ginac/expairseq.cpp

   1 /** @file expairseq.cpp
   2  *
   3  *  Implementation of sequences of expression pairs. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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 "expairseq.h"
  24 #include "lst.h"
  25 #include "add.h"
  26 #include "mul.h"
  27 #include "power.h"
  28 #include "relational.h"
  29 #include "wildcard.h"
  30 #include "archive.h"
  31 #include "operators.h"
  32 #include "utils.h"
  33 #include "hash_seed.h"
  34 #include "indexed.h"
  35 #include "compiler.h"
  36
  37 #include <algorithm>
  38 #include <iostream>
  39 #include <iterator>
  40 #include <memory>
  41 #include <stdexcept>
  42 #include <string>
  43
  44 namespace GiNaC {
  45
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(expairseq, basic,
  48   print_func<print_context>(&expairseq::do_print).
  49   print_func<print_tree>(&expairseq::do_print_tree))
  50
  51
  52 //////////
  53 // helper classes
  54 //////////
  55
  56 class epp_is_less
  57 {
  58 public:
  59         bool operator()(const epp &lh, const epp &rh) const
  60         {
  61                 return (*lh).is_less(*rh);
  62         }
  63 };
  64
  65 //////////
  66 // default constructor
  67 //////////
  68
  69 // public
  70
  71 expairseq::expairseq()
  72 {}
  73
  74 // protected
  75
  76 //////////
  77 // other constructors
  78 //////////
  79
  80 expairseq::expairseq(const ex &lh, const ex &rh)
  81 {
  82         construct_from_2_ex(lh,rh);
  83         GINAC_ASSERT(is_canonical());
  84 }
  85
  86 expairseq::expairseq(const exvector &v)
  87 {
  88         construct_from_exvector(v);
  89         GINAC_ASSERT(is_canonical());
  90 }
  91
  92 expairseq::expairseq(const epvector &v, const ex &oc, bool do_index_renaming)
  93   :  overall_coeff(oc)
  94 {
  95         GINAC_ASSERT(is_a<numeric>(oc));
  96         construct_from_epvector(v, do_index_renaming);
  97         GINAC_ASSERT(is_canonical());
  98 }
  99
 100 expairseq::expairseq(epvector && vp, const ex &oc, bool do_index_renaming)
 101   :  overall_coeff(oc)
 102 {
 103         GINAC_ASSERT(is_a<numeric>(oc));
 104         construct_from_epvector(std::move(vp), do_index_renaming);
 105         GINAC_ASSERT(is_canonical());
 106 }
 107
 108 //////////
 109 // archiving
 110 //////////
 111
 112 void expairseq::read_archive(const archive_node &n, lst &sym_lst)
 113 {
 114         inherited::read_archive(n, sym_lst);
 115         auto first = n.find_first("rest");
 116         auto last = n.find_last("coeff");
 117         ++last;
 118         seq.reserve((last-first)/2);
 119
 120         for (auto loc = first; loc < last;) {
 121                 ex rest;
 122                 ex coeff;
 123                 n.find_ex_by_loc(loc++, rest, sym_lst);
 124                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 125                 seq.push_back(expair(rest, coeff));
 126         }
 127
 128         n.find_ex("overall_coeff", overall_coeff, sym_lst);
 129
 130         canonicalize();
 131         GINAC_ASSERT(is_canonical());
 132 }
 133
 134 void expairseq::archive(archive_node &n) const
 135 {
 136         inherited::archive(n);
 137         for (auto & i : seq) {
 138                 n.add_ex("rest", i.rest);
 139                 n.add_ex("coeff", i.coeff);
 140         }
 141         n.add_ex("overall_coeff", overall_coeff);
 142 }
 143
 144
 145 //////////
 146 // functions overriding virtual functions from base classes
 147 //////////
 148
 149 // public
 150
 151 void expairseq::do_print(const print_context & c, unsigned level) const
 152 {
 153         c.s << "[[";
 154         printseq(c, ',', precedence(), level);
 155         c.s << "]]";
 156 }
 157
 158 void expairseq::do_print_tree(const print_tree & c, unsigned level) const
 159 {
 160         c.s << std::string(level, ' ') << class_name() << " @" << this
 161             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 162             << ", nops=" << nops()
 163             << std::endl;
 164         size_t num = seq.size();
 165         for (size_t i=0; i<num; ++i) {
 166                 seq[i].rest.print(c, level + c.delta_indent);
 167                 seq[i].coeff.print(c, level + c.delta_indent);
 168                 if (i != num - 1)
 169                         c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 170         }
 171         if (!overall_coeff.is_equal(default_overall_coeff())) {
 172                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl
 173                     << std::string(level + c.delta_indent, ' ') << "overall_coeff" << std::endl;
 174                 overall_coeff.print(c, level + c.delta_indent);
 175         }
 176         c.s << std::string(level + c.delta_indent,' ') << "=====" << std::endl;
 177 }
 178
 179 bool expairseq::info(unsigned inf) const
 180 {
 181         switch(inf) {
 182                 case info_flags::expanded:
 183                         return (flags & status_flags::expanded);
 184                 case info_flags::has_indices: {
 185                         if (flags & status_flags::has_indices)
 186                                 return true;
 187                         else if (flags & status_flags::has_no_indices)
 188                                 return false;
 189                         for (auto & i : seq) {
 190                                 if (i.rest.info(info_flags::has_indices)) {
 191                                         this->setflag(status_flags::has_indices);
 192                                         this->clearflag(status_flags::has_no_indices);
 193                                         return true;
 194                                 }
 195                         }
 196                         this->clearflag(status_flags::has_indices);
 197                         this->setflag(status_flags::has_no_indices);
 198                         return false;
 199                 }
 200         }
 201         return inherited::info(inf);
 202 }
 203
 204 size_t expairseq::nops() const
 205 {
 206         if (overall_coeff.is_equal(default_overall_coeff()))
 207                 return seq.size();
 208         else
 209                 return seq.size()+1;
 210 }
 211
 212 ex expairseq::op(size_t i) const
 213 {
 214         if (i < seq.size())
 215                 return recombine_pair_to_ex(seq[i]);
 216         GINAC_ASSERT(!overall_coeff.is_equal(default_overall_coeff()));
 217         return overall_coeff;
 218 }
 219
 220 ex expairseq::map(map_function &f) const
 221 {
 222         epvector v;
 223         v.reserve(seq.size()+1);
 224
 225         for (auto & it : seq)
 226                 v.push_back(split_ex_to_pair(f(recombine_pair_to_ex(it))));
 227
 228         if (overall_coeff.is_equal(default_overall_coeff()))
 229                 return thisexpairseq(std::move(v), default_overall_coeff(), true);
 230         else {
 231                 ex newcoeff = f(overall_coeff);
 232                 if(is_a<numeric>(newcoeff))
 233                         return thisexpairseq(std::move(v), newcoeff, true);
 234                 else {
 235                         v.push_back(split_ex_to_pair(newcoeff));
 236                         return thisexpairseq(std::move(v), default_overall_coeff(), true);
 237                 }
 238         }
 239 }
 240
 241 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 242 ex expairseq::eval() const
 243 {
 244         if (flags &status_flags::evaluated)
 245                 return *this;
 246
 247         const epvector evaled = evalchildren();
 248         if (!evaled.empty())
 249                 return dynallocate<expairseq>(std::move(evaled), overall_coeff).setflag(status_flags::evaluated);
 250         else
 251                 return this->hold();
 252 }
 253
 254 epvector* conjugateepvector(const epvector&epv)
 255 {
 256         epvector *newepv = nullptr;
 257         for (auto i=epv.begin(); i!=epv.end(); ++i) {
 258                 if (newepv) {
 259                         newepv->push_back(i->conjugate());
 260                         continue;
 261                 }
 262                 expair x = i->conjugate();
 263                 if (x.is_equal(*i)) {
 264                         continue;
 265                 }
 266                 newepv = new epvector;
 267                 newepv->reserve(epv.size());
 268                 for (epvector::const_iterator j=epv.begin(); j!=i; ++j) {
 269                         newepv->push_back(*j);
 270                 }
 271                 newepv->push_back(x);
 272         }
 273         return newepv;
 274 }
 275
 276 ex expairseq::conjugate() const
 277 {
 278         std::unique_ptr<epvector> newepv(conjugateepvector(seq));
 279         ex x = overall_coeff.conjugate();
 280         if (newepv) {
 281                 return thisexpairseq(std::move(*newepv), x);
 282         }
 283         if (are_ex_trivially_equal(x, overall_coeff)) {
 284                 return *this;
 285         }
 286         return thisexpairseq(seq, x);
 287 }
 288
 289 bool expairseq::match(const ex & pattern, exmap & repl_lst) const
 290 {
 291         // This differs from basic::match() because we want "a+b+c+d" to
 292         // match "d+*+b" with "*" being "a+c", and we want to honor commutativity
 293
 294         if (typeid(*this) == typeid(ex_to<basic>(pattern))) {
 295
 296                 // Check whether global wildcard (one that matches the "rest of the
 297                 // expression", like "*" above) is present
 298                 bool has_global_wildcard = false;
 299                 ex global_wildcard;
 300                 for (size_t i=0; i<pattern.nops(); i++) {
 301                         if (is_exactly_a<wildcard>(pattern.op(i))) {
 302                                 has_global_wildcard = true;
 303                                 global_wildcard = pattern.op(i);
 304                                 break;
 305                         }
 306                 }
 307
 308                 // Even if the expression does not match the pattern, some of
 309                 // its subexpressions could match it. For example, x^5*y^(-1)
 310                 // does not match the pattern $0^5, but its subexpression x^5
 311                 // does. So, save repl_lst in order to not add bogus entries.
 312                 exmap tmp_repl = repl_lst;
 313
 314                 // Unfortunately, this is an O(N^2) operation because we can't
 315                 // sort the pattern in a useful way...
 316
 317                 // Chop into terms
 318                 exvector ops;
 319                 ops.reserve(nops());
 320                 for (size_t i=0; i<nops(); i++)
 321                         ops.push_back(op(i));
 322
 323                 // Now, for every term of the pattern, look for a matching term in
 324                 // the expression and remove the match
 325                 for (size_t i=0; i<pattern.nops(); i++) {
 326                         ex p = pattern.op(i);
 327                         if (has_global_wildcard && p.is_equal(global_wildcard))
 328                                 continue;
 329                         auto it = ops.begin(), itend = ops.end();
 330                         while (it != itend) {
 331                                 if (it->match(p, tmp_repl)) {
 332                                         ops.erase(it);
 333                                         goto found;
 334                                 }
 335                                 ++it;
 336                         }
 337                         return false; // no match found
 338 found:          ;
 339                 }
 340
 341                 if (has_global_wildcard) {
 342
 343                         // Assign all the remaining terms to the global wildcard (unless
 344                         // it has already been matched before, in which case the matches
 345                         // must be equal)
 346                         size_t num = ops.size();
 347                         epvector vp;
 348                         vp.reserve(num);
 349                         for (size_t i=0; i<num; i++)
 350                                 vp.push_back(split_ex_to_pair(ops[i]));
 351                         ex rest = thisexpairseq(std::move(vp), default_overall_coeff());
 352                         for (auto & it : tmp_repl) {
 353                                 if (it.first.is_equal(global_wildcard)) {
 354                                         if (rest.is_equal(it.second)) {
 355                                                 repl_lst = tmp_repl;
 356                                                 return true;
 357                                         }
 358                                         return false;
 359                                 }
 360                         }
 361                         repl_lst = tmp_repl;
 362                         repl_lst[global_wildcard] = rest;
 363                         return true;
 364
 365                 } else {
 366
 367                         // No global wildcard, then the match fails if there are any
 368                         // unmatched terms left
 369                         if (ops.empty()) {
 370                                 repl_lst = tmp_repl;
 371                                 return true;
 372                         }
 373                         return false;
 374                 }
 375         }
 376         return inherited::match(pattern, repl_lst);
 377 }
 378
 379 ex expairseq::subs(const exmap & m, unsigned options) const
 380 {
 381         epvector subsed = subschildren(m, options);
 382         if (!subsed.empty())
 383                 return ex_to<basic>(thisexpairseq(std::move(subsed), overall_coeff, (options & subs_options::no_index_renaming) == 0));
 384         else if ((options & subs_options::algebraic) && is_exactly_a<mul>(*this))
 385                 return static_cast<const mul *>(this)->algebraic_subs_mul(m, options);
 386         else
 387                 return subs_one_level(m, options);
 388 }
 389
 390 // protected
 391
 392 int expairseq::compare_same_type(const basic &other) const
 393 {
 394         GINAC_ASSERT(is_a<expairseq>(other));
 395         const expairseq &o = static_cast<const expairseq &>(other);
 396
 397         int cmpval;
 398
 399         // compare number of elements
 400         if (seq.size() != o.seq.size())
 401                 return (seq.size()<o.seq.size()) ? -1 : 1;
 402
 403         // compare overall_coeff
 404         cmpval = overall_coeff.compare(o.overall_coeff);
 405         if (cmpval!=0)
 406                 return cmpval;
 407
 408         auto cit1 = seq.begin(), last1 = seq.end();
 409         auto cit2 = o.seq.begin(), last2 = o.seq.end();
 410         for (; (cit1!=last1) && (cit2!=last2); ++cit1, ++cit2) {
 411                 cmpval = (*cit1).compare(*cit2);
 412                 if (cmpval!=0) return cmpval;
 413         }
 414
 415         GINAC_ASSERT(cit1==last1);
 416         GINAC_ASSERT(cit2==last2);
 417
 418         return 0;
 419 }
 420
 421 bool expairseq::is_equal_same_type(const basic &other) const
 422 {
 423         const expairseq &o = static_cast<const expairseq &>(other);
 424
 425         // compare number of elements
 426         if (seq.size()!=o.seq.size())
 427                 return false;
 428
 429         // compare overall_coeff
 430         if (!overall_coeff.is_equal(o.overall_coeff))
 431                 return false;
 432
 433         auto cit2 = o.seq.begin();
 434         for (auto & cit1 : seq) {
 435                 if (!cit1.is_equal(*cit2))
 436                         return false;
 437                 ++cit2;
 438         }
 439
 440         return true;
 441 }
 442
 443 unsigned expairseq::return_type() const
 444 {
 445         return return_types::noncommutative_composite;
 446 }
 447
 448 unsigned expairseq::calchash() const
 449 {
 450         unsigned v = make_hash_seed(typeid(*this));
 451         for (auto & i : seq) {
 452                 v ^= i.rest.gethash();
 453                 v = rotate_left(v);
 454                 v ^= i.coeff.gethash();
 455         }
 456
 457         v ^= overall_coeff.gethash();
 458
 459         // store calculated hash value only if object is already evaluated
 460         if (flags &status_flags::evaluated) {
 461                 setflag(status_flags::hash_calculated);
 462                 hashvalue = v;
 463         }
 464
 465         return v;
 466 }
 467
 468 ex expairseq::expand(unsigned options) const
 469 {
 470         epvector expanded = expandchildren(options);
 471         if (!expanded.empty()) {
 472                 return thisexpairseq(std::move(expanded), overall_coeff);
 473         }
 474         return (options == 0) ? setflag(status_flags::expanded) : *this;
 475 }
 476
 477 //////////
 478 // new virtual functions which can be overridden by derived classes
 479 //////////
 480
 481 // protected
 482
 483 /** Create an object of this type.
 484  *  This method works similar to a constructor.  It is useful because expairseq
 485  *  has (at least) two possible different semantics but we want to inherit
 486  *  methods thus avoiding code duplication.  Sometimes a method in expairseq
 487  *  has to create a new one of the same semantics, which cannot be done by a
 488  *  ctor because the name (add, mul,...) is unknown on the expairseq level.  In
 489  *  order for this trick to work a derived class must of course override this
 490  *  definition. */
 491 ex expairseq::thisexpairseq(const epvector &v, const ex &oc, bool do_index_renaming) const
 492 {
 493         return expairseq(v, oc, do_index_renaming);
 494 }
 495
 496 ex expairseq::thisexpairseq(epvector && vp, const ex &oc, bool do_index_renaming) const
 497 {
 498         return expairseq(std::move(vp), oc, do_index_renaming);
 499 }
 500
 501 void expairseq::printpair(const print_context & c, const expair & p, unsigned upper_precedence) const
 502 {
 503         c.s << "[[";
 504         p.rest.print(c, precedence());
 505         c.s << ",";
 506         p.coeff.print(c, precedence());
 507         c.s << "]]";
 508 }
 509
 510 void expairseq::printseq(const print_context & c, char delim,
 511                          unsigned this_precedence,
 512                          unsigned upper_precedence) const
 513 {
 514         if (this_precedence <= upper_precedence)
 515                 c.s << "(";
 516         auto it = seq.begin(), it_last = seq.end() - 1;
 517         for (; it!=it_last; ++it) {
 518                 printpair(c, *it, this_precedence);
 519                 c.s << delim;
 520         }
 521         printpair(c, *it, this_precedence);
 522         if (!overall_coeff.is_equal(default_overall_coeff())) {
 523                 c.s << delim;
 524                 overall_coeff.print(c, this_precedence);
 525         }
 526
 527         if (this_precedence <= upper_precedence)
 528                 c.s << ")";
 529 }
 530
 531
 532 /** Form an expair from an ex, using the corresponding semantics.
 533  *  @see expairseq::recombine_pair_to_ex() */
 534 expair expairseq::split_ex_to_pair(const ex &e) const
 535 {
 536         return expair(e,_ex1);
 537 }
 538
 539
 540 expair expairseq::combine_ex_with_coeff_to_pair(const ex &e,
 541                                                 const ex &c) const
 542 {
 543         GINAC_ASSERT(is_exactly_a<numeric>(c));
 544
 545         return expair(e,c);
 546 }
 547
 548
 549 expair expairseq::combine_pair_with_coeff_to_pair(const expair &p,
 550                                                   const ex &c) const
 551 {
 552         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
 553         GINAC_ASSERT(is_exactly_a<numeric>(c));
 554
 555         return expair(p.rest,ex_to<numeric>(p.coeff).mul_dyn(ex_to<numeric>(c)));
 556 }
 557
 558
 559 /** Form an ex out of an expair, using the corresponding semantics.
 560  *  @see expairseq::split_ex_to_pair() */
 561 ex expairseq::recombine_pair_to_ex(const expair &p) const
 562 {
 563         return lst{p.rest, p.coeff};
 564 }
 565
 566 bool expairseq::expair_needs_further_processing(epp it)
 567 {
 568         return false;
 569 }
 570
 571 ex expairseq::default_overall_coeff() const
 572 {
 573         return _ex0;
 574 }
 575
 576 void expairseq::combine_overall_coeff(const ex &c)
 577 {
 578         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 579         GINAC_ASSERT(is_exactly_a<numeric>(c));
 580         overall_coeff = ex_to<numeric>(overall_coeff).add_dyn(ex_to<numeric>(c));
 581 }
 582
 583 void expairseq::combine_overall_coeff(const ex &c1, const ex &c2)
 584 {
 585         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
 586         GINAC_ASSERT(is_exactly_a<numeric>(c1));
 587         GINAC_ASSERT(is_exactly_a<numeric>(c2));
 588         overall_coeff = ex_to<numeric>(overall_coeff).
 589                         add_dyn(ex_to<numeric>(c1).mul(ex_to<numeric>(c2)));
 590 }
 591
 592 bool expairseq::can_make_flat(const expair &p) const
 593 {
 594         return true;
 595 }
 596
 597
 598 //////////
 599 // non-virtual functions in this class
 600 //////////
 601
 602 void expairseq::construct_from_2_ex_via_exvector(const ex &lh, const ex &rh)
 603 {
 604         const exvector v = {lh, rh};
 605         construct_from_exvector(std::move(v));
 606 }
 607
 608 void expairseq::construct_from_2_ex(const ex &lh, const ex &rh)
 609 {
 610         if (typeid(ex_to<basic>(lh)) == typeid(*this)) {
 611                 if (typeid(ex_to<basic>(rh)) == typeid(*this)) {
 612                         if (is_a<mul>(lh) && lh.info(info_flags::has_indices) &&
 613                                 rh.info(info_flags::has_indices)) {
 614                                 ex newrh=rename_dummy_indices_uniquely(lh, rh);
 615                                 construct_from_2_expairseq(ex_to<expairseq>(lh),
 616                                                            ex_to<expairseq>(newrh));
 617                         }
 618                         else
 619                                 construct_from_2_expairseq(ex_to<expairseq>(lh),
 620                                                            ex_to<expairseq>(rh));
 621                         return;
 622                 } else {
 623                         construct_from_expairseq_ex(ex_to<expairseq>(lh), rh);
 624                         return;
 625                 }
 626         } else if (typeid(ex_to<basic>(rh)) == typeid(*this)) {
 627                 construct_from_expairseq_ex(ex_to<expairseq>(rh),lh);
 628                 return;
 629         }
 630
 631         if (is_exactly_a<numeric>(lh)) {
 632                 if (is_exactly_a<numeric>(rh)) {
 633                         combine_overall_coeff(lh);
 634                         combine_overall_coeff(rh);
 635                 } else {
 636                         combine_overall_coeff(lh);
 637                         seq.push_back(split_ex_to_pair(rh));
 638                 }
 639         } else {
 640                 if (is_exactly_a<numeric>(rh)) {
 641                         combine_overall_coeff(rh);
 642                         seq.push_back(split_ex_to_pair(lh));
 643                 } else {
 644                         expair p1 = split_ex_to_pair(lh);
 645                         expair p2 = split_ex_to_pair(rh);
 646
 647                         int cmpval = p1.rest.compare(p2.rest);
 648                         if (cmpval==0) {
 649                                 p1.coeff = ex_to<numeric>(p1.coeff).add_dyn(ex_to<numeric>(p2.coeff));
 650                                 if (!ex_to<numeric>(p1.coeff).is_zero()) {
 651                                         // no further processing is necessary, since this
 652                                         // one element will usually be recombined in eval()
 653                                         seq.push_back(p1);
 654                                 }
 655                         } else {
 656                                 seq.reserve(2);
 657                                 if (cmpval<0) {
 658                                         seq.push_back(p1);
 659                                         seq.push_back(p2);
 660                                 } else {
 661                                         seq.push_back(p2);
 662                                         seq.push_back(p1);
 663                                 }
 664                         }
 665                 }
 666         }
 667 }
 668
 669 void expairseq::construct_from_2_expairseq(const expairseq &s1,
 670                                            const expairseq &s2)
 671 {
 672         combine_overall_coeff(s1.overall_coeff);
 673         combine_overall_coeff(s2.overall_coeff);
 674
 675         auto first1 = s1.seq.begin(), last1 = s1.seq.end();
 676         auto first2 = s2.seq.begin(), last2 = s2.seq.end();
 677
 678         seq.reserve(s1.seq.size()+s2.seq.size());
 679
 680         bool needs_further_processing=false;
 681
 682         while (first1!=last1 && first2!=last2) {
 683                 int cmpval = (*first1).rest.compare((*first2).rest);
 684
 685                 if (cmpval==0) {
 686                         // combine terms
 687                         const numeric &newcoeff = ex_to<numeric>(first1->coeff).
 688                                                    add(ex_to<numeric>(first2->coeff));
 689                         if (!newcoeff.is_zero()) {
 690                                 seq.push_back(expair(first1->rest,newcoeff));
 691                                 if (expair_needs_further_processing(seq.end()-1)) {
 692                                         needs_further_processing = true;
 693                                 }
 694                         }
 695                         ++first1;
 696                         ++first2;
 697                 } else if (cmpval<0) {
 698                         seq.push_back(*first1);
 699                         ++first1;
 700                 } else {
 701                         seq.push_back(*first2);
 702                         ++first2;
 703                 }
 704         }
 705
 706         while (first1!=last1) {
 707                 seq.push_back(*first1);
 708                 ++first1;
 709         }
 710         while (first2!=last2) {
 711                 seq.push_back(*first2);
 712                 ++first2;
 713         }
 714
 715         if (needs_further_processing) {
 716                 // Clear seq and start over.
 717                 epvector v = std::move(seq);
 718                 construct_from_epvector(std::move(v));
 719         }
 720 }
 721
 722 void expairseq::construct_from_expairseq_ex(const expairseq &s,
 723                                             const ex &e)
 724 {
 725         combine_overall_coeff(s.overall_coeff);
 726         if (is_exactly_a<numeric>(e)) {
 727                 combine_overall_coeff(e);
 728                 seq = s.seq;
 729                 return;
 730         }
 731
 732         auto first = s.seq.begin(), last = s.seq.end();
 733         expair p = split_ex_to_pair(e);
 734
 735         seq.reserve(s.seq.size()+1);
 736         bool p_pushed = false;
 737
 738         bool needs_further_processing=false;
 739
 740         // merge p into s.seq
 741         while (first!=last) {
 742                 int cmpval = (*first).rest.compare(p.rest);
 743                 if (cmpval==0) {
 744                         // combine terms
 745                         const numeric &newcoeff = ex_to<numeric>(first->coeff).
 746                                                    add(ex_to<numeric>(p.coeff));
 747                         if (!newcoeff.is_zero()) {
 748                                 seq.push_back(expair(first->rest,newcoeff));
 749                                 if (expair_needs_further_processing(seq.end()-1))
 750                                         needs_further_processing = true;
 751                         }
 752                         ++first;
 753                         p_pushed = true;
 754                         break;
 755                 } else if (cmpval<0) {
 756                         seq.push_back(*first);
 757                         ++first;
 758                 } else {
 759                         seq.push_back(p);
 760                         p_pushed = true;
 761                         break;
 762                 }
 763         }
 764
 765         if (p_pushed) {
 766                 // while loop exited because p was pushed, now push rest of s.seq
 767                 while (first!=last) {
 768                         seq.push_back(*first);
 769                         ++first;
 770                 }
 771         } else {
 772                 // while loop exited because s.seq was pushed, now push p
 773                 seq.push_back(p);
 774         }
 775
 776         if (needs_further_processing) {
 777                 // Clear seq and start over.
 778                 epvector v = std::move(seq);
 779                 construct_from_epvector(std::move(v));
 780         }
 781 }
 782
 783 void expairseq::construct_from_exvector(const exvector &v)
 784 {
 785         // simplifications: +(a,+(b,c),d) -> +(a,b,c,d) (associativity)
 786         //                  +(d,b,c,a) -> +(a,b,c,d) (canonicalization)
 787         //                  +(...,x,*(x,c1),*(x,c2)) -> +(...,*(x,1+c1+c2)) (c1, c2 numeric)
 788         //                  (same for (+,*) -> (*,^)
 789
 790         make_flat(v);
 791         canonicalize();
 792         combine_same_terms_sorted_seq();
 793 }
 794
 795 void expairseq::construct_from_epvector(const epvector &v, bool do_index_renaming)
 796 {
 797         // simplifications: +(a,+(b,c),d) -> +(a,b,c,d) (associativity)
 798         //                  +(d,b,c,a) -> +(a,b,c,d) (canonicalization)
 799         //                  +(...,x,*(x,c1),*(x,c2)) -> +(...,*(x,1+c1+c2)) (c1, c2 numeric)
 800         //                  same for (+,*) -> (*,^)
 801
 802         make_flat(v, do_index_renaming);
 803         canonicalize();
 804         combine_same_terms_sorted_seq();
 805 }
 806
 807 void expairseq::construct_from_epvector(epvector &&v, bool do_index_renaming)
 808 {
 809         // simplifications: +(a,+(b,c),d) -> +(a,b,c,d) (associativity)
 810         //                  +(d,b,c,a) -> +(a,b,c,d) (canonicalization)
 811         //                  +(...,x,*(x,c1),*(x,c2)) -> +(...,*(x,1+c1+c2)) (c1, c2 numeric)
 812         //                  same for (+,*) -> (*,^)
 813
 814         make_flat(std::move(v), do_index_renaming);
 815         canonicalize();
 816         combine_same_terms_sorted_seq();
 817 }
 818
 819 /** Combine this expairseq with argument exvector.
 820  *  It cares for associativity as well as for special handling of numerics. */
 821 void expairseq::make_flat(const exvector &v)
 822 {
 823         // count number of operands which are of same expairseq derived type
 824         // and their cumulative number of operands
 825         int nexpairseqs = 0;
 826         int noperands = 0;
 827         bool do_idx_rename = false;
 828
 829         for (auto & cit : v) {
 830                 if (typeid(ex_to<basic>(cit)) == typeid(*this)) {
 831                         ++nexpairseqs;
 832                         noperands += ex_to<expairseq>(cit).seq.size();
 833                 }
 834                 if (is_a<mul>(*this) && (!do_idx_rename) &&
 835                     cit.info(info_flags::has_indices))
 836                         do_idx_rename = true;
 837         }
 838
 839         // reserve seq and coeffseq which will hold all operands
 840         seq.reserve(v.size()+noperands-nexpairseqs);
 841
 842         // copy elements and split off numerical part
 843         make_flat_inserter mf(v, do_idx_rename);
 844         for (auto & cit : v) {
 845                 if (typeid(ex_to<basic>(cit)) == typeid(*this)) {
 846                         ex newfactor = mf.handle_factor(cit, _ex1);
 847                         const expairseq &subseqref = ex_to<expairseq>(newfactor);
 848                         combine_overall_coeff(subseqref.overall_coeff);
 849                         for (auto & cit_s : subseqref.seq) {
 850                                 seq.push_back(cit_s);
 851                         }
 852                 } else {
 853                         if (is_exactly_a<numeric>(cit))
 854                                 combine_overall_coeff(cit);
 855                         else {
 856                                 ex newfactor = mf.handle_factor(cit, _ex1);
 857                                 seq.push_back(split_ex_to_pair(newfactor));
 858                         }
 859                 }
 860         }
 861 }
 862
 863 /** Combine this expairseq with argument epvector.
 864  *  It cares for associativity as well as for special handling of numerics. */
 865 void expairseq::make_flat(const epvector &v, bool do_index_renaming)
 866 {
 867         // count number of operands which are of same expairseq derived type
 868         // and their cumulative number of operands
 869         int nexpairseqs = 0;
 870         int noperands = 0;
 871         bool really_need_rename_inds = false;
 872
 873         for (auto & cit : v) {
 874                 if (typeid(ex_to<basic>(cit.rest)) == typeid(*this)) {
 875                         ++nexpairseqs;
 876                         noperands += ex_to<expairseq>(cit.rest).seq.size();
 877                 }
 878                 if ((!really_need_rename_inds) && is_a<mul>(*this) &&
 879                     cit.rest.info(info_flags::has_indices))
 880                         really_need_rename_inds = true;
 881         }
 882         do_index_renaming = do_index_renaming && really_need_rename_inds;
 883
 884         // reserve seq and coeffseq which will hold all operands
 885         seq.reserve(v.size()+noperands-nexpairseqs);
 886         make_flat_inserter mf(v, do_index_renaming);
 887
 888         // copy elements and split off numerical part
 889         for (auto & cit : v) {
 890                 if (typeid(ex_to<basic>(cit.rest)) == typeid(*this) &&
 891                     this->can_make_flat(cit)) {
 892                         ex newrest = mf.handle_factor(cit.rest, cit.coeff);
 893                         const expairseq &subseqref = ex_to<expairseq>(newrest);
 894                         combine_overall_coeff(ex_to<numeric>(subseqref.overall_coeff),
 895                                               ex_to<numeric>(cit.coeff));
 896                         for (auto & cit_s : subseqref.seq) {
 897                                 seq.push_back(expair(cit_s.rest,
 898                                                      ex_to<numeric>(cit_s.coeff).mul_dyn(ex_to<numeric>(cit.coeff))));
 899                         }
 900                 } else {
 901                         if (cit.is_canonical_numeric())
 902                                 combine_overall_coeff(mf.handle_factor(cit.rest, _ex1));
 903                         else {
 904                                 ex rest = cit.rest;
 905                                 ex newrest = mf.handle_factor(rest, cit.coeff);
 906                                 if (are_ex_trivially_equal(newrest, rest))
 907                                         seq.push_back(cit);
 908                                 else
 909                                         seq.push_back(expair(newrest, cit.coeff));
 910                         }
 911                 }
 912         }
 913 }
 914
 915 /** Brings this expairseq into a sorted (canonical) form. */
 916 void expairseq::canonicalize()
 917 {
 918         std::sort(seq.begin(), seq.end(), expair_rest_is_less());
 919 }
 920
 921
 922 /** Compact a presorted expairseq by combining all matching expairs to one
 923  *  each.  On an add object, this is responsible for 2*x+3*x+y -> 5*x+y, for
 924  *  instance. */
 925 void expairseq::combine_same_terms_sorted_seq()
 926 {
 927         if (seq.size()<2)
 928                 return;
 929
 930         bool needs_further_processing = false;
 931
 932         auto itin1 = seq.begin();
 933         auto itin2 = itin1 + 1;
 934         auto itout = itin1;
 935         auto last = seq.end();
 936         // must_copy will be set to true the first time some combination is
 937         // possible from then on the sequence has changed and must be compacted
 938         bool must_copy = false;
 939         while (itin2!=last) {
 940                 if (itin1->rest.compare(itin2->rest)==0) {
 941                         itin1->coeff = ex_to<numeric>(itin1->coeff).
 942                                        add_dyn(ex_to<numeric>(itin2->coeff));
 943                         if (expair_needs_further_processing(itin1))
 944                                 needs_further_processing = true;
 945                         must_copy = true;
 946                 } else {
 947                         if (!ex_to<numeric>(itin1->coeff).is_zero()) {
 948                                 if (must_copy)
 949                                         *itout = *itin1;
 950                                 ++itout;
 951                         }
 952                         itin1 = itin2;
 953                 }
 954                 ++itin2;
 955         }
 956         if (!ex_to<numeric>(itin1->coeff).is_zero()) {
 957                 if (must_copy)
 958                         *itout = *itin1;
 959                 ++itout;
 960         }
 961         if (itout!=last)
 962                 seq.erase(itout,last);
 963
 964         if (needs_further_processing) {
 965                 // Clear seq and start over.
 966                 epvector v = std::move(seq);
 967                 construct_from_epvector(std::move(v));
 968         }
 969 }
 970
 971 /** Check if this expairseq is in sorted (canonical) form.  Useful mainly for
 972  *  debugging or in assertions since being sorted is an invariance. */
 973 bool expairseq::is_canonical() const
 974 {
 975         if (seq.size() <= 1)
 976                 return 1;
 977
 978         auto it = seq.begin(), itend = seq.end();
 979         auto it_last = it;
 980         for (++it; it!=itend; it_last=it, ++it) {
 981                 if (!(it_last->is_less(*it) || it_last->is_equal(*it))) {
 982                         if (!is_exactly_a<numeric>(it_last->rest) ||
 983                                 !is_exactly_a<numeric>(it->rest)) {
 984                                 // double test makes it easier to set a breakpoint...
 985                                 if (!is_exactly_a<numeric>(it_last->rest) ||
 986                                         !is_exactly_a<numeric>(it->rest)) {
 987                                         printpair(std::clog, *it_last, 0);
 988                                         std::clog << ">";
 989                                         printpair(std::clog, *it, 0);
 990                                         std::clog << "\n";
 991                                         std::clog << "pair1:" << std::endl;
 992                                         it_last->rest.print(print_tree(std::clog));
 993                                         it_last->coeff.print(print_tree(std::clog));
 994                                         std::clog << "pair2:" << std::endl;
 995                                         it->rest.print(print_tree(std::clog));
 996                                         it->coeff.print(print_tree(std::clog));
 997                                         return 0;
 998                                 }
 999                         }
1000                 }
1001         }
1002         return 1;
1003 }
1004
1005 /** Member-wise expand the expairs in this sequence.
1006  *
1007  *  @see expairseq::expand()
1008  *  @return epvector containing expanded pairs, empty if no members
1009  *    had to be changed. */
1010 epvector expairseq::expandchildren(unsigned options) const
1011 {
1012         auto cit = seq.begin(), last = seq.end();
1013         while (cit!=last) {
1014                 const ex expanded_ex = cit->rest.expand(options);
1015                 if (!are_ex_trivially_equal(cit->rest,expanded_ex)) {
1016
1017                         // something changed, copy seq, eval and return it
1018                         epvector s;
1019                         s.reserve(seq.size());
1020
1021                         // copy parts of seq which are known not to have changed
1022                         s.insert(s.begin(), seq.begin(), cit);
1023
1024                         // copy first changed element
1025                         s.push_back(expair(expanded_ex, cit->coeff));
1026                         ++cit;
1027
1028                         // copy rest
1029                         while (cit != last) {
1030                                 s.push_back(expair(cit->rest.expand(options), cit->coeff));
1031                                 ++cit;
1032                         }
1033                         return s;
1034                 }
1035
1036                 ++cit;
1037         }
1038
1039         return epvector(); // empty signalling nothing has changed
1040 }
1041
1042
1043 /** Member-wise evaluate the expairs in this sequence.
1044  *
1045  *  @see expairseq::eval()
1046  *  @return epvector containing evaluated pairs, empty if no members
1047  *    had to be changed. */
1048 epvector expairseq::evalchildren() const
1049 {
1050         auto cit = seq.begin(), last = seq.end();
1051         while (cit!=last) {
1052                 const expair evaled_pair = combine_ex_with_coeff_to_pair(cit->rest, cit->coeff);
1053                 if (unlikely(!evaled_pair.is_equal(*cit))) {
1054
1055                         // something changed: copy seq, eval and return it
1056                         epvector s;
1057                         s.reserve(seq.size());
1058
1059                         // copy parts of seq which are known not to have changed
1060                         s.insert(s.begin(), seq.begin(), cit);
1061
1062                         // copy first changed element
1063                         s.push_back(evaled_pair);
1064                         ++cit;
1065
1066                         // copy rest
1067                         while (cit != last) {
1068                                 s.push_back(combine_ex_with_coeff_to_pair(cit->rest, cit->coeff));
1069                                 ++cit;
1070                         }
1071                         return s;
1072                 }
1073
1074                 ++cit;
1075         }
1076
1077         return epvector(); // signalling nothing has changed
1078 }
1079
1080 /** Member-wise substitute in this sequence.
1081  *
1082  *  @see expairseq::subs()
1083  *  @return epvector containing expanded pairs, empty if no members
1084  *    had to be changed. */
1085 epvector expairseq::subschildren(const exmap & m, unsigned options) const
1086 {
1087         // When any of the objects to be substituted is a product or power
1088         // we have to recombine the pairs because the numeric coefficients may
1089         // be part of the search pattern.
1090         if (!(options & (subs_options::pattern_is_product | subs_options::pattern_is_not_product))) {
1091
1092                 // Search the list of substitutions and cache our findings
1093                 for (auto & it : m) {
1094                         if (is_exactly_a<mul>(it.first) || is_exactly_a<power>(it.first)) {
1095                                 options |= subs_options::pattern_is_product;
1096                                 break;
1097                         }
1098                 }
1099                 if (!(options & subs_options::pattern_is_product))
1100                         options |= subs_options::pattern_is_not_product;
1101         }
1102
1103         if (options & subs_options::pattern_is_product) {
1104
1105                 // Substitute in the recombined pairs
1106                 auto cit = seq.begin(), last = seq.end();
1107                 while (cit != last) {
1108
1109                         const ex &orig_ex = recombine_pair_to_ex(*cit);
1110                         const ex &subsed_ex = orig_ex.subs(m, options);
1111                         if (!are_ex_trivially_equal(orig_ex, subsed_ex)) {
1112
1113                                 // Something changed: copy seq, subs and return it
1114                                 epvector s;
1115                                 s.reserve(seq.size());
1116
1117                                 // Copy parts of seq which are known not to have changed
1118                                 s.insert(s.begin(), seq.begin(), cit);
1119
1120                                 // Copy first changed element
1121                                 s.push_back(split_ex_to_pair(subsed_ex));
1122                                 ++cit;
1123
1124                                 // Copy rest
1125                                 while (cit != last) {
1126                                         s.push_back(split_ex_to_pair(recombine_pair_to_ex(*cit).subs(m, options)));
1127                                         ++cit;
1128                                 }
1129                                 return s;
1130                         }
1131
1132                         ++cit;
1133                 }
1134
1135         } else {
1136
1137                 // Substitute only in the "rest" part of the pairs
1138                 auto cit = seq.begin(), last = seq.end();
1139                 while (cit != last) {
1140
1141                         const ex subsed_rest = cit->rest.subs(m, options);
1142                         const expair subsed_pair = combine_ex_with_coeff_to_pair(subsed_rest, cit->coeff);
1143                         if (!subsed_pair.is_equal(*cit)) {
1144
1145                                 // Something changed: copy seq, subs and return it
1146                                 epvector s;
1147                                 s.reserve(seq.size());
1148
1149                                 // Copy parts of seq which are known not to have changed
1150                                 s.insert(s.begin(), seq.begin(), cit);
1151
1152                                 // Copy first changed element
1153                                 s.push_back(subsed_pair);
1154                                 ++cit;
1155
1156                                 // Copy rest
1157                                 while (cit != last) {
1158                                         s.push_back(combine_ex_with_coeff_to_pair(cit->rest.subs(m, options), cit->coeff));
1159                                         ++cit;
1160                                 }
1161                                 return s;
1162                         }
1163
1164                         ++cit;
1165                 }
1166         }
1167
1168         // Nothing has changed
1169         return epvector();
1170 }
1171
1172 //////////
1173 // static member variables
1174 //////////
1175
1176 } // namespace GiNaC