ginac/mul.cpp

   1 /** @file mul.cpp
   2  *
   3  *  Implementation of GiNaC's products of expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2011 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::even:
 287                 case info_flags::crational_polynomial:
 288                 case info_flags::rational_function: {
 289                         epvector::const_iterator i = seq.begin(), end = seq.end();
 290                         while (i != end) {
 291                                 if (!(recombine_pair_to_ex(*i).info(inf)))
 292                                         return false;
 293                                 ++i;
 294                         }
 295                         if (overall_coeff.is_equal(*_num1_p) && inf == info_flags::even)
 296                                 return true;
 297                         return overall_coeff.info(inf);
 298                 }
 299                 case info_flags::algebraic: {
 300                         epvector::const_iterator i = seq.begin(), end = seq.end();
 301                         while (i != end) {
 302                                 if ((recombine_pair_to_ex(*i).info(inf)))
 303                                         return true;
 304                                 ++i;
 305                         }
 306                         return false;
 307                 }
 308                 case info_flags::positive:
 309                 case info_flags::negative: {
 310                         if ((inf==info_flags::positive) && (flags & status_flags::is_positive))
 311                                 return true;
 312                         else if ((inf==info_flags::negative) && (flags & status_flags::is_negative))
 313                                 return true;
 314                         if (flags & status_flags::purely_indefinite)
 315                                 return false;
 316
 317                         bool pos = true;
 318                         epvector::const_iterator i = seq.begin(), end = seq.end();
 319                         while (i != end) {
 320                                 const ex& factor = recombine_pair_to_ex(*i++);
 321                                 if (factor.info(info_flags::positive))
 322                                         continue;
 323                                 else if (factor.info(info_flags::negative))
 324                                         pos = !pos;
 325                                 else
 326                                         return false;
 327                         }
 328                         if (overall_coeff.info(info_flags::negative))
 329                                 pos = !pos;
 330                         setflag(pos ? status_flags::is_positive : status_flags::is_negative);
 331                         return (inf == info_flags::positive? pos : !pos);
 332                 }
 333                 case info_flags::nonnegative: {
 334                         if  (flags & status_flags::is_positive)
 335                                 return true;
 336                         bool pos = true;
 337                         epvector::const_iterator i = seq.begin(), end = seq.end();
 338                         while (i != end) {
 339                                 const ex& factor = recombine_pair_to_ex(*i++);
 340                                 if (factor.info(info_flags::nonnegative) || factor.info(info_flags::positive))
 341                                         continue;
 342                                 else if (factor.info(info_flags::negative))
 343                                         pos = !pos;
 344                                 else
 345                                         return false;
 346                         }
 347                         return (overall_coeff.info(info_flags::negative)? pos : !pos);
 348                 }
 349                 case info_flags::posint:
 350                 case info_flags::negint: {
 351                         bool pos = true;
 352                         epvector::const_iterator i = seq.begin(), end = seq.end();
 353                         while (i != end) {
 354                                 const ex& factor = recombine_pair_to_ex(*i++);
 355                                 if (factor.info(info_flags::posint))
 356                                         continue;
 357                                 else if (factor.info(info_flags::negint))
 358                                         pos = !pos;
 359                                 else
 360                                         return false;
 361                         }
 362                         if (overall_coeff.info(info_flags::negint))
 363                                 pos = !pos;
 364                         else if (!overall_coeff.info(info_flags::posint))
 365                                 return false;
 366                         return (inf ==info_flags::posint? pos : !pos);
 367                 }
 368                 case info_flags::nonnegint: {
 369                         bool pos = true;
 370                         epvector::const_iterator i = seq.begin(), end = seq.end();
 371                         while (i != end) {
 372                                 const ex& factor = recombine_pair_to_ex(*i++);
 373                                 if (factor.info(info_flags::nonnegint) || factor.info(info_flags::posint))
 374                                         continue;
 375                                 else if (factor.info(info_flags::negint))
 376                                         pos = !pos;
 377                                 else
 378                                         return false;
 379                         }
 380                         if (overall_coeff.info(info_flags::negint))
 381                                 pos = !pos;
 382                         else if (!overall_coeff.info(info_flags::posint))
 383                                 return false;
 384                         return pos;
 385                 }
 386                 case info_flags::indefinite: {
 387                         if (flags & status_flags::purely_indefinite)
 388                                 return true;
 389                         if (flags & (status_flags::is_positive | status_flags::is_negative))
 390                                 return false;
 391                         epvector::const_iterator i = seq.begin(), end = seq.end();
 392                         while (i != end) {
 393                                 const ex& term = recombine_pair_to_ex(*i);
 394                                 if (term.info(info_flags::positive) || term.info(info_flags::negative))
 395                                         return false;
 396                                 ++i;
 397                         }
 398                         setflag(status_flags::purely_indefinite);
 399                         return true;
 400                 }
 401         }
 402         return inherited::info(inf);
 403 }
 404
 405 bool mul::is_polynomial(const ex & var) const
 406 {
 407         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 408                 if (!i->rest.is_polynomial(var) ||
 409                     (i->rest.has(var) && !i->coeff.info(info_flags::nonnegint))) {
 410                         return false;
 411                 }
 412         }
 413         return true;
 414 }
 415
 416 int mul::degree(const ex & s) const
 417 {
 418         // Sum up degrees of factors
 419         int deg_sum = 0;
 420         epvector::const_iterator i = seq.begin(), end = seq.end();
 421         while (i != end) {
 422                 if (ex_to<numeric>(i->coeff).is_integer())
 423                         deg_sum += recombine_pair_to_ex(*i).degree(s);
 424                 else {
 425                         if (i->rest.has(s))
 426                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
 427                 }
 428                 ++i;
 429         }
 430         return deg_sum;
 431 }
 432
 433 int mul::ldegree(const ex & s) const
 434 {
 435         // Sum up degrees of factors
 436         int deg_sum = 0;
 437         epvector::const_iterator i = seq.begin(), end = seq.end();
 438         while (i != end) {
 439                 if (ex_to<numeric>(i->coeff).is_integer())
 440                         deg_sum += recombine_pair_to_ex(*i).ldegree(s);
 441                 else {
 442                         if (i->rest.has(s))
 443                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
 444                 }
 445                 ++i;
 446         }
 447         return deg_sum;
 448 }
 449
 450 ex mul::coeff(const ex & s, int n) const
 451 {
 452         exvector coeffseq;
 453         coeffseq.reserve(seq.size()+1);
 454
 455         if (n==0) {
 456                 // product of individual coeffs
 457                 // if a non-zero power of s is found, the resulting product will be 0
 458                 epvector::const_iterator i = seq.begin(), end = seq.end();
 459                 while (i != end) {
 460                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
 461                         ++i;
 462                 }
 463                 coeffseq.push_back(overall_coeff);
 464                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 465         }
 466
 467         epvector::const_iterator i = seq.begin(), end = seq.end();
 468         bool coeff_found = false;
 469         while (i != end) {
 470                 ex t = recombine_pair_to_ex(*i);
 471                 ex c = t.coeff(s, n);
 472                 if (!c.is_zero()) {
 473                         coeffseq.push_back(c);
 474                         coeff_found = 1;
 475                 } else {
 476                         coeffseq.push_back(t);
 477                 }
 478                 ++i;
 479         }
 480         if (coeff_found) {
 481                 coeffseq.push_back(overall_coeff);
 482                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
 483         }
 484
 485         return _ex0;
 486 }
 487
 488 /** Perform automatic term rewriting rules in this class.  In the following
 489  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 490  *  stand for such expressions that contain a plain number.
 491  *  - *(...,x;0) -> 0
 492  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
 493  *  - *(x;1) -> x
 494  *  - *(;c) -> c
 495  *
 496  *  @param level cut-off in recursive evaluation */
 497 ex mul::eval(int level) const
 498 {
 499         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
 500         if (evaled_seqp.get()) {
 501                 // do more evaluation later
 502                 return (new mul(evaled_seqp, overall_coeff))->
 503                            setflag(status_flags::dynallocated);
 504         }
 505
 506         if (flags & status_flags::evaluated) {
 507                 GINAC_ASSERT(seq.size()>0);
 508                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
 509                 return *this;
 510         }
 511
 512         size_t seq_size = seq.size();
 513         if (overall_coeff.is_zero()) {
 514                 // *(...,x;0) -> 0
 515                 return _ex0;
 516         } else if (seq_size==0) {
 517                 // *(;c) -> c
 518                 return overall_coeff;
 519         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
 520                 // *(x;1) -> x
 521                 return recombine_pair_to_ex(*(seq.begin()));
 522         } else if ((seq_size==1) &&
 523                    is_exactly_a<add>((*seq.begin()).rest) &&
 524                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
 525                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
 526                 const add & addref = ex_to<add>((*seq.begin()).rest);
 527                 std::auto_ptr<epvector> distrseq(new epvector);
 528                 distrseq->reserve(addref.seq.size());
 529                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
 530                 while (i != end) {
 531                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
 532                         ++i;
 533                 }
 534                 return (new add(distrseq,
 535                                 ex_to<numeric>(addref.overall_coeff).
 536                                 mul_dyn(ex_to<numeric>(overall_coeff)))
 537                        )->setflag(status_flags::dynallocated | status_flags::evaluated);
 538         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
 539                 // Strip the content and the unit part from each term. Thus
 540                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)^2
 541
 542                 epvector::const_iterator last = seq.end();
 543                 epvector::const_iterator i = seq.begin();
 544                 epvector::const_iterator j = seq.begin();
 545                 std::auto_ptr<epvector> s(new epvector);
 546                 numeric oc = *_num1_p;
 547                 bool something_changed = false;
 548                 while (i!=last) {
 549                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
 550                                 // power::eval has such a rule, no need to handle powers here
 551                                 ++i;
 552                                 continue;
 553                         }
 554
 555                         // XXX: What is the best way to check if the polynomial is a primitive?
 556                         numeric c = i->rest.integer_content();
 557                         const numeric lead_coeff =
 558                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
 559                         const bool canonicalizable = lead_coeff.is_integer();
 560
 561                         // XXX: The main variable is chosen in a random way, so this code
 562                         // does NOT transform the term into the canonical form (thus, in some
 563                         // very unlucky event it can even loop forever). Hopefully the main
 564                         // variable will be the same for all terms in *this
 565                         const bool unit_normal = lead_coeff.is_pos_integer();
 566                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
 567                                 ++i;
 568                                 continue;
 569                         }
 570
 571                         if (! something_changed) {
 572                                 s->reserve(seq_size);
 573                                 something_changed = true;
 574                         }
 575
 576                         while ((j!=i) && (j!=last)) {
 577                                 s->push_back(*j);
 578                                 ++j;
 579                         }
 580
 581                         if (! unit_normal)
 582                                 c = c.mul(*_num_1_p);
 583
 584                         oc = oc.mul(c);
 585
 586                         // divide add by the number in place to save at least 2 .eval() calls
 587                         const add& addref = ex_to<add>(i->rest);
 588                         add* primitive = new add(addref);
 589                         primitive->setflag(status_flags::dynallocated);
 590                         primitive->clearflag(status_flags::hash_calculated);
 591                         primitive->overall_coeff = ex_to<numeric>(primitive->overall_coeff).div_dyn(c);
 592                         for (epvector::iterator ai = primitive->seq.begin(); ai != primitive->seq.end(); ++ai)
 593                                 ai->coeff = ex_to<numeric>(ai->coeff).div_dyn(c);
 594
 595                         s->push_back(expair(*primitive, _ex1));
 596
 597                         ++i;
 598                         ++j;
 599                 }
 600                 if (something_changed) {
 601                         while (j!=last) {
 602                                 s->push_back(*j);
 603                                 ++j;
 604                         }
 605                         return (new mul(s, ex_to<numeric>(overall_coeff).mul_dyn(oc))
 606                                )->setflag(status_flags::dynallocated);
 607                 }
 608         }
 609
 610         return this->hold();
 611 }
 612
 613 ex mul::evalf(int level) const
 614 {
 615         if (level==1)
 616                 return mul(seq,overall_coeff);
 617
 618         if (level==-max_recursion_level)
 619                 throw(std::runtime_error("max recursion level reached"));
 620
 621         std::auto_ptr<epvector> s(new epvector);
 622         s->reserve(seq.size());
 623
 624         --level;
 625         epvector::const_iterator i = seq.begin(), end = seq.end();
 626         while (i != end) {
 627                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
 628                                                            i->coeff));
 629                 ++i;
 630         }
 631         return mul(s, overall_coeff.evalf(level));
 632 }
 633
 634 void mul::find_real_imag(ex & rp, ex & ip) const
 635 {
 636         rp = overall_coeff.real_part();
 637         ip = overall_coeff.imag_part();
 638         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 639                 ex factor = recombine_pair_to_ex(*i);
 640                 ex new_rp = factor.real_part();
 641                 ex new_ip = factor.imag_part();
 642                 if(new_ip.is_zero()) {
 643                         rp *= new_rp;
 644                         ip *= new_rp;
 645                 } else {
 646                         ex temp = rp*new_rp - ip*new_ip;
 647                         ip = ip*new_rp + rp*new_ip;
 648                         rp = temp;
 649                 }
 650         }
 651         rp = rp.expand();
 652         ip = ip.expand();
 653 }
 654
 655 ex mul::real_part() const
 656 {
 657         ex rp, ip;
 658         find_real_imag(rp, ip);
 659         return rp;
 660 }
 661
 662 ex mul::imag_part() const
 663 {
 664         ex rp, ip;
 665         find_real_imag(rp, ip);
 666         return ip;
 667 }
 668
 669 ex mul::evalm() const
 670 {
 671         // numeric*matrix
 672         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
 673          && is_a<matrix>(seq[0].rest))
 674                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
 675
 676         // Evaluate children first, look whether there are any matrices at all
 677         // (there can be either no matrices or one matrix; if there were more
 678         // than one matrix, it would be a non-commutative product)
 679         std::auto_ptr<epvector> s(new epvector);
 680         s->reserve(seq.size());
 681
 682         bool have_matrix = false;
 683         epvector::iterator the_matrix;
 684
 685         epvector::const_iterator i = seq.begin(), end = seq.end();
 686         while (i != end) {
 687                 const ex &m = recombine_pair_to_ex(*i).evalm();
 688                 s->push_back(split_ex_to_pair(m));
 689                 if (is_a<matrix>(m)) {
 690                         have_matrix = true;
 691                         the_matrix = s->end() - 1;
 692                 }
 693                 ++i;
 694         }
 695
 696         if (have_matrix) {
 697
 698                 // The product contained a matrix. We will multiply all other factors
 699                 // into that matrix.
 700                 matrix m = ex_to<matrix>(the_matrix->rest);
 701                 s->erase(the_matrix);
 702                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 703                 return m.mul_scalar(scalar);
 704
 705         } else
 706                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
 707 }
 708
 709 ex mul::eval_ncmul(const exvector & v) const
 710 {
 711         if (seq.empty())
 712                 return inherited::eval_ncmul(v);
 713
 714         // Find first noncommutative element and call its eval_ncmul()
 715         epvector::const_iterator i = seq.begin(), end = seq.end();
 716         while (i != end) {
 717                 if (i->rest.return_type() == return_types::noncommutative)
 718                         return i->rest.eval_ncmul(v);
 719                 ++i;
 720         }
 721         return inherited::eval_ncmul(v);
 722 }
 723
 724 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
 725 {
 726         ex origbase;
 727         int origexponent;
 728         int origexpsign;
 729
 730         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
 731                 origbase = origfactor.op(0);
 732                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
 733                 origexponent = expon > 0 ? expon : -expon;
 734                 origexpsign = expon > 0 ? 1 : -1;
 735         } else {
 736                 origbase = origfactor;
 737                 origexponent = 1;
 738                 origexpsign = 1;
 739         }
 740
 741         ex patternbase;
 742         int patternexponent;
 743         int patternexpsign;
 744
 745         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
 746                 patternbase = patternfactor.op(0);
 747                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
 748                 patternexponent = expon > 0 ? expon : -expon;
 749                 patternexpsign = expon > 0 ? 1 : -1;
 750         } else {
 751                 patternbase = patternfactor;
 752                 patternexponent = 1;
 753                 patternexpsign = 1;
 754         }
 755
 756         exmap saverepls = repls;
 757         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
 758                 return false;
 759         repls = saverepls;
 760
 761         int newnummatches = origexponent / patternexponent;
 762         if (newnummatches < nummatches)
 763                 nummatches = newnummatches;
 764         return true;
 765 }
 766
 767 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
 768   * list of replacements repls. This matching is in the sense of algebraic
 769   * substitutions. Matching starts with pat.op(factor) of the pattern because
 770   * the factors before this one have already been matched. The (possibly
 771   * updated) number of matches is in nummatches. subsed[i] is true for factors
 772   * that already have been replaced by previous substitutions and matched[i]
 773   * is true for factors that have been matched by the current match.
 774   */
 775 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
 776                                   int factor, int &nummatches, const std::vector<bool> &subsed,
 777                                   std::vector<bool> &matched)
 778 {
 779         GINAC_ASSERT(subsed.size() == e.nops());
 780         GINAC_ASSERT(matched.size() == e.nops());
 781
 782         if (factor == (int)pat.nops())
 783                 return true;
 784
 785         for (size_t i=0; i<e.nops(); ++i) {
 786                 if(subsed[i] || matched[i])
 787                         continue;
 788                 exmap newrepls = repls;
 789                 int newnummatches = nummatches;
 790                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
 791                         matched[i] = true;
 792                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
 793                                         newnummatches, subsed, matched)) {
 794                                 repls = newrepls;
 795                                 nummatches = newnummatches;
 796                                 return true;
 797                         }
 798                         else
 799                                 matched[i] = false;
 800                 }
 801         }
 802
 803         return false;
 804 }
 805
 806 bool mul::has(const ex & pattern, unsigned options) const
 807 {
 808         if(!(options&has_options::algebraic))
 809                 return basic::has(pattern,options);
 810         if(is_a<mul>(pattern)) {
 811                 exmap repls;
 812                 int nummatches = std::numeric_limits<int>::max();
 813                 std::vector<bool> subsed(nops(), false);
 814                 std::vector<bool> matched(nops(), false);
 815                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
 816                                 subsed, matched))
 817                         return true;
 818         }
 819         return basic::has(pattern, options);
 820 }
 821
 822 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
 823 {
 824         std::vector<bool> subsed(nops(), false);
 825         ex divide_by = 1;
 826         ex multiply_by = 1;
 827
 828         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 829
 830                 if (is_exactly_a<mul>(it->first)) {
 831 retry1:
 832                         int nummatches = std::numeric_limits<int>::max();
 833                         std::vector<bool> currsubsed(nops(), false);
 834                         exmap repls;
 835
 836                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
 837                                 continue;
 838
 839                         for (size_t j=0; j<subsed.size(); j++)
 840                                 if (currsubsed[j])
 841                                         subsed[j] = true;
 842                         ex subsed_pattern
 843                                 = it->first.subs(repls, subs_options::no_pattern);
 844                         divide_by *= power(subsed_pattern, nummatches);
 845                         ex subsed_result
 846                                 = it->second.subs(repls, subs_options::no_pattern);
 847                         multiply_by *= power(subsed_result, nummatches);
 848                         goto retry1;
 849
 850                 } else {
 851
 852                         for (size_t j=0; j<this->nops(); j++) {
 853                                 int nummatches = std::numeric_limits<int>::max();
 854                                 exmap repls;
 855                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
 856                                         subsed[j] = true;
 857                                         ex subsed_pattern
 858                                                 = it->first.subs(repls, subs_options::no_pattern);
 859                                         divide_by *= power(subsed_pattern, nummatches);
 860                                         ex subsed_result
 861                                                 = it->second.subs(repls, subs_options::no_pattern);
 862                                         multiply_by *= power(subsed_result, nummatches);
 863                                 }
 864                         }
 865                 }
 866         }
 867
 868         bool subsfound = false;
 869         for (size_t i=0; i<subsed.size(); i++) {
 870                 if (subsed[i]) {
 871                         subsfound = true;
 872                         break;
 873                 }
 874         }
 875         if (!subsfound)
 876                 return subs_one_level(m, options | subs_options::algebraic);
 877
 878         return ((*this)/divide_by)*multiply_by;
 879 }
 880
 881 ex mul::conjugate() const
 882 {
 883         // The base class' method is wrong here because we have to be careful at
 884         // branch cuts. power::conjugate takes care of that already, so use it.
 885         epvector *newepv = 0;
 886         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 887                 if (newepv) {
 888                         newepv->push_back(split_ex_to_pair(recombine_pair_to_ex(*i).conjugate()));
 889                         continue;
 890                 }
 891                 ex x = recombine_pair_to_ex(*i);
 892                 ex c = x.conjugate();
 893                 if (c.is_equal(x)) {
 894                         continue;
 895                 }
 896                 newepv = new epvector;
 897                 newepv->reserve(seq.size());
 898                 for (epvector::const_iterator j=seq.begin(); j!=i; ++j) {
 899                         newepv->push_back(*j);
 900                 }
 901                 newepv->push_back(split_ex_to_pair(c));
 902         }
 903         ex x = overall_coeff.conjugate();
 904         if (!newepv && are_ex_trivially_equal(x, overall_coeff)) {
 905                 return *this;
 906         }
 907         ex result = thisexpairseq(newepv ? *newepv : seq, x);
 908         delete newepv;
 909         return result;
 910 }
 911
 912
 913 // protected
 914
 915 /** Implementation of ex::diff() for a product.  It applies the product rule.
 916  *  @see ex::diff */
 917 ex mul::derivative(const symbol & s) const
 918 {
 919         size_t num = seq.size();
 920         exvector addseq;
 921         addseq.reserve(num);
 922
 923         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
 924         epvector mulseq = seq;
 925         epvector::const_iterator i = seq.begin(), end = seq.end();
 926         epvector::iterator i2 = mulseq.begin();
 927         while (i != end) {
 928                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
 929                                              i->rest.diff(s));
 930                 ep.swap(*i2);
 931                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
 932                 ep.swap(*i2);
 933                 ++i; ++i2;
 934         }
 935         return (new add(addseq))->setflag(status_flags::dynallocated);
 936 }
 937
 938 int mul::compare_same_type(const basic & other) const
 939 {
 940         return inherited::compare_same_type(other);
 941 }
 942
 943 unsigned mul::return_type() const
 944 {
 945         if (seq.empty()) {
 946                 // mul without factors: should not happen, but commutates
 947                 return return_types::commutative;
 948         }
 949
 950         bool all_commutative = true;
 951         epvector::const_iterator noncommutative_element; // point to first found nc element
 952
 953         epvector::const_iterator i = seq.begin(), end = seq.end();
 954         while (i != end) {
 955                 unsigned rt = i->rest.return_type();
 956                 if (rt == return_types::noncommutative_composite)
 957                         return rt; // one ncc -> mul also ncc
 958                 if ((rt == return_types::noncommutative) && (all_commutative)) {
 959                         // first nc element found, remember position
 960                         noncommutative_element = i;
 961                         all_commutative = false;
 962                 }
 963                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
 964                         // another nc element found, compare type_infos
 965                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
 966                                         // different types -> mul is ncc
 967                                         return return_types::noncommutative_composite;
 968                         }
 969                 }
 970                 ++i;
 971         }
 972         // all factors checked
 973         return all_commutative ? return_types::commutative : return_types::noncommutative;
 974 }
 975
 976 return_type_t mul::return_type_tinfo() const
 977 {
 978         if (seq.empty())
 979                 return make_return_type_t<mul>(); // mul without factors: should not happen
 980
 981         // return type_info of first noncommutative element
 982         epvector::const_iterator i = seq.begin(), end = seq.end();
 983         while (i != end) {
 984                 if (i->rest.return_type() == return_types::noncommutative)
 985                         return i->rest.return_type_tinfo();
 986                 ++i;
 987         }
 988         // no noncommutative element found, should not happen
 989         return make_return_type_t<mul>();
 990 }
 991
 992 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
 993 {
 994         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
 995 }
 996
 997 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
 998 {
 999         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
1000 }
1001
1002 expair mul::split_ex_to_pair(const ex & e) const
1003 {
1004         if (is_exactly_a<power>(e)) {
1005                 const power & powerref = ex_to<power>(e);
1006                 if (is_exactly_a<numeric>(powerref.exponent))
1007                         return expair(powerref.basis,powerref.exponent);
1008         }
1009         return expair(e,_ex1);
1010 }
1011
1012 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
1013                                           const ex & c) const
1014 {
1015         // to avoid duplication of power simplification rules,
1016         // we create a temporary power object
1017         // otherwise it would be hard to correctly evaluate
1018         // expression like (4^(1/3))^(3/2)
1019         if (c.is_equal(_ex1))
1020                 return split_ex_to_pair(e);
1021
1022         return split_ex_to_pair(power(e,c));
1023 }
1024
1025 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
1026                                             const ex & c) const
1027 {
1028         // to avoid duplication of power simplification rules,
1029         // we create a temporary power object
1030         // otherwise it would be hard to correctly evaluate
1031         // expression like (4^(1/3))^(3/2)
1032         if (c.is_equal(_ex1))
1033                 return p;
1034
1035         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
1036 }
1037
1038 ex mul::recombine_pair_to_ex(const expair & p) const
1039 {
1040         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
1041                 return p.rest;
1042         else
1043                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
1044 }
1045
1046 bool mul::expair_needs_further_processing(epp it)
1047 {
1048         if (is_exactly_a<mul>(it->rest) &&
1049             ex_to<numeric>(it->coeff).is_integer()) {
1050                 // combined pair is product with integer power -> expand it
1051                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
1052                 return true;
1053         }
1054         if (is_exactly_a<numeric>(it->rest)) {
1055                 if (it->coeff.is_equal(_ex1)) {
1056                         // pair has coeff 1 and must be moved to the end
1057                         return true;
1058                 }
1059                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
1060                 if (!ep.is_equal(*it)) {
1061                         // combined pair is a numeric power which can be simplified
1062                         *it = ep;
1063                         return true;
1064                 }
1065         }
1066         return false;
1067 }
1068
1069 ex mul::default_overall_coeff() const
1070 {
1071         return _ex1;
1072 }
1073
1074 void mul::combine_overall_coeff(const ex & c)
1075 {
1076         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1077         GINAC_ASSERT(is_exactly_a<numeric>(c));
1078         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
1079 }
1080
1081 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
1082 {
1083         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1084         GINAC_ASSERT(is_exactly_a<numeric>(c1));
1085         GINAC_ASSERT(is_exactly_a<numeric>(c2));
1086         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
1087 }
1088
1089 bool mul::can_make_flat(const expair & p) const
1090 {
1091         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
1092         // this assertion will probably fail somewhere
1093         // it would require a more careful make_flat, obeying the power laws
1094         // probably should return true only if p.coeff is integer
1095         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
1096 }
1097
1098 bool mul::can_be_further_expanded(const ex & e)
1099 {
1100         if (is_exactly_a<mul>(e)) {
1101                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
1102                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
1103                                 return true;
1104                 }
1105         } else if (is_exactly_a<power>(e)) {
1106                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
1107                         return true;
1108         }
1109         return false;
1110 }
1111
1112 ex mul::expand(unsigned options) const
1113 {
1114         {
1115         // trivial case: expanding the monomial (~ 30% of all calls)
1116                 epvector::const_iterator i = seq.begin(), seq_end = seq.end();
1117                 while ((i != seq.end()) &&  is_a<symbol>(i->rest) && i->coeff.info(info_flags::integer))
1118                         ++i;
1119                 if (i == seq_end) {
1120                         setflag(status_flags::expanded);
1121                         return *this;
1122                 }
1123         }
1124
1125         // do not rename indices if the object has no indices at all
1126         if ((!(options & expand_options::expand_rename_idx)) &&
1127                         this->info(info_flags::has_indices))
1128                 options |= expand_options::expand_rename_idx;
1129
1130         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1131
1132         // First, expand the children
1133         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
1134         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
1135
1136         // Now, look for all the factors that are sums and multiply each one out
1137         // with the next one that is found while collecting the factors which are
1138         // not sums
1139         ex last_expanded = _ex1;
1140
1141         epvector non_adds;
1142         non_adds.reserve(expanded_seq.size());
1143
1144         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
1145                 if (is_exactly_a<add>(cit->rest) &&
1146                         (cit->coeff.is_equal(_ex1))) {
1147                         if (is_exactly_a<add>(last_expanded)) {
1148
1149                                 // Expand a product of two sums, aggressive version.
1150                                 // Caring for the overall coefficients in separate loops can
1151                                 // sometimes give a performance gain of up to 15%!
1152
1153                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
1154                                 // add2 is for the inner loop and should be the bigger of the two sums
1155                                 // in the presence of asymptotically good sorting:
1156                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
1157                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
1158                                 const epvector::const_iterator add1begin = add1.seq.begin();
1159                                 const epvector::const_iterator add1end   = add1.seq.end();
1160                                 const epvector::const_iterator add2begin = add2.seq.begin();
1161                                 const epvector::const_iterator add2end   = add2.seq.end();
1162                                 epvector distrseq;
1163                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
1164
1165                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1166                                 if (!add1.overall_coeff.is_zero()) {
1167                                         if (add1.overall_coeff.is_equal(_ex1))
1168                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
1169                                         else
1170                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
1171                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1172                                 }
1173
1174                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1175                                 if (!add2.overall_coeff.is_zero()) {
1176                                         if (add2.overall_coeff.is_equal(_ex1))
1177                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
1178                                         else
1179                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
1180                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1181                                 }
1182
1183                                 // Compute the new overall coefficient and put it together:
1184                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
1185
1186                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1187                                 lst dummy_subs;
1188
1189                                 if (!skip_idx_rename) {
1190                                         for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
1191                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1192                                                 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1193                                         }
1194                                         for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
1195                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1196                                                 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1197                                         }
1198
1199                                         sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1200                                         sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1201                                         dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1202                                 }
1203
1204                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1205                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
1206                                         // We really have to combine terms here in order to compactify
1207                                         // the result.  Otherwise it would become waayy tooo bigg.
1208                                         numeric oc(*_num0_p);
1209                                         epvector distrseq2;
1210                                         distrseq2.reserve(add1.seq.size());
1211                                         const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1212                                                         i2->rest :
1213                                                         i2->rest.subs(ex_to<lst>(dummy_subs.op(0)),
1214                                                                 ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1215                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1216                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1217                                                 // since that would violate an invariant of expairseq:
1218                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
1219                                                 if (is_exactly_a<numeric>(rest)) {
1220                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1221                                                 } else {
1222                                                         distrseq2.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1223                                                 }
1224                                         }
1225                                         tmp_accu += (new add(distrseq2, oc))->setflag(status_flags::dynallocated);
1226                                 }
1227                                 last_expanded = tmp_accu;
1228                         } else {
1229                                 if (!last_expanded.is_equal(_ex1))
1230                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1231                                 last_expanded = cit->rest;
1232                         }
1233
1234                 } else {
1235                         non_adds.push_back(*cit);
1236                 }
1237         }
1238
1239         // Now the only remaining thing to do is to multiply the factors which
1240         // were not sums into the "last_expanded" sum
1241         if (is_exactly_a<add>(last_expanded)) {
1242                 size_t n = last_expanded.nops();
1243                 exvector distrseq;
1244                 distrseq.reserve(n);
1245                 exvector va;
1246                 if (! skip_idx_rename) {
1247                         va = get_all_dummy_indices_safely(mul(non_adds));
1248                         sort(va.begin(), va.end(), ex_is_less());
1249                 }
1250
1251                 for (size_t i=0; i<n; ++i) {
1252                         epvector factors = non_adds;
1253                         if (skip_idx_rename)
1254                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1255                         else
1256                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1257                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1258                         if (can_be_further_expanded(term)) {
1259                                 distrseq.push_back(term.expand());
1260                         } else {
1261                                 if (options == 0)
1262                                         ex_to<basic>(term).setflag(status_flags::expanded);
1263                                 distrseq.push_back(term);
1264                         }
1265                 }
1266
1267                 return ((new add(distrseq))->
1268                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1269         }
1270
1271         non_adds.push_back(split_ex_to_pair(last_expanded));
1272         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1273         if (can_be_further_expanded(result)) {
1274                 return result.expand();
1275         } else {
1276                 if (options == 0)
1277                         ex_to<basic>(result).setflag(status_flags::expanded);
1278                 return result;
1279         }
1280 }
1281
1282
1283 //////////
1284 // new virtual functions which can be overridden by derived classes
1285 //////////
1286
1287 // none
1288
1289 //////////
1290 // non-virtual functions in this class
1291 //////////
1292
1293
1294 /** Member-wise expand the expairs representing this sequence.  This must be
1295  *  overridden from expairseq::expandchildren() and done iteratively in order
1296  *  to allow for early cancallations and thus safe memory.
1297  *
1298  *  @see mul::expand()
1299  *  @return pointer to epvector containing expanded representation or zero
1300  *  pointer, if sequence is unchanged. */
1301 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1302 {
1303         const epvector::const_iterator last = seq.end();
1304         epvector::const_iterator cit = seq.begin();
1305         while (cit!=last) {
1306                 const ex & factor = recombine_pair_to_ex(*cit);
1307                 const ex & expanded_factor = factor.expand(options);
1308                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1309
1310                         // something changed, copy seq, eval and return it
1311                         std::auto_ptr<epvector> s(new epvector);
1312                         s->reserve(seq.size());
1313
1314                         // copy parts of seq which are known not to have changed
1315                         epvector::const_iterator cit2 = seq.begin();
1316                         while (cit2!=cit) {
1317                                 s->push_back(*cit2);
1318                                 ++cit2;
1319                         }
1320
1321                         // copy first changed element
1322                         s->push_back(split_ex_to_pair(expanded_factor));
1323                         ++cit2;
1324
1325                         // copy rest
1326                         while (cit2!=last) {
1327                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1328                                 ++cit2;
1329                         }
1330                         return s;
1331                 }
1332                 ++cit;
1333         }
1334
1335         return std::auto_ptr<epvector>(0); // nothing has changed
1336 }
1337
1338 GINAC_BIND_UNARCHIVER(mul);
1339
1340 } // namespace GiNaC