ginac/pseries.cpp

   1 /** @file pseries.cpp
   2  *
   3  *  Implementation of class for extended truncated power series and
   4  *  methods for series expansion. */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include "pseries.h"
  25 #include "add.h"
  26 #include "inifcns.h" // for Order function
  27 #include "lst.h"
  28 #include "mul.h"
  29 #include "power.h"
  30 #include "relational.h"
  31 #include "operators.h"
  32 #include "symbol.h"
  33 #include "integral.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36
  37 #include <limits>
  38 #include <numeric>
  39 #include <stdexcept>
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
  44   print_func<print_context>(&pseries::do_print).
  45   print_func<print_latex>(&pseries::do_print_latex).
  46   print_func<print_tree>(&pseries::do_print_tree).
  47   print_func<print_python>(&pseries::do_print_python).
  48   print_func<print_python_repr>(&pseries::do_print_python_repr))
  49
  50
  51 /*
  52  *  Default constructor
  53  */
  54
  55 pseries::pseries() { }
  56
  57
  58 /*
  59  *  Other ctors
  60  */
  61
  62 /** Construct pseries from a vector of coefficients and powers.
  63  *  expair.rest holds the coefficient, expair.coeff holds the power.
  64  *  The powers must be integers (positive or negative) and in ascending order;
  65  *  the last coefficient can be Order(_ex1) to represent a truncated,
  66  *  non-terminating series.
  67  *
  68  *  @param rel_  expansion variable and point (must hold a relational)
  69  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  70  *  @return newly constructed pseries */
  71 pseries::pseries(const ex &rel_, const epvector &ops_)
  72   : seq(ops_)
  73 {
  74 #ifdef DO_GINAC_ASSERT
  75         auto i = seq.begin();
  76         while (i != seq.end()) {
  77                 auto ip1 = i+1;
  78                 if (ip1 != seq.end())
  79                         GINAC_ASSERT(!is_order_function(i->rest));
  80                 else
  81                         break;
  82                 GINAC_ASSERT(is_a<numeric>(i->coeff));
  83                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
  84                 ++i;
  85         }
  86 #endif // def DO_GINAC_ASSERT
  87         GINAC_ASSERT(is_a<relational>(rel_));
  88         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  89         point = rel_.rhs();
  90         var = rel_.lhs();
  91 }
  92 pseries::pseries(const ex &rel_, epvector &&ops_)
  93   : seq(std::move(ops_))
  94 {
  95 #ifdef DO_GINAC_ASSERT
  96         auto i = seq.begin();
  97         while (i != seq.end()) {
  98                 auto ip1 = i+1;
  99                 if (ip1 != seq.end())
 100                         GINAC_ASSERT(!is_order_function(i->rest));
 101                 else
 102                         break;
 103                 GINAC_ASSERT(is_a<numeric>(i->coeff));
 104                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
 105                 ++i;
 106         }
 107 #endif // def DO_GINAC_ASSERT
 108         GINAC_ASSERT(is_a<relational>(rel_));
 109         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
 110         point = rel_.rhs();
 111         var = rel_.lhs();
 112 }
 113
 114
 115 /*
 116  *  Archiving
 117  */
 118
 119 void pseries::read_archive(const archive_node &n, lst &sym_lst)
 120 {
 121         inherited::read_archive(n, sym_lst);
 122         auto range = n.find_property_range("coeff", "power");
 123         seq.reserve((range.end-range.begin)/2);
 124
 125         for (auto loc = range.begin; loc < range.end;) {
 126                 ex rest;
 127                 ex coeff;
 128                 n.find_ex_by_loc(loc++, rest, sym_lst);
 129                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 130                 seq.emplace_back(expair(rest, coeff));
 131         }
 132
 133         n.find_ex("var", var, sym_lst);
 134         n.find_ex("point", point, sym_lst);
 135 }
 136
 137 void pseries::archive(archive_node &n) const
 138 {
 139         inherited::archive(n);
 140         for (auto & it : seq) {
 141                 n.add_ex("coeff", it.rest);
 142                 n.add_ex("power", it.coeff);
 143         }
 144         n.add_ex("var", var);
 145         n.add_ex("point", point);
 146 }
 147
 148
 149 //////////
 150 // functions overriding virtual functions from base classes
 151 //////////
 152
 153 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
 154 {
 155         if (precedence() <= level)
 156                 c.s << '(';
 157
 158         // objects of type pseries must not have any zero entries, so the
 159         // trivial (zero) pseries needs a special treatment here:
 160         if (seq.empty())
 161                 c.s << '0';
 162
 163         auto i = seq.begin(), end = seq.end();
 164         while (i != end) {
 165
 166                 // print a sign, if needed
 167                 if (i != seq.begin())
 168                         c.s << '+';
 169
 170                 if (!is_order_function(i->rest)) {
 171
 172                         // print 'rest', i.e. the expansion coefficient
 173                         if (i->rest.info(info_flags::numeric) &&
 174                                 i->rest.info(info_flags::positive)) {
 175                                 i->rest.print(c);
 176                         } else {
 177                                 c.s << openbrace << '(';
 178                                 i->rest.print(c);
 179                                 c.s << ')' << closebrace;
 180                         }
 181
 182                         // print 'coeff', something like (x-1)^42
 183                         if (!i->coeff.is_zero()) {
 184                                 c.s << mul_sym;
 185                                 if (!point.is_zero()) {
 186                                         c.s << openbrace << '(';
 187                                         (var-point).print(c);
 188                                         c.s << ')' << closebrace;
 189                                 } else
 190                                         var.print(c);
 191                                 if (i->coeff.compare(_ex1)) {
 192                                         c.s << pow_sym;
 193                                         c.s << openbrace;
 194                                         if (i->coeff.info(info_flags::negative)) {
 195                                                 c.s << '(';
 196                                                 i->coeff.print(c);
 197                                                 c.s << ')';
 198                                         } else
 199                                                 i->coeff.print(c);
 200                                         c.s << closebrace;
 201                                 }
 202                         }
 203                 } else
 204                         Order(pow(var - point, i->coeff)).print(c);
 205                 ++i;
 206         }
 207
 208         if (precedence() <= level)
 209                 c.s << ')';
 210 }
 211
 212 void pseries::do_print(const print_context & c, unsigned level) const
 213 {
 214         print_series(c, "", "", "*", "^", level);
 215 }
 216
 217 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 218 {
 219         print_series(c, "{", "}", " ", "^", level);
 220 }
 221
 222 void pseries::do_print_python(const print_python & c, unsigned level) const
 223 {
 224         print_series(c, "", "", "*", "**", level);
 225 }
 226
 227 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 228 {
 229         c.s << std::string(level, ' ') << class_name() << " @" << this
 230             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 231             << std::endl;
 232         size_t num = seq.size();
 233         for (size_t i=0; i<num; ++i) {
 234                 seq[i].rest.print(c, level + c.delta_indent);
 235                 seq[i].coeff.print(c, level + c.delta_indent);
 236                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 237         }
 238         var.print(c, level + c.delta_indent);
 239         point.print(c, level + c.delta_indent);
 240 }
 241
 242 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 243 {
 244         c.s << class_name() << "(relational(";
 245         var.print(c);
 246         c.s << ',';
 247         point.print(c);
 248         c.s << "),[";
 249         size_t num = seq.size();
 250         for (size_t i=0; i<num; ++i) {
 251                 if (i)
 252                         c.s << ',';
 253                 c.s << '(';
 254                 seq[i].rest.print(c);
 255                 c.s << ',';
 256                 seq[i].coeff.print(c);
 257                 c.s << ')';
 258         }
 259         c.s << "])";
 260 }
 261
 262 int pseries::compare_same_type(const basic & other) const
 263 {
 264         GINAC_ASSERT(is_a<pseries>(other));
 265         const pseries &o = static_cast<const pseries &>(other);
 266
 267         // first compare the lengths of the series...
 268         if (seq.size()>o.seq.size())
 269                 return 1;
 270         if (seq.size()<o.seq.size())
 271                 return -1;
 272
 273         // ...then the expansion point...
 274         int cmpval = var.compare(o.var);
 275         if (cmpval)
 276                 return cmpval;
 277         cmpval = point.compare(o.point);
 278         if (cmpval)
 279                 return cmpval;
 280
 281         // ...and if that failed the individual elements
 282         auto it = seq.begin(), o_it = o.seq.begin();
 283         while (it!=seq.end() && o_it!=o.seq.end()) {
 284                 cmpval = it->compare(*o_it);
 285                 if (cmpval)
 286                         return cmpval;
 287                 ++it;
 288                 ++o_it;
 289         }
 290
 291         // so they are equal.
 292         return 0;
 293 }
 294
 295 /** Return the number of operands including a possible order term. */
 296 size_t pseries::nops() const
 297 {
 298         return seq.size();
 299 }
 300
 301 /** Return the ith term in the series when represented as a sum. */
 302 ex pseries::op(size_t i) const
 303 {
 304         if (i >= seq.size())
 305                 throw (std::out_of_range("op() out of range"));
 306
 307         if (is_order_function(seq[i].rest))
 308                 return Order(pow(var-point, seq[i].coeff));
 309         return seq[i].rest * pow(var - point, seq[i].coeff);
 310 }
 311
 312 /** Return degree of highest power of the series.  This is usually the exponent
 313  *  of the Order term.  If s is not the expansion variable of the series, the
 314  *  series is examined termwise. */
 315 int pseries::degree(const ex &s) const
 316 {
 317         if (seq.empty())
 318                 return 0;
 319
 320         if (var.is_equal(s))
 321                 // Return last/greatest exponent
 322                 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 323
 324         int max_pow = std::numeric_limits<int>::min();
 325         for (auto & it : seq)
 326                 max_pow = std::max(max_pow, it.rest.degree(s));
 327         return max_pow;
 328 }
 329
 330 /** Return degree of lowest power of the series.  This is usually the exponent
 331  *  of the leading term.  If s is not the expansion variable of the series, the
 332  *  series is examined termwise.  If s is the expansion variable but the
 333  *  expansion point is not zero the series is not expanded to find the degree.
 334  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 335 int pseries::ldegree(const ex &s) const
 336 {
 337         if (seq.empty())
 338                 return 0;
 339
 340         if (var.is_equal(s))
 341                 // Return first/smallest exponent
 342                 return ex_to<numeric>((seq.begin())->coeff).to_int();
 343
 344         int min_pow = std::numeric_limits<int>::max();
 345         for (auto & it : seq)
 346                 min_pow = std::min(min_pow, it.rest.degree(s));
 347         return min_pow;
 348 }
 349
 350 /** Return coefficient of degree n in power series if s is the expansion
 351  *  variable.  If the expansion point is nonzero, by definition the n=1
 352  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 353  *  the expansion took place in the s in the first place).
 354  *  If s is not the expansion variable, an attempt is made to convert the
 355  *  series to a polynomial and return the corresponding coefficient from
 356  *  there. */
 357 ex pseries::coeff(const ex &s, int n) const
 358 {
 359         if (var.is_equal(s)) {
 360                 if (seq.empty())
 361                         return _ex0;
 362
 363                 // Binary search in sequence for given power
 364                 numeric looking_for = numeric(n);
 365                 int lo = 0, hi = seq.size() - 1;
 366                 while (lo <= hi) {
 367                         int mid = (lo + hi) / 2;
 368                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 369                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 370                         switch (cmp) {
 371                                 case -1:
 372                                         lo = mid + 1;
 373                                         break;
 374                                 case 0:
 375                                         return seq[mid].rest;
 376                                 case 1:
 377                                         hi = mid - 1;
 378                                         break;
 379                                 default:
 380                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 381                         }
 382                 }
 383                 return _ex0;
 384         } else
 385                 return convert_to_poly().coeff(s, n);
 386 }
 387
 388 /** Does nothing. */
 389 ex pseries::collect(const ex &s, bool distributed) const
 390 {
 391         return *this;
 392 }
 393
 394 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 395 ex pseries::eval() const
 396 {
 397         if (flags & status_flags::evaluated) {
 398                 return *this;
 399         }
 400
 401         // Construct a new series with evaluated coefficients
 402         epvector new_seq;
 403         new_seq.reserve(seq.size());
 404         for (auto & it : seq)
 405                 new_seq.push_back(expair(it.rest, it.coeff));
 406
 407         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 408 }
 409
 410 /** Evaluate coefficients numerically. */
 411 ex pseries::evalf() const
 412 {
 413         // Construct a new series with evaluated coefficients
 414         epvector new_seq;
 415         new_seq.reserve(seq.size());
 416         for (auto & it : seq)
 417                 new_seq.push_back(expair(it.rest, it.coeff));
 418
 419         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 420 }
 421
 422 ex pseries::conjugate() const
 423 {
 424         if(!var.info(info_flags::real))
 425                 return conjugate_function(*this).hold();
 426
 427         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
 428         ex newpoint = point.conjugate();
 429
 430         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 431                 return *this;
 432         }
 433
 434         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
 435 }
 436
 437 ex pseries::real_part() const
 438 {
 439         if(!var.info(info_flags::real))
 440                 return real_part_function(*this).hold();
 441         ex newpoint = point.real_part();
 442         if(newpoint != point)
 443                 return real_part_function(*this).hold();
 444
 445         epvector v;
 446         v.reserve(seq.size());
 447         for (auto & it : seq)
 448                 v.push_back(expair((it.rest).real_part(), it.coeff));
 449         return dynallocate<pseries>(var==point, std::move(v));
 450 }
 451
 452 ex pseries::imag_part() const
 453 {
 454         if(!var.info(info_flags::real))
 455                 return imag_part_function(*this).hold();
 456         ex newpoint = point.real_part();
 457         if(newpoint != point)
 458                 return imag_part_function(*this).hold();
 459
 460         epvector v;
 461         v.reserve(seq.size());
 462         for (auto & it : seq)
 463                 v.push_back(expair((it.rest).imag_part(), it.coeff));
 464         return dynallocate<pseries>(var==point, std::move(v));
 465 }
 466
 467 ex pseries::eval_integ() const
 468 {
 469         std::unique_ptr<epvector> newseq(nullptr);
 470         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 471                 if (newseq) {
 472                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 473                         continue;
 474                 }
 475                 ex newterm = i->rest.eval_integ();
 476                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 477                         newseq.reset(new epvector);
 478                         newseq->reserve(seq.size());
 479                         for (auto j=seq.begin(); j!=i; ++j)
 480                                 newseq->push_back(*j);
 481                         newseq->push_back(expair(newterm, i->coeff));
 482                 }
 483         }
 484
 485         ex newpoint = point.eval_integ();
 486         if (newseq || !are_ex_trivially_equal(newpoint, point))
 487                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
 488         return *this;
 489 }
 490
 491 ex pseries::evalm() const
 492 {
 493         // evalm each coefficient
 494         epvector newseq;
 495         bool something_changed = false;
 496         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 497                 if (something_changed) {
 498                         ex newcoeff = i->rest.evalm();
 499                         if (!newcoeff.is_zero())
 500                                 newseq.push_back(expair(newcoeff, i->coeff));
 501                 } else {
 502                         ex newcoeff = i->rest.evalm();
 503                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 504                                 something_changed = true;
 505                                 newseq.reserve(seq.size());
 506                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 507                                 if (!newcoeff.is_zero())
 508                                         newseq.push_back(expair(newcoeff, i->coeff));
 509                         }
 510                 }
 511         }
 512         if (something_changed)
 513                 return dynallocate<pseries>(var==point, std::move(newseq));
 514         else
 515                 return *this;
 516 }
 517
 518 ex pseries::subs(const exmap & m, unsigned options) const
 519 {
 520         // If expansion variable is being substituted, convert the series to a
 521         // polynomial and do the substitution there because the result might
 522         // no longer be a power series
 523         if (m.find(var) != m.end())
 524                 return convert_to_poly(true).subs(m, options);
 525
 526         // Otherwise construct a new series with substituted coefficients and
 527         // expansion point
 528         epvector newseq;
 529         newseq.reserve(seq.size());
 530         for (auto & it : seq)
 531                 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
 532         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
 533 }
 534
 535 /** Implementation of ex::expand() for a power series.  It expands all the
 536  *  terms individually and returns the resulting series as a new pseries. */
 537 ex pseries::expand(unsigned options) const
 538 {
 539         epvector newseq;
 540         for (auto & it : seq) {
 541                 ex restexp = it.rest.expand();
 542                 if (!restexp.is_zero())
 543                         newseq.push_back(expair(restexp, it.coeff));
 544         }
 545         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
 546 }
 547
 548 /** Implementation of ex::diff() for a power series.
 549  *  @see ex::diff */
 550 ex pseries::derivative(const symbol & s) const
 551 {
 552         epvector new_seq;
 553
 554         if (s == var) {
 555
 556                 // FIXME: coeff might depend on var
 557                 for (auto & it : seq) {
 558                         if (is_order_function(it.rest)) {
 559                                 new_seq.push_back(expair(it.rest, it.coeff - 1));
 560                         } else {
 561                                 ex c = it.rest * it.coeff;
 562                                 if (!c.is_zero())
 563                                         new_seq.push_back(expair(c, it.coeff - 1));
 564                         }
 565                 }
 566
 567         } else {
 568
 569                 for (auto & it : seq) {
 570                         if (is_order_function(it.rest)) {
 571                                 new_seq.push_back(it);
 572                         } else {
 573                                 ex c = it.rest.diff(s);
 574                                 if (!c.is_zero())
 575                                         new_seq.push_back(expair(c, it.coeff));
 576                         }
 577                 }
 578         }
 579
 580         return pseries(relational(var,point), std::move(new_seq));
 581 }
 582
 583 ex pseries::convert_to_poly(bool no_order) const
 584 {
 585         ex e;
 586         for (auto & it : seq) {
 587                 if (is_order_function(it.rest)) {
 588                         if (!no_order)
 589                                 e += Order(pow(var - point, it.coeff));
 590                 } else
 591                         e += it.rest * pow(var - point, it.coeff);
 592         }
 593         return e;
 594 }
 595
 596 bool pseries::is_terminating() const
 597 {
 598         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 599 }
 600
 601 ex pseries::coeffop(size_t i) const
 602 {
 603         if (i >= nops())
 604                 throw (std::out_of_range("coeffop() out of range"));
 605         return seq[i].rest;
 606 }
 607
 608 ex pseries::exponop(size_t i) const
 609 {
 610         if (i >= nops())
 611                 throw (std::out_of_range("exponop() out of range"));
 612         return seq[i].coeff;
 613 }
 614
 615
 616 /*
 617  *  Implementations of series expansion
 618  */
 619
 620 /** Default implementation of ex::series(). This performs Taylor expansion.
 621  *  @see ex::series */
 622 ex basic::series(const relational & r, int order, unsigned options) const
 623 {
 624         epvector seq;
 625         const symbol &s = ex_to<symbol>(r.lhs());
 626
 627         // default for order-values that make no sense for Taylor expansion
 628         if ((order <= 0) && this->has(s)) {
 629                 seq.push_back(expair(Order(_ex1), order));
 630                 return pseries(r, std::move(seq));
 631         }
 632
 633         // do Taylor expansion
 634         numeric fac = 1;
 635         ex deriv = *this;
 636         ex coeff = deriv.subs(r, subs_options::no_pattern);
 637
 638         if (!coeff.is_zero()) {
 639                 seq.push_back(expair(coeff, _ex0));
 640         }
 641
 642         int n;
 643         for (n=1; n<order; ++n) {
 644                 fac = fac.div(n);
 645                 // We need to test for zero in order to see if the series terminates.
 646                 // The problem is that there is no such thing as a perfect test for
 647                 // zero.  Expanding the term occasionally helps a little...
 648                 deriv = deriv.diff(s).expand();
 649                 if (deriv.is_zero())  // Series terminates
 650                         return pseries(r, std::move(seq));
 651
 652                 coeff = deriv.subs(r, subs_options::no_pattern);
 653                 if (!coeff.is_zero())
 654                         seq.push_back(expair(fac * coeff, n));
 655         }
 656
 657         // Higher-order terms, if present
 658         deriv = deriv.diff(s);
 659         if (!deriv.expand().is_zero())
 660                 seq.push_back(expair(Order(_ex1), n));
 661         return pseries(r, std::move(seq));
 662 }
 663
 664
 665 /** Implementation of ex::series() for symbols.
 666  *  @see ex::series */
 667 ex symbol::series(const relational & r, int order, unsigned options) const
 668 {
 669         epvector seq;
 670         const ex point = r.rhs();
 671         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 672
 673         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 674                 if (order > 0 && !point.is_zero())
 675                         seq.push_back(expair(point, _ex0));
 676                 if (order > 1)
 677                         seq.push_back(expair(_ex1, _ex1));
 678                 else
 679                         seq.push_back(expair(Order(_ex1), numeric(order)));
 680         } else
 681                 seq.push_back(expair(*this, _ex0));
 682         return pseries(r, std::move(seq));
 683 }
 684
 685
 686 /** Add one series object to another, producing a pseries object that
 687  *  represents the sum.
 688  *
 689  *  @param other  pseries object to add with
 690  *  @return the sum as a pseries */
 691 ex pseries::add_series(const pseries &other) const
 692 {
 693         // Adding two series with different variables or expansion points
 694         // results in an empty (constant) series
 695         if (!is_compatible_to(other)) {
 696                 epvector nul { expair(Order(_ex1), _ex0) };
 697                 return pseries(relational(var,point), std::move(nul));
 698         }
 699
 700         // Series addition
 701         epvector new_seq;
 702         auto a = seq.begin(), a_end = seq.end();
 703         auto b = other.seq.begin(), b_end = other.seq.end();
 704         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 705         for (;;) {
 706                 // If a is empty, fill up with elements from b and stop
 707                 if (a == a_end) {
 708                         while (b != b_end) {
 709                                 new_seq.push_back(*b);
 710                                 ++b;
 711                         }
 712                         break;
 713                 } else
 714                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 715
 716                 // If b is empty, fill up with elements from a and stop
 717                 if (b == b_end) {
 718                         while (a != a_end) {
 719                                 new_seq.push_back(*a);
 720                                 ++a;
 721                         }
 722                         break;
 723                 } else
 724                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 725
 726                 // a and b are non-empty, compare powers
 727                 if (pow_a < pow_b) {
 728                         // a has lesser power, get coefficient from a
 729                         new_seq.push_back(*a);
 730                         if (is_order_function((*a).rest))
 731                                 break;
 732                         ++a;
 733                 } else if (pow_b < pow_a) {
 734                         // b has lesser power, get coefficient from b
 735                         new_seq.push_back(*b);
 736                         if (is_order_function((*b).rest))
 737                                 break;
 738                         ++b;
 739                 } else {
 740                         // Add coefficient of a and b
 741                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 742                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 743                                 break;  // Order term ends the sequence
 744                         } else {
 745                                 ex sum = (*a).rest + (*b).rest;
 746                                 if (!(sum.is_zero()))
 747                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 748                                 ++a;
 749                                 ++b;
 750                         }
 751                 }
 752         }
 753         return pseries(relational(var,point), std::move(new_seq));
 754 }
 755
 756
 757 /** Implementation of ex::series() for sums. This performs series addition when
 758  *  adding pseries objects.
 759  *  @see ex::series */
 760 ex add::series(const relational & r, int order, unsigned options) const
 761 {
 762         ex acc; // Series accumulator
 763
 764         // Get first term from overall_coeff
 765         acc = overall_coeff.series(r, order, options);
 766
 767         // Add remaining terms
 768         for (auto & it : seq) {
 769                 ex op;
 770                 if (is_exactly_a<pseries>(it.rest))
 771                         op = it.rest;
 772                 else
 773                         op = it.rest.series(r, order, options);
 774                 if (!it.coeff.is_equal(_ex1))
 775                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
 776
 777                 // Series addition
 778                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 779         }
 780         return acc;
 781 }
 782
 783
 784 /** Multiply a pseries object with a numeric constant, producing a pseries
 785  *  object that represents the product.
 786  *
 787  *  @param other  constant to multiply with
 788  *  @return the product as a pseries */
 789 ex pseries::mul_const(const numeric &other) const
 790 {
 791         epvector new_seq;
 792         new_seq.reserve(seq.size());
 793
 794         for (auto & it : seq) {
 795                 if (!is_order_function(it.rest))
 796                         new_seq.push_back(expair(it.rest * other, it.coeff));
 797                 else
 798                         new_seq.push_back(it);
 799         }
 800         return pseries(relational(var,point), std::move(new_seq));
 801 }
 802
 803
 804 /** Multiply one pseries object to another, producing a pseries object that
 805  *  represents the product.
 806  *
 807  *  @param other  pseries object to multiply with
 808  *  @return the product as a pseries */
 809 ex pseries::mul_series(const pseries &other) const
 810 {
 811         // Multiplying two series with different variables or expansion points
 812         // results in an empty (constant) series
 813         if (!is_compatible_to(other)) {
 814                 epvector nul { expair(Order(_ex1), _ex0) };
 815                 return pseries(relational(var,point), std::move(nul));
 816         }
 817
 818         if (seq.empty() || other.seq.empty()) {
 819                 return dynallocate<pseries>(var==point, epvector());
 820         }
 821
 822         // Series multiplication
 823         epvector new_seq;
 824         const int a_max = degree(var);
 825         const int b_max = other.degree(var);
 826         const int a_min = ldegree(var);
 827         const int b_min = other.ldegree(var);
 828         const int cdeg_min = a_min + b_min;
 829         int cdeg_max = a_max + b_max;
 830
 831         int higher_order_a = std::numeric_limits<int>::max();
 832         int higher_order_b = std::numeric_limits<int>::max();
 833         if (is_order_function(coeff(var, a_max)))
 834                 higher_order_a = a_max + b_min;
 835         if (is_order_function(other.coeff(var, b_max)))
 836                 higher_order_b = b_max + a_min;
 837         const int higher_order_c = std::min(higher_order_a, higher_order_b);
 838         if (cdeg_max >= higher_order_c)
 839                 cdeg_max = higher_order_c - 1;
 840
 841         std::map<int, ex> rest_map_a, rest_map_b;
 842         for (const auto& it : seq)
 843                 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 844
 845         if (other.var.is_equal(var))
 846                 for (const auto& it : other.seq)
 847                         rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 848
 849         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 850                 ex co = _ex0;
 851                 // c(i)=a(0)b(i)+...+a(i)b(0)
 852                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 853                         const auto& ita = rest_map_a.find(i);
 854                         if (ita == rest_map_a.end())
 855                                 continue;
 856                         const auto& itb = rest_map_b.find(cdeg-i);
 857                         if (itb == rest_map_b.end())
 858                                 continue;
 859                         if (!is_order_function(ita->second) && !is_order_function(itb->second))
 860                                 co += ita->second * itb->second;
 861                 }
 862                 if (!co.is_zero())
 863                         new_seq.push_back(expair(co, numeric(cdeg)));
 864         }
 865         if (higher_order_c < std::numeric_limits<int>::max())
 866                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 867         return pseries(relational(var, point), std::move(new_seq));
 868 }
 869
 870
 871 /** Implementation of ex::series() for product. This performs series
 872  *  multiplication when multiplying series.
 873  *  @see ex::series */
 874 ex mul::series(const relational & r, int order, unsigned options) const
 875 {
 876         pseries acc; // Series accumulator
 877
 878         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 879         const ex& sym = r.lhs();
 880
 881         // holds ldegrees of the series of individual factors
 882         std::vector<int> ldegrees;
 883         std::vector<bool> ldegree_redo;
 884
 885         // find minimal degrees
 886         // first round: obtain a bound up to which minimal degrees have to be
 887         // considered
 888         for (auto & it : seq) {
 889
 890                 ex expon = it.coeff;
 891                 int factor = 1;
 892                 ex buf;
 893                 if (expon.info(info_flags::integer)) {
 894                         buf = it.rest;
 895                         factor = ex_to<numeric>(expon).to_int();
 896                 } else {
 897                         buf = recombine_pair_to_ex(it);
 898                 }
 899
 900                 int real_ldegree = 0;
 901                 bool flag_redo = false;
 902                 try {
 903                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 904                 } catch (std::runtime_error) {}
 905
 906                 if (real_ldegree == 0) {
 907                         if ( factor < 0 ) {
 908                                 // This case must terminate, otherwise we would have division by
 909                                 // zero.
 910                                 int orderloop = 0;
 911                                 do {
 912                                         orderloop++;
 913                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 914                                 } while (real_ldegree == orderloop);
 915                         } else {
 916                                 // Here it is possible that buf does not have a ldegree, therefore
 917                                 // check only if ldegree is negative, otherwise reconsider the case
 918                                 // in the second round.
 919                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 920                                 if (real_ldegree == 0)
 921                                         flag_redo = true;
 922                         }
 923                 }
 924
 925                 ldegrees.push_back(factor * real_ldegree);
 926                 ldegree_redo.push_back(flag_redo);
 927         }
 928
 929         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 930         // Second round: determine the remaining positive ldegrees by the series
 931         // method.
 932         // here we can ignore ldegrees larger than degbound
 933         size_t j = 0;
 934         for (auto & it : seq) {
 935                 if ( ldegree_redo[j] ) {
 936                         ex expon = it.coeff;
 937                         int factor = 1;
 938                         ex buf;
 939                         if (expon.info(info_flags::integer)) {
 940                                 buf = it.rest;
 941                                 factor = ex_to<numeric>(expon).to_int();
 942                         } else {
 943                                 buf = recombine_pair_to_ex(it);
 944                         }
 945                         int real_ldegree = 0;
 946                         int orderloop = 0;
 947                         do {
 948                                 orderloop++;
 949                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 950                         } while ((real_ldegree == orderloop)
 951                               && (factor*real_ldegree < degbound));
 952                         ldegrees[j] = factor * real_ldegree;
 953                         degbound -= factor * real_ldegree;
 954                 }
 955                 j++;
 956         }
 957
 958         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 959
 960         if (degsum > order) {
 961                 return dynallocate<pseries>(r, epvector{{Order(_ex1), order}});
 962         }
 963
 964         // Multiply with remaining terms
 965         auto itd = ldegrees.begin();
 966         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 967
 968                 // do series expansion with adjusted order
 969                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 970
 971                 // Series multiplication
 972                 if (it == seq.begin())
 973                         acc = ex_to<pseries>(op);
 974                 else
 975                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 976         }
 977
 978         return acc.mul_const(ex_to<numeric>(overall_coeff));
 979 }
 980
 981
 982 /** Compute the p-th power of a series.
 983  *
 984  *  @param p  power to compute
 985  *  @param deg  truncation order of series calculation */
 986 ex pseries::power_const(const numeric &p, int deg) const
 987 {
 988         // method:
 989         // (due to Leonhard Euler)
 990         // let A(x) be this series and for the time being let it start with a
 991         // constant (later we'll generalize):
 992         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 993         // We want to compute
 994         //     C(x) = A(x)^p
 995         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 996         // Taking the derivative on both sides and multiplying with A(x) one
 997         // immediately arrives at
 998         //     C'(x)*A(x) = p*C(x)*A'(x)
 999         // Multiplying this out and comparing coefficients we get the recurrence
1000         // formula
1001         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1002         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1003         // which can easily be solved given the starting value c_0 = (a_0)^p.
1004         // For the more general case where the leading coefficient of A(x) is not
1005         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1006         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
1007         // then of course x^(p*m) but the recurrence formula still holds.
1008
1009         if (seq.empty()) {
1010                 // as a special case, handle the empty (zero) series honoring the
1011                 // usual power laws such as implemented in power::eval()
1012                 if (p.real().is_zero())
1013                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1014                 else if (p.real().is_negative())
1015                         throw pole_error("pseries::power_const(): division by zero",1);
1016                 else
1017                         return *this;
1018         }
1019
1020         const int ldeg = ldegree(var);
1021         if (!(p*ldeg).is_integer())
1022                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1023
1024         // adjust number of coefficients
1025         int numcoeff = deg - (p*ldeg).to_int();
1026         if (numcoeff <= 0) {
1027                 epvector epv { expair(Order(_ex1), deg) };
1028                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1029         }
1030
1031         // O(x^n)^(-m) is undefined
1032         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1033                 throw pole_error("pseries::power_const(): division by zero",1);
1034
1035         // Compute coefficients of the powered series
1036         exvector co;
1037         co.reserve(numcoeff);
1038         co.push_back(pow(coeff(var, ldeg), p));
1039         for (int i=1; i<numcoeff; ++i) {
1040                 ex sum = _ex0;
1041                 for (int j=1; j<=i; ++j) {
1042                         ex c = coeff(var, j + ldeg);
1043                         if (is_order_function(c)) {
1044                                 co.push_back(Order(_ex1));
1045                                 break;
1046                         } else
1047                                 sum += (p * j - (i - j)) * co[i - j] * c;
1048                 }
1049                 co.push_back(sum / coeff(var, ldeg) / i);
1050         }
1051
1052         // Construct new series (of non-zero coefficients)
1053         epvector new_seq;
1054         bool higher_order = false;
1055         for (int i=0; i<numcoeff; ++i) {
1056                 if (!co[i].is_zero())
1057                         new_seq.push_back(expair(co[i], p * ldeg + i));
1058                 if (is_order_function(co[i])) {
1059                         higher_order = true;
1060                         break;
1061                 }
1062         }
1063         if (!higher_order)
1064                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1065
1066         return pseries(relational(var,point), std::move(new_seq));
1067 }
1068
1069
1070 /** Return a new pseries object with the powers shifted by deg. */
1071 pseries pseries::shift_exponents(int deg) const
1072 {
1073         epvector newseq = seq;
1074         for (auto & it : newseq)
1075                 it.coeff += deg;
1076         return pseries(relational(var, point), std::move(newseq));
1077 }
1078
1079
1080 /** Implementation of ex::series() for powers. This performs Laurent expansion
1081  *  of reciprocals of series at singularities.
1082  *  @see ex::series */
1083 ex power::series(const relational & r, int order, unsigned options) const
1084 {
1085         // If basis is already a series, just power it
1086         if (is_exactly_a<pseries>(basis))
1087                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1088
1089         // Basis is not a series, may there be a singularity?
1090         bool must_expand_basis = false;
1091         try {
1092                 basis.subs(r, subs_options::no_pattern);
1093         } catch (pole_error) {
1094                 must_expand_basis = true;
1095         }
1096
1097         bool exponent_is_regular = true;
1098         try {
1099                 exponent.subs(r, subs_options::no_pattern);
1100         } catch (pole_error) {
1101                 exponent_is_regular = false;
1102         }
1103
1104         if (!exponent_is_regular) {
1105                 ex l = exponent*log(basis);
1106                 // this == exp(l);
1107                 ex le = l.series(r, order, options);
1108                 // Note: expanding exp(l) won't help, since that will attempt
1109                 // Taylor expansion, and fail (because exponent is "singular")
1110                 // Still l itself might be expanded in Taylor series.
1111                 // Examples:
1112                 // sin(x)/x*log(cos(x))
1113                 // 1/x*log(1 + x)
1114                 return exp(le).series(r, order, options);
1115                 // Note: if l happens to have a Laurent expansion (with
1116                 // negative powers of (var - point)), expanding exp(le)
1117                 // will barf (which is The Right Thing).
1118         }
1119
1120         // Is the expression of type something^(-int)?
1121         if (!must_expand_basis && !exponent.info(info_flags::negint)
1122          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1123                 return basic::series(r, order, options);
1124
1125         // Is the expression of type 0^something?
1126         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1127          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1128                 return basic::series(r, order, options);
1129
1130         // Singularity encountered, is the basis equal to (var - point)?
1131         if (basis.is_equal(r.lhs() - r.rhs())) {
1132                 epvector new_seq;
1133                 if (ex_to<numeric>(exponent).to_int() < order)
1134                         new_seq.push_back(expair(_ex1, exponent));
1135                 else
1136                         new_seq.push_back(expair(Order(_ex1), exponent));
1137                 return pseries(r, std::move(new_seq));
1138         }
1139
1140         // No, expand basis into series
1141
1142         numeric numexp;
1143         if (is_a<numeric>(exponent)) {
1144                 numexp = ex_to<numeric>(exponent);
1145         } else {
1146                 numexp = 0;
1147         }
1148         const ex& sym = r.lhs();
1149         // find existing minimal degree
1150         ex eb = basis.expand();
1151         int real_ldegree = 0;
1152         if (eb.info(info_flags::rational_function))
1153                 real_ldegree = eb.ldegree(sym-r.rhs());
1154         if (real_ldegree == 0) {
1155                 int orderloop = 0;
1156                 do {
1157                         orderloop++;
1158                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1159                 } while (real_ldegree == orderloop);
1160         }
1161
1162         if (!(real_ldegree*numexp).is_integer())
1163                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1164         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1165
1166         ex result;
1167         try {
1168                 result = ex_to<pseries>(e).power_const(numexp, order);
1169         } catch (pole_error) {
1170                 epvector ser { expair(Order(_ex1), order) };
1171                 result = pseries(r, std::move(ser));
1172         }
1173
1174         return result;
1175 }
1176
1177
1178 /** Re-expansion of a pseries object. */
1179 ex pseries::series(const relational & r, int order, unsigned options) const
1180 {
1181         const ex p = r.rhs();
1182         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1183         const symbol &s = ex_to<symbol>(r.lhs());
1184
1185         if (var.is_equal(s) && point.is_equal(p)) {
1186                 if (order > degree(s))
1187                         return *this;
1188                 else {
1189                         epvector new_seq;
1190                         for (auto & it : seq) {
1191                                 int o = ex_to<numeric>(it.coeff).to_int();
1192                                 if (o >= order) {
1193                                         new_seq.push_back(expair(Order(_ex1), o));
1194                                         break;
1195                                 }
1196                                 new_seq.push_back(it);
1197                         }
1198                         return pseries(r, std::move(new_seq));
1199                 }
1200         } else
1201                 return convert_to_poly().series(r, order, options);
1202 }
1203
1204 ex integral::series(const relational & r, int order, unsigned options) const
1205 {
1206         if (x.subs(r) != x)
1207                 throw std::logic_error("Cannot series expand wrt dummy variable");
1208
1209         // Expanding integrand with r substituted taken in boundaries.
1210         ex fseries = f.series(r, order, options);
1211         epvector fexpansion;
1212         fexpansion.reserve(fseries.nops());
1213         for (size_t i=0; i<fseries.nops(); ++i) {
1214                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1215                 currcoeff = (currcoeff == Order(_ex1))
1216                         ? currcoeff
1217                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1218                 if (currcoeff != 0)
1219                         fexpansion.push_back(
1220                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1221         }
1222
1223         // Expanding lower boundary
1224         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1225         ex aseries = (a-a.subs(r)).series(r, order, options);
1226         fseries = f.series(x == (a.subs(r)), order, options);
1227         for (size_t i=0; i<fseries.nops(); ++i) {
1228                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1229                 if (is_order_function(currcoeff))
1230                         break;
1231                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1232                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1233                 currcoeff = currcoeff.series(r, orderforf);
1234                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1235                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1236                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1237                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1238         }
1239
1240         // Expanding upper boundary
1241         ex bseries = (b-b.subs(r)).series(r, order, options);
1242         fseries = f.series(x == (b.subs(r)), order, options);
1243         for (size_t i=0; i<fseries.nops(); ++i) {
1244                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1245                 if (is_order_function(currcoeff))
1246                         break;
1247                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1248                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1249                 currcoeff = currcoeff.series(r, orderforf);
1250                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1251                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1252                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1253                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1254         }
1255
1256         return result;
1257 }
1258
1259
1260 /** Compute the truncated series expansion of an expression.
1261  *  This function returns an expression containing an object of class pseries
1262  *  to represent the series. If the series does not terminate within the given
1263  *  truncation order, the last term of the series will be an order term.
1264  *
1265  *  @param r  expansion relation, lhs holds variable and rhs holds point
1266  *  @param order  truncation order of series calculations
1267  *  @param options  of class series_options
1268  *  @return an expression holding a pseries object */
1269 ex ex::series(const ex & r, int order, unsigned options) const
1270 {
1271         ex e;
1272         relational rel_;
1273
1274         if (is_a<relational>(r))
1275                 rel_ = ex_to<relational>(r);
1276         else if (is_a<symbol>(r))
1277                 rel_ = relational(r,_ex0);
1278         else
1279                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1280
1281         e = bp->series(rel_, order, options);
1282         return e;
1283 }
1284
1285 GINAC_BIND_UNARCHIVER(pseries);
1286
1287 } // namespace GiNaC