                 ep.swap(*i2);
 813                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 814                 ep.swap(*i2);
 815                 ++i; ++i2;
 816         }
 817         return (new add(addseq))->setflag(status_flags::dynallocated);
 818 }
 819
 820 int mul::compare_same_type(const basic & other) const
 821 {
 822         return inherited::compare_same_type(other);
 823 }
 824
 825 unsigned mul::return_type() const
 826 {
 827         if (seq.empty()) {
 828                 // mul without factors: should not happen, but commutates
 829                 return return_types::commutative;
 830         }
 831
 832         bool all_commutative = true;
 833         epvector::const_iterator noncommutative_element; // point to first found nc element
 834
 835         epvector::const_iterator i = seq.begin(), end = seq.end();
 836         while (i != end) {
 837                 unsigned rt = i->rest.return_type();
 838                 if (rt == return_types::noncommutative_composite)
 839                         return rt; // one ncc -> mul also ncc
 840                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 841                         // first nc element found, remember position
 842                         noncommutative_element = i;
 843                         all_commutative = false;
 844                 }
 845                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 846                         // another nc element found, compare type_infos
 847                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 848                                         // different types -> mul is ncc
 849                                         return return_types::noncommutative_composite;
 850                         }
 851                 }
 852                 ++i;
 853         }
 854         // all factors checked
 855         return all_commutative ? return_types::commutative : return_types::noncommutative;
 856 }
 857
 858 tinfo_t mul::return_type_tinfo() const
 859 {
 860         if (seq.empty())
 861                 return this;  // mul without factors: should not happen
 862
 863         // return type_info of first noncommutative element
 864         epvector::const_iterator i = seq.begin(), end = seq.end();
 865         while (i != end) {
 866                 if (i->rest.return_type() == return_types::noncommutative)
 867                         return i->rest.return_type_tinfo();
 868                 ++i;
 869         }
 870         // no noncommutative element found, should not happen
 871         return this;
 872 }
 873
 874 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 875 {
 876         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 877 }
 878
 879 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
 880 {
 881         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 882 }
 883
 884 expair mul::split_ex_to_pair(const ex & e) const
 885 {
 886         if (is_exactly_a<power>(e)) {
 887                 const power & powerref = ex_to<power>(e);
 888                 if (is_exactly_a<numeric>(powerref.exponent))
 889                         return expair(powerref.basis,powerref.exponent);
 890         }
 891         return expair(e,_ex1);
 892 }
 893
 894 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
 895                                           const ex & c) const
 896 {
 897         // to avoid duplication of power simplification rules,
 898         // we create a temporary power object
 899         // otherwise it would be hard to correctly evaluate
 900         // expression like (4^(1/3))^(3/2)
 901         if (c.is_equal(_ex1))
 902                 return split_ex_to_pair(e);
 903
 904         return split_ex_to_pair(power(e,c));
 905 }
 906
 907 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
 908                                             const ex & c) const
 909 {
 910         // to avoid duplication of power simplification rules,
 911         // we create a temporary power object
 912         // otherwise it would be hard to correctly evaluate
 913         // expression like (4^(1/3))^(3/2)
 914         if (c.is_equal(_ex1))
 915                 return p;
 916
 917         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
 918 }
 919
 920 ex mul::recombine_pair_to_ex(const expair & p) const
 921 {
 922         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
 923                 return p.rest;
 924         else
 925                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
 926 }
 927
 928 bool mul::expair_needs_further_processing(epp it)
 929 {
 930         if (is_exactly_a<mul>(it->rest) &&
 931                 ex_to<numeric>(it->coeff).is_integer()) {
 932                 // combined pair is product with integer power -> expand it
 933                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
 934                 return true;
 935         }
 936         if (is_exactly_a<numeric>(it->rest)) {
 937                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
 938                 if (!ep.is_equal(*it)) {
 939                         // combined pair is a numeric power which can be simplified
 940                         *it = ep;
 941                         return true;
 942                 }
 943                 if (it->coeff.is_equal(_ex1)) {
 944                         // combined pair has coeff 1 and must be moved to the end
 945                         return true;
 946                 }
 947         }
 948         return false;
 949 }
 950
 951 ex mul::default_overall_coeff() const
 952 {
 953         return _ex1;
 954 }
 955
 956 void mul::combine_overall_coeff(const ex & c)
 957 {
 958         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 959         GINAC_ASSERT(is_exactly_a<numeric>(c));
 960         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
 961 }
 962
 963 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
 964 {
 965         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 966         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 967         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 968         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
 969 }
 970
 971 bool mul::can_make_flat(const expair & p) const
 972 {
 973         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 974         // this assertion will probably fail somewhere
 975         // it would require a more careful make_flat, obeying the power laws
 976         // probably should return true only if p.coeff is integer
 977         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
 978 }
 979
 980 bool mul::can_be_further_expanded(const ex & e)
 981 {
 982         if (is_exactly_a<mul>(e)) {
 983                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
 984                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
 985                                 return true;
 986                 }
 987         } else if (is_exactly_a<power>(e)) {
 988                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
 989                         return true;
 990         }
 991         return false;
 992 }
 993
 994 ex mul::expand(unsigned options) const
 995 {
 996         {
 997         // trivial case: expanding the monomial (~ 30% of all calls)
 998                 epvector::const_iterator i = seq.begin(), seq_end = seq.end();
 999                 while ((i != seq.end()) &&  is_a<symbol>(i->rest) && i->coeff.info(info_flags::integer))
1000                         ++i;
1001                 if (i == seq_end)
1002                         return (new mul(*this))->setflag(status_flags::dynallocated | status_flags::expanded);
1003         }
1004
1005         // do not rename indices if the object has no indices at all
1006         if ((!(options & expand_options::expand_rename_idx)) &&
1007                         this->info(info_flags::has_indices))
1008                 options |= expand_options::expand_rename_idx;
1009
1010         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1011
1012         // First, expand the children
1013         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
1014         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
1015
1016         // Now, look for all the factors that are sums and multiply each one out
1017         // with the next one that is found while collecting the factors which are
1018         // not sums
1019         ex last_expanded = _ex1;
1020
1021         epvector non_adds;
1022         non_adds.reserve(expanded_seq.size());
1023
1024         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
1025                 if (is_exactly_a<add>(cit->rest) &&
1026                         (cit->coeff.is_equal(_ex1))) {
1027                         if (is_exactly_a<add>(last_expanded)) {
1028
1029                                 // Expand a product of two sums, aggressive version.
1030                                 // Caring for the overall coefficients in separate loops can
1031                                 // sometimes give a performance gain of up to 15%!
1032
1033                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
1034                                 // add2 is for the inner loop and should be the bigger of the two sums
1035                                 // in the presence of asymptotically good sorting:
1036                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
1037                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
1038                                 const epvector::const_iterator add1begin = add1.seq.begin();
1039                                 const epvector::const_iterator add1end   = add1.seq.end();
1040                                 const epvector::const_iterator add2begin = add2.seq.begin();
1041                                 const epvector::const_iterator add2end   = add2.seq.end();
1042                                 epvector distrseq;
1043                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
1044
1045                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1046                                 if (!add1.overall_coeff.is_zero()) {
1047                                         if (add1.overall_coeff.is_equal(_ex1))
1048                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
1049                                         else
1050                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
1051                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1052                                 }
1053
1054                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1055                                 if (!add2.overall_coeff.is_zero()) {
1056                                         if (add2.overall_coeff.is_equal(_ex1))
1057                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
1058                                         else
1059                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
1060                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1061                                 }
1062
1063                                 // Compute the new overall coefficient and put it together:
1064                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
1065
1066                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1067                                 lst dummy_subs;
1068
1069                                 if (!skip_idx_rename) {
1070                                         for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
1071                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1072                                                 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1073                                         }
1074                                         for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
1075                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1076                                                 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1077                                         }
1078
1079                                         sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1080                                         sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1081                                         dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1082                                 }
1083
1084                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1085                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
1086                                         // We really have to combine terms here in order to compactify
1087                                         // the result.  Otherwise it would become waayy tooo bigg.
1088                                         numeric oc(*_num0_p);
1089                                         epvector distrseq2;
1090                                         distrseq2.reserve(add1.seq.size());
1091                                         const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1092                                                         i2->rest :
1093                                                         i2->rest.subs(ex_to<lst>(dummy_subs.op(0)),
1094                                                                 ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1095                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1096                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1097                                                 // since that would violate an invariant of expairseq:
1098                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
1099                                                 if (is_exactly_a<numeric>(rest)) {
1100                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1101                                                 } else {
1102                                                         distrseq2.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1103                                                 }
1104                                         }
1105                                         tmp_accu += (new add(distrseq2, oc))->setflag(status_flags::dynallocated);
1106                                 }
1107                                 last_expanded = tmp_accu;
1108                         } else {
1109                                 if (!last_expanded.is_equal(_ex1))
1110                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1111                                 last_expanded = cit->rest;
1112                         }
1113
1114                 } else {
1115                         non_adds.push_back(*cit);
1116                 }
1117         }
1118
1119         // Now the only remaining thing to do is to multiply the factors which
1120         // were not sums into the "last_expanded" sum
1121         if (is_exactly_a<add>(last_expanded)) {
1122                 size_t n = last_expanded.nops();
1123                 exvector distrseq;
1124                 distrseq.reserve(n);
1125                 exvector va;
1126                 if (! skip_idx_rename) {
1127                         va = get_all_dummy_indices_safely(mul(non_adds));
1128                         sort(va.begin(), va.end(), ex_is_less());
1129                 }
1130
1131                 for (size_t i=0; i<n; ++i) {
1132                         epvector factors = non_adds;
1133                         if (skip_idx_rename)
1134                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1135                         else
1136                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1137                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1138                         if (can_be_further_expanded(term)) {
1139                                 distrseq.push_back(term.expand());
1140                         } else {
1141                                 if (options == 0)
1142                                         ex_to<basic>(term).setflag(status_flags::expanded);
1143                                 distrseq.push_back(term);
1144                         }
1145                 }
1146
1147                 return ((new add(distrseq))->
1148                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1149         }
1150
1151         non_adds.push_back(split_ex_to_pair(last_expanded));
1152         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1153         if (can_be_further_expanded(result)) {
1154                 return result.expand();
1155         } else {
1156                 if (options == 0)
1157                         ex_to<basic>(result).setflag(status_flags::expanded);
1158                 return result;
1159         }
1160 }
1161
1162
1163 //////////
1164 // new virtual functions which can be overridden by derived classes
1165 //////////
1166
1167 // none
1168
1169 //////////
1170 // non-virtual functions in this class
1171 //////////
1172
1173
1174 /** Member-wise expand the expairs representing this sequence.  This must be
1175  *  overridden from expairseq::expandchildren() and done iteratively in order
1176  *  to allow for early cancallations and thus safe memory.
1177  *
1178  *  @see mul::expand()
1179  *  @return pointer to epvector containing expanded representation or zero
1180  *  pointer, if sequence is unchanged. */
1181 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1182 {
1183         const epvector::const_iterator last = seq.end();
1184         epvector::const_iterator cit = seq.begin();
1185         while (cit!=last) {
1186                 const ex & factor = recombine_pair_to_ex(*cit);
1187                 const ex & expanded_factor = factor.expand(options);
1188                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1189
1190                         // something changed, copy seq, eval and return it
1191                         std::auto_ptr<epvector> s(new epvector);
1192                         s->reserve(seq.size());
1193
1194                         // copy parts of seq which are known not to have changed
1195                         epvector::const_iterator cit2 = seq.begin();
1196                         while (cit2!=cit) {
1197                                 s->push_back(*cit2);
1198                                 ++cit2;
1199                         }
1200
1201                         // copy first changed element
1202                         s->push_back(split_ex_to_pair(expanded_factor));
1203                         ++cit2;
1204
1205                         // copy rest
1206                         while (cit2!=last) {
1207                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1208                                 ++cit2;
1209                         }
1210                         return s;
1211                 }
1212                 ++cit;
1213         }
1214
1215         return std::auto_ptr<epvector>(0); // nothing has changed
1216 }
1217
1218 } // namespace GiNaC