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