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) 1999-2011 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_) : seq(ops_)
  72 {
  73         GINAC_ASSERT(is_a<relational>(rel_));
  74         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  75         point = rel_.rhs();
  76         var = rel_.lhs();
  77 }
  78
  79
  80 /*
  81  *  Archiving
  82  */
  83
  84 void pseries::read_archive(const archive_node &n, lst &sym_lst)
  85 {
  86         inherited::read_archive(n, sym_lst);
  87         archive_node::archive_node_cit first = n.find_first("coeff");
  88         archive_node::archive_node_cit last = n.find_last("power");
  89         ++last;
  90         seq.reserve((last-first)/2);
  91
  92         for (archive_node::archive_node_cit loc = first; loc < last;) {
  93                 ex rest;
  94                 ex coeff;
  95                 n.find_ex_by_loc(loc++, rest, sym_lst);
  96                 n.find_ex_by_loc(loc++, coeff, sym_lst);
  97                 seq.push_back(expair(rest, coeff));
  98         }
  99
 100         n.find_ex("var", var, sym_lst);
 101         n.find_ex("point", point, sym_lst);
 102 }
 103
 104 void pseries::archive(archive_node &n) const
 105 {
 106         inherited::archive(n);
 107         epvector::const_iterator i = seq.begin(), iend = seq.end();
 108         while (i != iend) {
 109                 n.add_ex("coeff", i->rest);
 110                 n.add_ex("power", i->coeff);
 111                 ++i;
 112         }
 113         n.add_ex("var", var);
 114         n.add_ex("point", point);
 115 }
 116
 117
 118 //////////
 119 // functions overriding virtual functions from base classes
 120 //////////
 121
 122 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
 123 {
 124         if (precedence() <= level)
 125                 c.s << '(';
 126
 127         // objects of type pseries must not have any zero entries, so the
 128         // trivial (zero) pseries needs a special treatment here:
 129         if (seq.empty())
 130                 c.s << '0';
 131
 132         epvector::const_iterator i = seq.begin(), end = seq.end();
 133         while (i != end) {
 134
 135                 // print a sign, if needed
 136                 if (i != seq.begin())
 137                         c.s << '+';
 138
 139                 if (!is_order_function(i->rest)) {
 140
 141                         // print 'rest', i.e. the expansion coefficient
 142                         if (i->rest.info(info_flags::numeric) &&
 143                                 i->rest.info(info_flags::positive)) {
 144                                 i->rest.print(c);
 145                         } else {
 146                                 c.s << openbrace << '(';
 147                                 i->rest.print(c);
 148                                 c.s << ')' << closebrace;
 149                         }
 150
 151                         // print 'coeff', something like (x-1)^42
 152                         if (!i->coeff.is_zero()) {
 153                                 c.s << mul_sym;
 154                                 if (!point.is_zero()) {
 155                                         c.s << openbrace << '(';
 156                                         (var-point).print(c);
 157                                         c.s << ')' << closebrace;
 158                                 } else
 159                                         var.print(c);
 160                                 if (i->coeff.compare(_ex1)) {
 161                                         c.s << pow_sym;
 162                                         c.s << openbrace;
 163                                         if (i->coeff.info(info_flags::negative)) {
 164                                                 c.s << '(';
 165                                                 i->coeff.print(c);
 166                                                 c.s << ')';
 167                                         } else
 168                                                 i->coeff.print(c);
 169                                         c.s << closebrace;
 170                                 }
 171                         }
 172                 } else
 173                         Order(power(var-point,i->coeff)).print(c);
 174                 ++i;
 175         }
 176
 177         if (precedence() <= level)
 178                 c.s << ')';
 179 }
 180
 181 void pseries::do_print(const print_context & c, unsigned level) const
 182 {
 183         print_series(c, "", "", "*", "^", level);
 184 }
 185
 186 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 187 {
 188         print_series(c, "{", "}", " ", "^", level);
 189 }
 190
 191 void pseries::do_print_python(const print_python & c, unsigned level) const
 192 {
 193         print_series(c, "", "", "*", "**", level);
 194 }
 195
 196 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 197 {
 198         c.s << std::string(level, ' ') << class_name() << " @" << this
 199             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 200             << std::endl;
 201         size_t num = seq.size();
 202         for (size_t i=0; i<num; ++i) {
 203                 seq[i].rest.print(c, level + c.delta_indent);
 204                 seq[i].coeff.print(c, level + c.delta_indent);
 205                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 206         }
 207         var.print(c, level + c.delta_indent);
 208         point.print(c, level + c.delta_indent);
 209 }
 210
 211 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 212 {
 213         c.s << class_name() << "(relational(";
 214         var.print(c);
 215         c.s << ',';
 216         point.print(c);
 217         c.s << "),[";
 218         size_t num = seq.size();
 219         for (size_t i=0; i<num; ++i) {
 220                 if (i)
 221                         c.s << ',';
 222                 c.s << '(';
 223                 seq[i].rest.print(c);
 224                 c.s << ',';
 225                 seq[i].coeff.print(c);
 226                 c.s << ')';
 227         }
 228         c.s << "])";
 229 }
 230
 231 int pseries::compare_same_type(const basic & other) const
 232 {
 233         GINAC_ASSERT(is_a<pseries>(other));
 234         const pseries &o = static_cast<const pseries &>(other);
 235
 236         // first compare the lengths of the series...
 237         if (seq.size()>o.seq.size())
 238                 return 1;
 239         if (seq.size()<o.seq.size())
 240                 return -1;
 241
 242         // ...then the expansion point...
 243         int cmpval = var.compare(o.var);
 244         if (cmpval)
 245                 return cmpval;
 246         cmpval = point.compare(o.point);
 247         if (cmpval)
 248                 return cmpval;
 249
 250         // ...and if that failed the individual elements
 251         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 252         while (it!=seq.end() && o_it!=o.seq.end()) {
 253                 cmpval = it->compare(*o_it);
 254                 if (cmpval)
 255                         return cmpval;
 256                 ++it;
 257                 ++o_it;
 258         }
 259
 260         // so they are equal.
 261         return 0;
 262 }
 263
 264 /** Return the number of operands including a possible order term. */
 265 size_t pseries::nops() const
 266 {
 267         return seq.size();
 268 }
 269
 270 /** Return the ith term in the series when represented as a sum. */
 271 ex pseries::op(size_t i) const
 272 {
 273         if (i >= seq.size())
 274                 throw (std::out_of_range("op() out of range"));
 275
 276         if (is_order_function(seq[i].rest))
 277                 return Order(power(var-point, seq[i].coeff));
 278         return seq[i].rest * power(var - point, seq[i].coeff);
 279 }
 280
 281 /** Return degree of highest power of the series.  This is usually the exponent
 282  *  of the Order term.  If s is not the expansion variable of the series, the
 283  *  series is examined termwise. */
 284 int pseries::degree(const ex &s) const
 285 {
 286         if (var.is_equal(s)) {
 287                 // Return last exponent
 288                 if (seq.size())
 289                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 290                 else
 291                         return 0;
 292         } else {
 293                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 294                 if (it == itend)
 295                         return 0;
 296                 int max_pow = std::numeric_limits<int>::min();
 297                 while (it != itend) {
 298                         int pow = it->rest.degree(s);
 299                         if (pow > max_pow)
 300                                 max_pow = pow;
 301                         ++it;
 302                 }
 303                 return max_pow;
 304         }
 305 }
 306
 307 /** Return degree of lowest power of the series.  This is usually the exponent
 308  *  of the leading term.  If s is not the expansion variable of the series, the
 309  *  series is examined termwise.  If s is the expansion variable but the
 310  *  expansion point is not zero the series is not expanded to find the degree.
 311  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 312 int pseries::ldegree(const ex &s) const
 313 {
 314         if (var.is_equal(s)) {
 315                 // Return first exponent
 316                 if (seq.size())
 317                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 318                 else
 319                         return 0;
 320         } else {
 321                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 322                 if (it == itend)
 323                         return 0;
 324                 int min_pow = std::numeric_limits<int>::max();
 325                 while (it != itend) {
 326                         int pow = it->rest.ldegree(s);
 327                         if (pow < min_pow)
 328                                 min_pow = pow;
 329                         ++it;
 330                 }
 331                 return min_pow;
 332         }
 333 }
 334
 335 /** Return coefficient of degree n in power series if s is the expansion
 336  *  variable.  If the expansion point is nonzero, by definition the n=1
 337  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 338  *  the expansion took place in the s in the first place).
 339  *  If s is not the expansion variable, an attempt is made to convert the
 340  *  series to a polynomial and return the corresponding coefficient from
 341  *  there. */
 342 ex pseries::coeff(const ex &s, int n) const
 343 {
 344         if (var.is_equal(s)) {
 345                 if (seq.empty())
 346                         return _ex0;
 347
 348                 // Binary search in sequence for given power
 349                 numeric looking_for = numeric(n);
 350                 int lo = 0, hi = seq.size() - 1;
 351                 while (lo <= hi) {
 352                         int mid = (lo + hi) / 2;
 353                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 354                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 355                         switch (cmp) {
 356                                 case -1:
 357                                         lo = mid + 1;
 358                                         break;
 359                                 case 0:
 360                                         return seq[mid].rest;
 361                                 case 1:
 362                                         hi = mid - 1;
 363                                         break;
 364                                 default:
 365                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 366                         }
 367                 }
 368                 return _ex0;
 369         } else
 370                 return convert_to_poly().coeff(s, n);
 371 }
 372
 373 /** Does nothing. */
 374 ex pseries::collect(const ex &s, bool distributed) const
 375 {
 376         return *this;
 377 }
 378
 379 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 380 ex pseries::eval(int level) const
 381 {
 382         if (level == 1)
 383                 return this->hold();
 384
 385         if (level == -max_recursion_level)
 386                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 387
 388         // Construct a new series with evaluated coefficients
 389         epvector new_seq;
 390         new_seq.reserve(seq.size());
 391         epvector::const_iterator it = seq.begin(), itend = seq.end();
 392         while (it != itend) {
 393                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 394                 ++it;
 395         }
 396         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 397 }
 398
 399 /** Evaluate coefficients numerically. */
 400 ex pseries::evalf(int level) const
 401 {
 402         if (level == 1)
 403                 return *this;
 404
 405         if (level == -max_recursion_level)
 406                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 407
 408         // Construct a new series with evaluated coefficients
 409         epvector new_seq;
 410         new_seq.reserve(seq.size());
 411         epvector::const_iterator it = seq.begin(), itend = seq.end();
 412         while (it != itend) {
 413                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 414                 ++it;
 415         }
 416         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 417 }
 418
 419 ex pseries::conjugate() const
 420 {
 421         if(!var.info(info_flags::real))
 422                 return conjugate_function(*this).hold();
 423
 424         epvector * newseq = conjugateepvector(seq);
 425         ex newpoint = point.conjugate();
 426
 427         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 428                 return *this;
 429         }
 430
 431         ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
 432         delete newseq;
 433         return result;
 434 }
 435
 436 ex pseries::real_part() const
 437 {
 438         if(!var.info(info_flags::real))
 439                 return real_part_function(*this).hold();
 440         ex newpoint = point.real_part();
 441         if(newpoint != point)
 442                 return real_part_function(*this).hold();
 443
 444         epvector v;
 445         v.reserve(seq.size());
 446         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 447                 v.push_back(expair((i->rest).real_part(), i->coeff));
 448         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 449 }
 450
 451 ex pseries::imag_part() const
 452 {
 453         if(!var.info(info_flags::real))
 454                 return imag_part_function(*this).hold();
 455         ex newpoint = point.real_part();
 456         if(newpoint != point)
 457                 return imag_part_function(*this).hold();
 458
 459         epvector v;
 460         v.reserve(seq.size());
 461         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 462                 v.push_back(expair((i->rest).imag_part(), i->coeff));
 463         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 464 }
 465
 466 ex pseries::eval_integ() const
 467 {
 468         epvector *newseq = NULL;
 469         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 470                 if (newseq) {
 471                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 472                         continue;
 473                 }
 474                 ex newterm = i->rest.eval_integ();
 475                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 476                         newseq = new epvector;
 477                         newseq->reserve(seq.size());
 478                         for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
 479                                 newseq->push_back(*j);
 480                         newseq->push_back(expair(newterm, i->coeff));
 481                 }
 482         }
 483
 484         ex newpoint = point.eval_integ();
 485         if (newseq || !are_ex_trivially_equal(newpoint, point))
 486                 return (new pseries(var==newpoint, *newseq))
 487                        ->setflag(status_flags::dynallocated);
 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 (epvector::const_iterator 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                 }
 502                 else {
 503                         ex newcoeff = i->rest.evalm();
 504                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 505                                 something_changed = true;
 506                                 newseq.reserve(seq.size());
 507                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 508                                 if (!newcoeff.is_zero())
 509                                         newseq.push_back(expair(newcoeff, i->coeff));
 510                         }
 511                 }
 512         }
 513         if (something_changed)
 514                 return (new pseries(var==point, newseq))->setflag(status_flags::dynallocated);
 515         else
 516                 return *this;
 517 }
 518
 519 ex pseries::subs(const exmap & m, unsigned options) const
 520 {
 521         // If expansion variable is being substituted, convert the series to a
 522         // polynomial and do the substitution there because the result might
 523         // no longer be a power series
 524         if (m.find(var) != m.end())
 525                 return convert_to_poly(true).subs(m, options);
 526
 527         // Otherwise construct a new series with substituted coefficients and
 528         // expansion point
 529         epvector newseq;
 530         newseq.reserve(seq.size());
 531         epvector::const_iterator it = seq.begin(), itend = seq.end();
 532         while (it != itend) {
 533                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
 534                 ++it;
 535         }
 536         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
 537 }
 538
 539 /** Implementation of ex::expand() for a power series.  It expands all the
 540  *  terms individually and returns the resulting series as a new pseries. */
 541 ex pseries::expand(unsigned options) const
 542 {
 543         epvector newseq;
 544         epvector::const_iterator i = seq.begin(), end = seq.end();
 545         while (i != end) {
 546                 ex restexp = i->rest.expand();
 547                 if (!restexp.is_zero())
 548                         newseq.push_back(expair(restexp, i->coeff));
 549                 ++i;
 550         }
 551         return (new pseries(relational(var,point), newseq))
 552                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 553 }
 554
 555 /** Implementation of ex::diff() for a power series.
 556  *  @see ex::diff */
 557 ex pseries::derivative(const symbol & s) const
 558 {
 559         epvector new_seq;
 560         epvector::const_iterator it = seq.begin(), itend = seq.end();
 561
 562         if (s == var) {
 563
 564                 // FIXME: coeff might depend on var
 565                 while (it != itend) {
 566                         if (is_order_function(it->rest)) {
 567                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 568                         } else {
 569                                 ex c = it->rest * it->coeff;
 570                                 if (!c.is_zero())
 571                                         new_seq.push_back(expair(c, it->coeff - 1));
 572                         }
 573                         ++it;
 574                 }
 575
 576         } else {
 577
 578                 while (it != itend) {
 579                         if (is_order_function(it->rest)) {
 580                                 new_seq.push_back(*it);
 581                         } else {
 582                                 ex c = it->rest.diff(s);
 583                                 if (!c.is_zero())
 584                                         new_seq.push_back(expair(c, it->coeff));
 585                         }
 586                         ++it;
 587                 }
 588         }
 589
 590         return pseries(relational(var,point), new_seq);
 591 }
 592
 593 ex pseries::convert_to_poly(bool no_order) const
 594 {
 595         ex e;
 596         epvector::const_iterator it = seq.begin(), itend = seq.end();
 597
 598         while (it != itend) {
 599                 if (is_order_function(it->rest)) {
 600                         if (!no_order)
 601                                 e += Order(power(var - point, it->coeff));
 602                 } else
 603                         e += it->rest * power(var - point, it->coeff);
 604                 ++it;
 605         }
 606         return e;
 607 }
 608
 609 bool pseries::is_terminating() const
 610 {
 611         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 612 }
 613
 614 ex pseries::coeffop(size_t i) const
 615 {
 616         if (i >=nops())
 617                 throw (std::out_of_range("coeffop() out of range"));
 618         return seq[i].rest;
 619 }
 620
 621 ex pseries::exponop(size_t i) const
 622 {
 623         if (i >= nops())
 624                 throw (std::out_of_range("exponop() out of range"));
 625         return seq[i].coeff;
 626 }
 627
 628
 629 /*
 630  *  Implementations of series expansion
 631  */
 632
 633 /** Default implementation of ex::series(). This performs Taylor expansion.
 634  *  @see ex::series */
 635 ex basic::series(const relational & r, int order, unsigned options) const
 636 {
 637         epvector seq;
 638         const symbol &s = ex_to<symbol>(r.lhs());
 639
 640         // default for order-values that make no sense for Taylor expansion
 641         if ((order <= 0) && this->has(s)) {
 642                 seq.push_back(expair(Order(_ex1), order));
 643                 return pseries(r, seq);
 644         }
 645
 646         // do Taylor expansion
 647         numeric fac = 1;
 648         ex deriv = *this;
 649         ex coeff = deriv.subs(r, subs_options::no_pattern);
 650
 651         if (!coeff.is_zero()) {
 652                 seq.push_back(expair(coeff, _ex0));
 653         }
 654
 655         int n;
 656         for (n=1; n<order; ++n) {
 657                 fac = fac.mul(n);
 658                 // We need to test for zero in order to see if the series terminates.
 659                 // The problem is that there is no such thing as a perfect test for
 660                 // zero.  Expanding the term occasionally helps a little...
 661                 deriv = deriv.diff(s).expand();
 662                 if (deriv.is_zero())  // Series terminates
 663                         return pseries(r, seq);
 664
 665                 coeff = deriv.subs(r, subs_options::no_pattern);
 666                 if (!coeff.is_zero())
 667                         seq.push_back(expair(fac.inverse() * coeff, n));
 668         }
 669
 670         // Higher-order terms, if present
 671         deriv = deriv.diff(s);
 672         if (!deriv.expand().is_zero())
 673                 seq.push_back(expair(Order(_ex1), n));
 674         return pseries(r, seq);
 675 }
 676
 677
 678 /** Implementation of ex::series() for symbols.
 679  *  @see ex::series */
 680 ex symbol::series(const relational & r, int order, unsigned options) const
 681 {
 682         epvector seq;
 683         const ex point = r.rhs();
 684         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 685
 686         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 687                 if (order > 0 && !point.is_zero())
 688                         seq.push_back(expair(point, _ex0));
 689                 if (order > 1)
 690                         seq.push_back(expair(_ex1, _ex1));
 691                 else
 692                         seq.push_back(expair(Order(_ex1), numeric(order)));
 693         } else
 694                 seq.push_back(expair(*this, _ex0));
 695         return pseries(r, seq);
 696 }
 697
 698
 699 /** Add one series object to another, producing a pseries object that
 700  *  represents the sum.
 701  *
 702  *  @param other  pseries object to add with
 703  *  @return the sum as a pseries */
 704 ex pseries::add_series(const pseries &other) const
 705 {
 706         // Adding two series with different variables or expansion points
 707         // results in an empty (constant) series
 708         if (!is_compatible_to(other)) {
 709                 epvector nul;
 710                 nul.push_back(expair(Order(_ex1), _ex0));
 711                 return pseries(relational(var,point), nul);
 712         }
 713
 714         // Series addition
 715         epvector new_seq;
 716         epvector::const_iterator a = seq.begin();
 717         epvector::const_iterator b = other.seq.begin();
 718         epvector::const_iterator a_end = seq.end();
 719         epvector::const_iterator b_end = other.seq.end();
 720         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 721         for (;;) {
 722                 // If a is empty, fill up with elements from b and stop
 723                 if (a == a_end) {
 724                         while (b != b_end) {
 725                                 new_seq.push_back(*b);
 726                                 ++b;
 727                         }
 728                         break;
 729                 } else
 730                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 731
 732                 // If b is empty, fill up with elements from a and stop
 733                 if (b == b_end) {
 734                         while (a != a_end) {
 735                                 new_seq.push_back(*a);
 736                                 ++a;
 737                         }
 738                         break;
 739                 } else
 740                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 741
 742                 // a and b are non-empty, compare powers
 743                 if (pow_a < pow_b) {
 744                         // a has lesser power, get coefficient from a
 745                         new_seq.push_back(*a);
 746                         if (is_order_function((*a).rest))
 747                                 break;
 748                         ++a;
 749                 } else if (pow_b < pow_a) {
 750                         // b has lesser power, get coefficient from b
 751                         new_seq.push_back(*b);
 752                         if (is_order_function((*b).rest))
 753                                 break;
 754                         ++b;
 755                 } else {
 756                         // Add coefficient of a and b
 757                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 758                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 759                                 break;  // Order term ends the sequence
 760                         } else {
 761                                 ex sum = (*a).rest + (*b).rest;
 762                                 if (!(sum.is_zero()))
 763                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 764                                 ++a;
 765                                 ++b;
 766                         }
 767                 }
 768         }
 769         return pseries(relational(var,point), new_seq);
 770 }
 771
 772
 773 /** Implementation of ex::series() for sums. This performs series addition when
 774  *  adding pseries objects.
 775  *  @see ex::series */
 776 ex add::series(const relational & r, int order, unsigned options) const
 777 {
 778         ex acc; // Series accumulator
 779
 780         // Get first term from overall_coeff
 781         acc = overall_coeff.series(r, order, options);
 782
 783         // Add remaining terms
 784         epvector::const_iterator it = seq.begin();
 785         epvector::const_iterator itend = seq.end();
 786         for (; it!=itend; ++it) {
 787                 ex op;
 788                 if (is_exactly_a<pseries>(it->rest))
 789                         op = it->rest;
 790                 else
 791                         op = it->rest.series(r, order, options);
 792                 if (!it->coeff.is_equal(_ex1))
 793                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 794
 795                 // Series addition
 796                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 797         }
 798         return acc;
 799 }
 800
 801
 802 /** Multiply a pseries object with a numeric constant, producing a pseries
 803  *  object that represents the product.
 804  *
 805  *  @param other  constant to multiply with
 806  *  @return the product as a pseries */
 807 ex pseries::mul_const(const numeric &other) const
 808 {
 809         epvector new_seq;
 810         new_seq.reserve(seq.size());
 811
 812         epvector::const_iterator it = seq.begin(), itend = seq.end();
 813         while (it != itend) {
 814                 if (!is_order_function(it->rest))
 815                         new_seq.push_back(expair(it->rest * other, it->coeff));
 816                 else
 817                         new_seq.push_back(*it);
 818                 ++it;
 819         }
 820         return pseries(relational(var,point), new_seq);
 821 }
 822
 823
 824 /** Multiply one pseries object to another, producing a pseries object that
 825  *  represents the product.
 826  *
 827  *  @param other  pseries object to multiply with
 828  *  @return the product as a pseries */
 829 ex pseries::mul_series(const pseries &other) const
 830 {
 831         // Multiplying two series with different variables or expansion points
 832         // results in an empty (constant) series
 833         if (!is_compatible_to(other)) {
 834                 epvector nul;
 835                 nul.push_back(expair(Order(_ex1), _ex0));
 836                 return pseries(relational(var,point), nul);
 837         }
 838
 839         if (seq.empty() || other.seq.empty()) {
 840                 return (new pseries(var==point, epvector()))
 841                        ->setflag(status_flags::dynallocated);
 842         }
 843
 844         // Series multiplication
 845         epvector new_seq;
 846         int a_max = degree(var);
 847         int b_max = other.degree(var);
 848         int a_min = ldegree(var);
 849         int b_min = other.ldegree(var);
 850         int cdeg_min = a_min + b_min;
 851         int cdeg_max = a_max + b_max;
 852
 853         int higher_order_a = std::numeric_limits<int>::max();
 854         int higher_order_b = std::numeric_limits<int>::max();
 855         if (is_order_function(coeff(var, a_max)))
 856                 higher_order_a = a_max + b_min;
 857         if (is_order_function(other.coeff(var, b_max)))
 858                 higher_order_b = b_max + a_min;
 859         int higher_order_c = std::min(higher_order_a, higher_order_b);
 860         if (cdeg_max >= higher_order_c)
 861                 cdeg_max = higher_order_c - 1;
 862
 863         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 864                 ex co = _ex0;
 865                 // c(i)=a(0)b(i)+...+a(i)b(0)
 866                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 867                         ex a_coeff = coeff(var, i);
 868                         ex b_coeff = other.coeff(var, cdeg-i);
 869                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 870                                 co += a_coeff * b_coeff;
 871                 }
 872                 if (!co.is_zero())
 873                         new_seq.push_back(expair(co, numeric(cdeg)));
 874         }
 875         if (higher_order_c < std::numeric_limits<int>::max())
 876                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 877         return pseries(relational(var, point), new_seq);
 878 }
 879
 880
 881 /** Implementation of ex::series() for product. This performs series
 882  *  multiplication when multiplying series.
 883  *  @see ex::series */
 884 ex mul::series(const relational & r, int order, unsigned options) const
 885 {
 886         pseries acc; // Series accumulator
 887
 888         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 889         const ex& sym = r.lhs();
 890
 891         // holds ldegrees of the series of individual factors
 892         std::vector<int> ldegrees;
 893         std::vector<bool> ldegree_redo;
 894
 895         // find minimal degrees
 896         const epvector::const_iterator itbeg = seq.begin();
 897         const epvector::const_iterator itend = seq.end();
 898         // first round: obtain a bound up to which minimal degrees have to be
 899         // considered
 900         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 901
 902                 ex expon = it->coeff;
 903                 int factor = 1;
 904                 ex buf;
 905                 if (expon.info(info_flags::integer)) {
 906                         buf = it->rest;
 907                         factor = ex_to<numeric>(expon).to_int();
 908                 } else {
 909                         buf = recombine_pair_to_ex(*it);
 910                 }
 911
 912                 int real_ldegree = 0;
 913                 bool flag_redo = false;
 914                 try {
 915                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 916                 } catch (std::runtime_error) {}
 917
 918                 if (real_ldegree == 0) {
 919                         if ( factor < 0 ) {
 920                                 // This case must terminate, otherwise we would have division by
 921                                 // zero.
 922                                 int orderloop = 0;
 923                                 do {
 924                                         orderloop++;
 925                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 926                                 } while (real_ldegree == orderloop);
 927                         } else {
 928                                 // Here it is possible that buf does not have a ldegree, therefore
 929                                 // check only if ldegree is negative, otherwise reconsider the case
 930                                 // in the second round.
 931                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 932                                 if (real_ldegree == 0)
 933                                         flag_redo = true;
 934                         }
 935                 }
 936
 937                 ldegrees.push_back(factor * real_ldegree);
 938                 ldegree_redo.push_back(flag_redo);
 939         }
 940
 941         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 942         // Second round: determine the remaining positive ldegrees by the series
 943         // method.
 944         // here we can ignore ldegrees larger than degbound
 945         size_t j = 0;
 946         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 947                 if ( ldegree_redo[j] ) {
 948                         ex expon = it->coeff;
 949                         int factor = 1;
 950                         ex buf;
 951                         if (expon.info(info_flags::integer)) {
 952                                 buf = it->rest;
 953                                 factor = ex_to<numeric>(expon).to_int();
 954                         } else {
 955                                 buf = recombine_pair_to_ex(*it);
 956                         }
 957                         int real_ldegree = 0;
 958                         int orderloop = 0;
 959                         do {
 960                                 orderloop++;
 961                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 962                         } while ((real_ldegree == orderloop)
 963                                         && ( factor*real_ldegree < degbound));
 964                         ldegrees[j] = factor * real_ldegree;
 965                         degbound -= factor * real_ldegree;
 966                 }
 967                 j++;
 968         }
 969
 970         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 971
 972         if (degsum >= order) {
 973                 epvector epv;
 974                 epv.push_back(expair(Order(_ex1), order));
 975                 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
 976         }
 977
 978         // Multiply with remaining terms
 979         std::vector<int>::const_iterator itd = ldegrees.begin();
 980         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
 981
 982                 // do series expansion with adjusted order
 983                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 984
 985                 // Series multiplication
 986                 if (it == itbeg)
 987                         acc = ex_to<pseries>(op);
 988                 else
 989                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 990         }
 991
 992         return acc.mul_const(ex_to<numeric>(overall_coeff));
 993 }
 994
 995
 996 /** Compute the p-th power of a series.
 997  *
 998  *  @param p  power to compute
 999  *  @param deg  truncation order of series calculation */
1000 ex pseries::power_const(const numeric &p, int deg) const
1001 {
1002         // method:
1003         // (due to Leonhard Euler)
1004         // let A(x) be this series and for the time being let it start with a
1005         // constant (later we'll generalize):
1006         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
1007         // We want to compute
1008         //     C(x) = A(x)^p
1009         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
1010         // Taking the derivative on both sides and multiplying with A(x) one
1011         // immediately arrives at
1012         //     C'(x)*A(x) = p*C(x)*A'(x)
1013         // Multiplying this out and comparing coefficients we get the recurrence
1014         // formula
1015         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1016         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1017         // which can easily be solved given the starting value c_0 = (a_0)^p.
1018         // For the more general case where the leading coefficient of A(x) is not
1019         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1020         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
1021         // then of course x^(p*m) but the recurrence formula still holds.
1022
1023         if (seq.empty()) {
1024                 // as a special case, handle the empty (zero) series honoring the
1025                 // usual power laws such as implemented in power::eval()
1026                 if (p.real().is_zero())
1027                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1028                 else if (p.real().is_negative())
1029                         throw pole_error("pseries::power_const(): division by zero",1);
1030                 else
1031                         return *this;
1032         }
1033
1034         const int ldeg = ldegree(var);
1035         if (!(p*ldeg).is_integer())
1036                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1037
1038         // adjust number of coefficients
1039         int numcoeff = deg - (p*ldeg).to_int();
1040         if (numcoeff <= 0) {
1041                 epvector epv;
1042                 epv.reserve(1);
1043                 epv.push_back(expair(Order(_ex1), deg));
1044                 return (new pseries(relational(var,point), epv))
1045                        ->setflag(status_flags::dynallocated);
1046         }
1047
1048         // O(x^n)^(-m) is undefined
1049         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1050                 throw pole_error("pseries::power_const(): division by zero",1);
1051
1052         // Compute coefficients of the powered series
1053         exvector co;
1054         co.reserve(numcoeff);
1055         co.push_back(power(coeff(var, ldeg), p));
1056         for (int i=1; i<numcoeff; ++i) {
1057                 ex sum = _ex0;
1058                 for (int j=1; j<=i; ++j) {
1059                         ex c = coeff(var, j + ldeg);
1060                         if (is_order_function(c)) {
1061                                 co.push_back(Order(_ex1));
1062                                 break;
1063                         } else
1064                                 sum += (p * j - (i - j)) * co[i - j] * c;
1065                 }
1066                 co.push_back(sum / coeff(var, ldeg) / i);
1067         }
1068
1069         // Construct new series (of non-zero coefficients)
1070         epvector new_seq;
1071         bool higher_order = false;
1072         for (int i=0; i<numcoeff; ++i) {
1073                 if (!co[i].is_zero())
1074                         new_seq.push_back(expair(co[i], p * ldeg + i));
1075                 if (is_order_function(co[i])) {
1076                         higher_order = true;
1077                         break;
1078                 }
1079         }
1080         if (!higher_order)
1081                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1082
1083         return pseries(relational(var,point), new_seq);
1084 }
1085
1086
1087 /** Return a new pseries object with the powers shifted by deg. */
1088 pseries pseries::shift_exponents(int deg) const
1089 {
1090         epvector newseq = seq;
1091         epvector::iterator i = newseq.begin(), end  = newseq.end();
1092         while (i != end) {
1093                 i->coeff += deg;
1094                 ++i;
1095         }
1096         return pseries(relational(var, point), newseq);
1097 }
1098
1099
1100 /** Implementation of ex::series() for powers. This performs Laurent expansion
1101  *  of reciprocals of series at singularities.
1102  *  @see ex::series */
1103 ex power::series(const relational & r, int order, unsigned options) const
1104 {
1105         // If basis is already a series, just power it
1106         if (is_exactly_a<pseries>(basis))
1107                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1108
1109         // Basis is not a series, may there be a singularity?
1110         bool must_expand_basis = false;
1111         try {
1112                 basis.subs(r, subs_options::no_pattern);
1113         } catch (pole_error) {
1114                 must_expand_basis = true;
1115         }
1116
1117         bool exponent_is_regular = true;
1118         try {
1119                 exponent.subs(r, subs_options::no_pattern);
1120         } catch (pole_error) {
1121                 exponent_is_regular = false;
1122         }
1123
1124         if (!exponent_is_regular) {
1125                 ex l = exponent*log(basis);
1126                 // this == exp(l);
1127                 ex le = l.series(r, order, options);
1128                 // Note: expanding exp(l) won't help, since that will attempt
1129                 // Taylor expansion, and fail (because exponent is "singular")
1130                 // Still l itself might be expanded in Taylor series.
1131                 // Examples:
1132                 // sin(x)/x*log(cos(x))
1133                 // 1/x*log(1 + x)
1134                 return exp(le).series(r, order, options);
1135                 // Note: if l happens to have a Laurent expansion (with
1136                 // negative powers of (var - point)), expanding exp(le)
1137                 // will barf (which is The Right Thing).
1138         }
1139
1140         // Is the expression of type something^(-int)?
1141         if (!must_expand_basis && !exponent.info(info_flags::negint)
1142          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1143                 return basic::series(r, order, options);
1144
1145         // Is the expression of type 0^something?
1146         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1147          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1148                 return basic::series(r, order, options);
1149
1150         // Singularity encountered, is the basis equal to (var - point)?
1151         if (basis.is_equal(r.lhs() - r.rhs())) {
1152                 epvector new_seq;
1153                 if (ex_to<numeric>(exponent).to_int() < order)
1154                         new_seq.push_back(expair(_ex1, exponent));
1155                 else
1156                         new_seq.push_back(expair(Order(_ex1), exponent));
1157                 return pseries(r, new_seq);
1158         }
1159
1160         // No, expand basis into series
1161
1162         numeric numexp;
1163         if (is_a<numeric>(exponent)) {
1164                 numexp = ex_to<numeric>(exponent);
1165         } else {
1166                 numexp = 0;
1167         }
1168         const ex& sym = r.lhs();
1169         // find existing minimal degree
1170         ex eb = basis.expand();
1171         int real_ldegree = 0;
1172         if (eb.info(info_flags::rational_function))
1173                 real_ldegree = eb.ldegree(sym-r.rhs());
1174         if (real_ldegree == 0) {
1175                 int orderloop = 0;
1176                 do {
1177                         orderloop++;
1178                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1179                 } while (real_ldegree == orderloop);
1180         }
1181
1182         if (!(real_ldegree*numexp).is_integer())
1183                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1184         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1185
1186         ex result;
1187         try {
1188                 result = ex_to<pseries>(e).power_const(numexp, order);
1189         } catch (pole_error) {
1190                 epvector ser;
1191                 ser.push_back(expair(Order(_ex1), order));
1192                 result = pseries(r, ser);
1193         }
1194
1195         return result;
1196 }
1197
1198
1199 /** Re-expansion of a pseries object. */
1200 ex pseries::series(const relational & r, int order, unsigned options) const
1201 {
1202         const ex p = r.rhs();
1203         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1204         const symbol &s = ex_to<symbol>(r.lhs());
1205
1206         if (var.is_equal(s) && point.is_equal(p)) {
1207                 if (order > degree(s))
1208                         return *this;
1209                 else {
1210                         epvector new_seq;
1211                         epvector::const_iterator it = seq.begin(), itend = seq.end();
1212                         while (it != itend) {
1213                                 int o = ex_to<numeric>(it->coeff).to_int();
1214                                 if (o >= order) {
1215                                         new_seq.push_back(expair(Order(_ex1), o));
1216                                         break;
1217                                 }
1218                                 new_seq.push_back(*it);
1219                                 ++it;
1220                         }
1221                         return pseries(r, new_seq);
1222                 }
1223         } else
1224                 return convert_to_poly().series(r, order, options);
1225 }
1226
1227 ex integral::series(const relational & r, int order, unsigned options) const
1228 {
1229         if (x.subs(r) != x)
1230                 throw std::logic_error("Cannot series expand wrt dummy variable");
1231
1232         // Expanding integrant with r substituted taken in boundaries.
1233         ex fseries = f.series(r, order, options);
1234         epvector fexpansion;
1235         fexpansion.reserve(fseries.nops());
1236         for (size_t i=0; i<fseries.nops(); ++i) {
1237                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1238                 currcoeff = (currcoeff == Order(_ex1))
1239                         ? currcoeff
1240                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1241                 if (currcoeff != 0)
1242                         fexpansion.push_back(
1243                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1244         }
1245
1246         // Expanding lower boundary
1247         ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1248         ex aseries = (a-a.subs(r)).series(r, order, options);
1249         fseries = f.series(x == (a.subs(r)), order, options);
1250         for (size_t i=0; i<fseries.nops(); ++i) {
1251                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1252                 if (is_order_function(currcoeff))
1253                         break;
1254                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1255                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1256                 currcoeff = currcoeff.series(r, orderforf);
1257                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1258                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1259                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1260                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1261         }
1262
1263         // Expanding upper boundary
1264         ex bseries = (b-b.subs(r)).series(r, order, options);
1265         fseries = f.series(x == (b.subs(r)), order, options);
1266         for (size_t i=0; i<fseries.nops(); ++i) {
1267                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1268                 if (is_order_function(currcoeff))
1269                         break;
1270                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1271                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1272                 currcoeff = currcoeff.series(r, orderforf);
1273                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1274                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1275                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1276                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1277         }
1278
1279         return result;
1280 }
1281
1282
1283 /** Compute the truncated series expansion of an expression.
1284  *  This function returns an expression containing an object of class pseries
1285  *  to represent the series. If the series does not terminate within the given
1286  *  truncation order, the last term of the series will be an order term.
1287  *
1288  *  @param r  expansion relation, lhs holds variable and rhs holds point
1289  *  @param order  truncation order of series calculations
1290  *  @param options  of class series_options
1291  *  @return an expression holding a pseries object */
1292 ex ex::series(const ex & r, int order, unsigned options) const
1293 {
1294         ex e;
1295         relational rel_;
1296
1297         if (is_a<relational>(r))
1298                 rel_ = ex_to<relational>(r);
1299         else if (is_a<symbol>(r))
1300                 rel_ = relational(r,_ex0);
1301         else
1302                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1303
1304         e = bp->series(rel_, order, options);
1305         return e;
1306 }
1307
1308 GINAC_BIND_UNARCHIVER(pseries);
1309
1310 } // namespace GiNaC