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