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) 1999-2006 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 <numeric>
  25 #include <stdexcept>
  26
  27 #include "pseries.h"
  28 #include "add.h"
  29 #include "inifcns.h" // for Order function
  30 #include "lst.h"
  31 #include "mul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "operators.h"
  35 #include "symbol.h"
  36 #include "integral.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
  43   print_func<print_context>(&pseries::do_print).
  44   print_func<print_latex>(&pseries::do_print_latex).
  45   print_func<print_tree>(&pseries::do_print_tree).
  46   print_func<print_python>(&pseries::do_print_python).
  47   print_func<print_python_repr>(&pseries::do_print_python_repr))
  48
  49
  50 /*
  51  *  Default constructor
  52  */
  53
  54 pseries::pseries() : inherited(&pseries::tinfo_static) { }
  55
  56
  57 /*
  58  *  Other ctors
  59  */
  60
  61 /** Construct pseries from a vector of coefficients and powers.
  62  *  expair.rest holds the coefficient, expair.coeff holds the power.
  63  *  The powers must be integers (positive or negative) and in ascending order;
  64  *  the last coefficient can be Order(_ex1) to represent a truncated,
  65  *  non-terminating series.
  66  *
  67  *  @param rel_  expansion variable and point (must hold a relational)
  68  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  69  *  @return newly constructed pseries */
  70 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(&pseries::tinfo_static), seq(ops_)
  71 {
  72         GINAC_ASSERT(is_a<relational>(rel_));
  73         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  74         point = rel_.rhs();
  75         var = rel_.lhs();
  76 }
  77
  78
  79 /*
  80  *  Archiving
  81  */
  82
  83 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  84 {
  85         archive_node::archive_node_cit first = n.find_first("coeff");
  86         archive_node::archive_node_cit last = n.find_last("power");
  87         ++last;
  88         seq.reserve((last-first)/2);
  89
  90         for (archive_node::archive_node_cit loc = first; loc < last;) {
  91                 ex rest;
  92                 ex coeff;
  93                 n.find_ex_by_loc(loc++, rest, sym_lst);
  94                 n.find_ex_by_loc(loc++, coeff, sym_lst);
  95                 seq.push_back(expair(rest, coeff));
  96         }
  97
  98         n.find_ex("var", var, sym_lst);
  99         n.find_ex("point", point, sym_lst);
 100 }
 101
 102 void pseries::archive(archive_node &n) const
 103 {
 104         inherited::archive(n);
 105         epvector::const_iterator i = seq.begin(), iend = seq.end();
 106         while (i != iend) {
 107                 n.add_ex("coeff", i->rest);
 108                 n.add_ex("power", i->coeff);
 109                 ++i;
 110         }
 111         n.add_ex("var", var);
 112         n.add_ex("point", point);
 113 }
 114
 115 DEFAULT_UNARCHIVE(pseries)
 116
 117 //////////
 118 // functions overriding virtual functions from base classes
 119 //////////
 120
 121 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
 122 {
 123         if (precedence() <= level)
 124                 c.s << '(';
 125
 126         // objects of type pseries must not have any zero entries, so the
 127         // trivial (zero) pseries needs a special treatment here:
 128         if (seq.empty())
 129                 c.s << '0';
 130
 131         epvector::const_iterator i = seq.begin(), end = seq.end();
 132         while (i != end) {
 133
 134                 // print a sign, if needed
 135                 if (i != seq.begin())
 136                         c.s << '+';
 137
 138                 if (!is_order_function(i->rest)) {
 139
 140                         // print 'rest', i.e. the expansion coefficient
 141                         if (i->rest.info(info_flags::numeric) &&
 142                                 i->rest.info(info_flags::positive)) {
 143                                 i->rest.print(c);
 144                         } else {
 145                                 c.s << openbrace << '(';
 146                                 i->rest.print(c);
 147                                 c.s << ')' << closebrace;
 148                         }
 149
 150                         // print 'coeff', something like (x-1)^42
 151                         if (!i->coeff.is_zero()) {
 152                                 c.s << mul_sym;
 153                                 if (!point.is_zero()) {
 154                                         c.s << openbrace << '(';
 155                                         (var-point).print(c);
 156                                         c.s << ')' << closebrace;
 157                                 } else
 158                                         var.print(c);
 159                                 if (i->coeff.compare(_ex1)) {
 160                                         c.s << pow_sym;
 161                                         c.s << openbrace;
 162                                         if (i->coeff.info(info_flags::negative)) {
 163                                                 c.s << '(';
 164                                                 i->coeff.print(c);
 165                                                 c.s << ')';
 166                                         } else
 167                                                 i->coeff.print(c);
 168                                         c.s << closebrace;
 169                                 }
 170                         }
 171                 } else
 172                         Order(power(var-point,i->coeff)).print(c);
 173                 ++i;
 174         }
 175
 176         if (precedence() <= level)
 177                 c.s << ')';
 178 }
 179
 180 void pseries::do_print(const print_context & c, unsigned level) const
 181 {
 182         print_series(c, "", "", "*", "^", level);
 183 }
 184
 185 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 186 {
 187         print_series(c, "{", "}", " ", "^", level);
 188 }
 189
 190 void pseries::do_print_python(const print_python & c, unsigned level) const
 191 {
 192         print_series(c, "", "", "*", "**", level);
 193 }
 194
 195 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 196 {
 197         c.s << std::string(level, ' ') << class_name() << " @" << this
 198             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 199             << std::endl;
 200         size_t num = seq.size();
 201         for (size_t i=0; i<num; ++i) {
 202                 seq[i].rest.print(c, level + c.delta_indent);
 203                 seq[i].coeff.print(c, level + c.delta_indent);
 204                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 205         }
 206         var.print(c, level + c.delta_indent);
 207         point.print(c, level + c.delta_indent);
 208 }
 209
 210 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 211 {
 212         c.s << class_name() << "(relational(";
 213         var.print(c);
 214         c.s << ',';
 215         point.print(c);
 216         c.s << "),[";
 217         size_t num = seq.size();
 218         for (size_t i=0; i<num; ++i) {
 219                 if (i)
 220                         c.s << ',';
 221                 c.s << '(';
 222                 seq[i].rest.print(c);
 223                 c.s << ',';
 224                 seq[i].coeff.print(c);
 225                 c.s << ')';
 226         }
 227         c.s << "])";
 228 }
 229
 230 int pseries::compare_same_type(const basic & other) const
 231 {
 232         GINAC_ASSERT(is_a<pseries>(other));
 233         const pseries &o = static_cast<const pseries &>(other);
 234
 235         // first compare the lengths of the series...
 236         if (seq.size()>o.seq.size())
 237                 return 1;
 238         if (seq.size()<o.seq.size())
 239                 return -1;
 240
 241         // ...then the expansion point...
 242         int cmpval = var.compare(o.var);
 243         if (cmpval)
 244                 return cmpval;
 245         cmpval = point.compare(o.point);
 246         if (cmpval)
 247                 return cmpval;
 248
 249         // ...and if that failed the individual elements
 250         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 251         while (it!=seq.end() && o_it!=o.seq.end()) {
 252                 cmpval = it->compare(*o_it);
 253                 if (cmpval)
 254                         return cmpval;
 255                 ++it;
 256                 ++o_it;
 257         }
 258
 259         // so they are equal.
 260         return 0;
 261 }
 262
 263 /** Return the number of operands including a possible order term. */
 264 size_t pseries::nops() const
 265 {
 266         return seq.size();
 267 }
 268
 269 /** Return the ith term in the series when represented as a sum. */
 270 ex pseries::op(size_t i) const
 271 {
 272         if (i >= seq.size())
 273                 throw (std::out_of_range("op() out of range"));
 274
 275         if (is_order_function(seq[i].rest))
 276                 return Order(power(var-point, seq[i].coeff));
 277         return seq[i].rest * power(var - point, seq[i].coeff);
 278 }
 279
 280 /** Return degree of highest power of the series.  This is usually the exponent
 281  *  of the Order term.  If s is not the expansion variable of the series, the
 282  *  series is examined termwise. */
 283 int pseries::degree(const ex &s) const
 284 {
 285         if (var.is_equal(s)) {
 286                 // Return last exponent
 287                 if (seq.size())
 288                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 289                 else
 290                         return 0;
 291         } else {
 292                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 293                 if (it == itend)
 294                         return 0;
 295                 int max_pow = INT_MIN;
 296                 while (it != itend) {
 297                         int pow = it->rest.degree(s);
 298                         if (pow > max_pow)
 299                                 max_pow = pow;
 300                         ++it;
 301                 }
 302                 return max_pow;
 303         }
 304 }
 305
 306 /** Return degree of lowest power of the series.  This is usually the exponent
 307  *  of the leading term.  If s is not the expansion variable of the series, the
 308  *  series is examined termwise.  If s is the expansion variable but the
 309  *  expansion point is not zero the series is not expanded to find the degree.
 310  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 311 int pseries::ldegree(const ex &s) const
 312 {
 313         if (var.is_equal(s)) {
 314                 // Return first exponent
 315                 if (seq.size())
 316                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 317                 else
 318                         return 0;
 319         } else {
 320                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 321                 if (it == itend)
 322                         return 0;
 323                 int min_pow = INT_MAX;
 324                 while (it != itend) {
 325                         int pow = it->rest.ldegree(s);
 326                         if (pow < min_pow)
 327                                 min_pow = pow;
 328                         ++it;
 329                 }
 330                 return min_pow;
 331         }
 332 }
 333
 334 /** Return coefficient of degree n in power series if s is the expansion
 335  *  variable.  If the expansion point is nonzero, by definition the n=1
 336  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 337  *  the expansion took place in the s in the first place).
 338  *  If s is not the expansion variable, an attempt is made to convert the
 339  *  series to a polynomial and return the corresponding coefficient from
 340  *  there. */
 341 ex pseries::coeff(const ex &s, int n) const
 342 {
 343         if (var.is_equal(s)) {
 344                 if (seq.empty())
 345                         return _ex0;
 346
 347                 // Binary search in sequence for given power
 348                 numeric looking_for = numeric(n);
 349                 int lo = 0, hi = seq.size() - 1;
 350                 while (lo <= hi) {
 351                         int mid = (lo + hi) / 2;
 352                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 353                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 354                         switch (cmp) {
 355                                 case -1:
 356                                         lo = mid + 1;
 357                                         break;
 358                                 case 0:
 359                                         return seq[mid].rest;
 360                                 case 1:
 361                                         hi = mid - 1;
 362                                         break;
 363                                 default:
 364                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 365                         }
 366                 }
 367                 return _ex0;
 368         } else
 369                 return convert_to_poly().coeff(s, n);
 370 }
 371
 372 /** Does nothing. */
 373 ex pseries::collect(const ex &s, bool distributed) const
 374 {
 375         return *this;
 376 }
 377
 378 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 379 ex pseries::eval(int level) const
 380 {
 381         if (level == 1)
 382                 return this->hold();
 383
 384         if (level == -max_recursion_level)
 385                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 386
 387         // Construct a new series with evaluated coefficients
 388         epvector new_seq;
 389         new_seq.reserve(seq.size());
 390         epvector::const_iterator it = seq.begin(), itend = seq.end();
 391         while (it != itend) {
 392                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 393                 ++it;
 394         }
 395         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 396 }
 397
 398 /** Evaluate coefficients numerically. */
 399 ex pseries::evalf(int level) const
 400 {
 401         if (level == 1)
 402                 return *this;
 403
 404         if (level == -max_recursion_level)
 405                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 406
 407         // Construct a new series with evaluated coefficients
 408         epvector new_seq;
 409         new_seq.reserve(seq.size());
 410         epvector::const_iterator it = seq.begin(), itend = seq.end();
 411         while (it != itend) {
 412                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 413                 ++it;
 414         }
 415         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 416 }
 417
 418 ex pseries::conjugate() const
 419 {
 420         if(!var.info(info_flags::real))
 421                 return conjugate_function(*this).hold();
 422
 423         epvector * newseq = conjugateepvector(seq);
 424         ex newpoint = point.conjugate();
 425
 426         if (!newseq     && are_ex_trivially_equal(point, newpoint)) {
 427                 return *this;
 428         }
 429
 430         ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
 431         if (newseq) {
 432                 delete newseq;
 433         }
 434         return result;
 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(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 448                 v.push_back(expair((i->rest).real_part(), i->coeff));
 449         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 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(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 463                 v.push_back(expair((i->rest).imag_part(), i->coeff));
 464         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 465 }
 466
 467 ex pseries::eval_integ() const
 468 {
 469         epvector *newseq = NULL;
 470         for (epvector::const_iterator 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 = new epvector;
 478                         newseq->reserve(seq.size());
 479                         for (epvector::const_iterator 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 (new pseries(var==newpoint, *newseq))
 488                        ->setflag(status_flags::dynallocated);
 489         return *this;
 490 }
 491
 492 ex pseries::subs(const exmap & m, unsigned options) const
 493 {
 494         // If expansion variable is being substituted, convert the series to a
 495         // polynomial and do the substitution there because the result might
 496         // no longer be a power series
 497         if (m.find(var) != m.end())
 498                 return convert_to_poly(true).subs(m, options);
 499
 500         // Otherwise construct a new series with substituted coefficients and
 501         // expansion point
 502         epvector newseq;
 503         newseq.reserve(seq.size());
 504         epvector::const_iterator it = seq.begin(), itend = seq.end();
 505         while (it != itend) {
 506                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
 507                 ++it;
 508         }
 509         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
 510 }
 511
 512 /** Implementation of ex::expand() for a power series.  It expands all the
 513  *  terms individually and returns the resulting series as a new pseries. */
 514 ex pseries::expand(unsigned options) const
 515 {
 516         epvector newseq;
 517         epvector::const_iterator i = seq.begin(), end = seq.end();
 518         while (i != end) {
 519                 ex restexp = i->rest.expand();
 520                 if (!restexp.is_zero())
 521                         newseq.push_back(expair(restexp, i->coeff));
 522                 ++i;
 523         }
 524         return (new pseries(relational(var,point), newseq))
 525                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 526 }
 527
 528 /** Implementation of ex::diff() for a power series.
 529  *  @see ex::diff */
 530 ex pseries::derivative(const symbol & s) const
 531 {
 532         epvector new_seq;
 533         epvector::const_iterator it = seq.begin(), itend = seq.end();
 534
 535         if (s == var) {
 536
 537                 // FIXME: coeff might depend on var
 538                 while (it != itend) {
 539                         if (is_order_function(it->rest)) {
 540                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 541                         } else {
 542                                 ex c = it->rest * it->coeff;
 543                                 if (!c.is_zero())
 544                                         new_seq.push_back(expair(c, it->coeff - 1));
 545                         }
 546                         ++it;
 547                 }
 548
 549         } else {
 550
 551                 while (it != itend) {
 552                         if (is_order_function(it->rest)) {
 553                                 new_seq.push_back(*it);
 554                         } else {
 555                                 ex c = it->rest.diff(s);
 556                                 if (!c.is_zero())
 557                                         new_seq.push_back(expair(c, it->coeff));
 558                         }
 559                         ++it;
 560                 }
 561         }
 562
 563         return pseries(relational(var,point), new_seq);
 564 }
 565
 566 ex pseries::convert_to_poly(bool no_order) const
 567 {
 568         ex e;
 569         epvector::const_iterator it = seq.begin(), itend = seq.end();
 570
 571         while (it != itend) {
 572                 if (is_order_function(it->rest)) {
 573                         if (!no_order)
 574                                 e += Order(power(var - point, it->coeff));
 575                 } else
 576                         e += it->rest * power(var - point, it->coeff);
 577                 ++it;
 578         }
 579         return e;
 580 }
 581
 582 bool pseries::is_terminating() const
 583 {
 584         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 585 }
 586
 587 ex pseries::coeffop(size_t i) const
 588 {
 589         if (i >=nops())
 590                 throw (std::out_of_range("coeffop() out of range"));
 591         return seq[i].rest;
 592 }
 593
 594 ex pseries::exponop(size_t i) const
 595 {
 596         if (i >= nops())
 597                 throw (std::out_of_range("exponop() out of range"));
 598         return seq[i].coeff;
 599 }
 600
 601
 602 /*
 603  *  Implementations of series expansion
 604  */
 605
 606 /** Default implementation of ex::series(). This performs Taylor expansion.
 607  *  @see ex::series */
 608 ex basic::series(const relational & r, int order, unsigned options) const
 609 {
 610         epvector seq;
 611         const symbol &s = ex_to<symbol>(r.lhs());
 612
 613         // default for order-values that make no sense for Taylor expansion
 614         if ((order <= 0) && this->has(s)) {
 615                 seq.push_back(expair(Order(_ex1), order));
 616                 return pseries(r, seq);
 617         }
 618
 619         // do Taylor expansion
 620         numeric fac = 1;
 621         ex deriv = *this;
 622         ex coeff = deriv.subs(r, subs_options::no_pattern);
 623
 624         if (!coeff.is_zero()) {
 625                 seq.push_back(expair(coeff, _ex0));
 626         }
 627
 628         int n;
 629         for (n=1; n<order; ++n) {
 630                 fac = fac.mul(n);
 631                 // We need to test for zero in order to see if the series terminates.
 632                 // The problem is that there is no such thing as a perfect test for
 633                 // zero.  Expanding the term occasionally helps a little...
 634                 deriv = deriv.diff(s).expand();
 635                 if (deriv.is_zero())  // Series terminates
 636                         return pseries(r, seq);
 637
 638                 coeff = deriv.subs(r, subs_options::no_pattern);
 639                 if (!coeff.is_zero())
 640                         seq.push_back(expair(fac.inverse() * coeff, n));
 641         }
 642
 643         // Higher-order terms, if present
 644         deriv = deriv.diff(s);
 645         if (!deriv.expand().is_zero())
 646                 seq.push_back(expair(Order(_ex1), n));
 647         return pseries(r, seq);
 648 }
 649
 650
 651 /** Implementation of ex::series() for symbols.
 652  *  @see ex::series */
 653 ex symbol::series(const relational & r, int order, unsigned options) const
 654 {
 655         epvector seq;
 656         const ex point = r.rhs();
 657         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 658
 659         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 660                 if (order > 0 && !point.is_zero())
 661                         seq.push_back(expair(point, _ex0));
 662                 if (order > 1)
 663                         seq.push_back(expair(_ex1, _ex1));
 664                 else
 665                         seq.push_back(expair(Order(_ex1), numeric(order)));
 666         } else
 667                 seq.push_back(expair(*this, _ex0));
 668         return pseries(r, seq);
 669 }
 670
 671
 672 /** Add one series object to another, producing a pseries object that
 673  *  represents the sum.
 674  *
 675  *  @param other  pseries object to add with
 676  *  @return the sum as a pseries */
 677 ex pseries::add_series(const pseries &other) const
 678 {
 679         // Adding two series with different variables or expansion points
 680         // results in an empty (constant) series
 681         if (!is_compatible_to(other)) {
 682                 epvector nul;
 683                 nul.push_back(expair(Order(_ex1), _ex0));
 684                 return pseries(relational(var,point), nul);
 685         }
 686
 687         // Series addition
 688         epvector new_seq;
 689         epvector::const_iterator a = seq.begin();
 690         epvector::const_iterator b = other.seq.begin();
 691         epvector::const_iterator a_end = seq.end();
 692         epvector::const_iterator b_end = other.seq.end();
 693         int pow_a = INT_MAX, pow_b = INT_MAX;
 694         for (;;) {
 695                 // If a is empty, fill up with elements from b and stop
 696                 if (a == a_end) {
 697                         while (b != b_end) {
 698                                 new_seq.push_back(*b);
 699                                 ++b;
 700                         }
 701                         break;
 702                 } else
 703                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 704
 705                 // If b is empty, fill up with elements from a and stop
 706                 if (b == b_end) {
 707                         while (a != a_end) {
 708                                 new_seq.push_back(*a);
 709                                 ++a;
 710                         }
 711                         break;
 712                 } else
 713                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 714
 715                 // a and b are non-empty, compare powers
 716                 if (pow_a < pow_b) {
 717                         // a has lesser power, get coefficient from a
 718                         new_seq.push_back(*a);
 719                         if (is_order_function((*a).rest))
 720                                 break;
 721                         ++a;
 722                 } else if (pow_b < pow_a) {
 723                         // b has lesser power, get coefficient from b
 724                         new_seq.push_back(*b);
 725                         if (is_order_function((*b).rest))
 726                                 break;
 727                         ++b;
 728                 } else {
 729                         // Add coefficient of a and b
 730                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 731                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 732                                 break;  // Order term ends the sequence
 733                         } else {
 734                                 ex sum = (*a).rest + (*b).rest;
 735                                 if (!(sum.is_zero()))
 736                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 737                                 ++a;
 738                                 ++b;
 739                         }
 740                 }
 741         }
 742         return pseries(relational(var,point), new_seq);
 743 }
 744
 745
 746 /** Implementation of ex::series() for sums. This performs series addition when
 747  *  adding pseries objects.
 748  *  @see ex::series */
 749 ex add::series(const relational & r, int order, unsigned options) const
 750 {
 751         ex acc; // Series accumulator
 752
 753         // Get first term from overall_coeff
 754         acc = overall_coeff.series(r, order, options);
 755
 756         // Add remaining terms
 757         epvector::const_iterator it = seq.begin();
 758         epvector::const_iterator itend = seq.end();
 759         for (; it!=itend; ++it) {
 760                 ex op;
 761                 if (is_exactly_a<pseries>(it->rest))
 762                         op = it->rest;
 763                 else
 764                         op = it->rest.series(r, order, options);
 765                 if (!it->coeff.is_equal(_ex1))
 766                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 767
 768                 // Series addition
 769                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 770         }
 771         return acc;
 772 }
 773
 774
 775 /** Multiply a pseries object with a numeric constant, producing a pseries
 776  *  object that represents the product.
 777  *
 778  *  @param other  constant to multiply with
 779  *  @return the product as a pseries */
 780 ex pseries::mul_const(const numeric &other) const
 781 {
 782         epvector new_seq;
 783         new_seq.reserve(seq.size());
 784
 785         epvector::const_iterator it = seq.begin(), itend = seq.end();
 786         while (it != itend) {
 787                 if (!is_order_function(it->rest))
 788                         new_seq.push_back(expair(it->rest * other, it->coeff));
 789                 else
 790                         new_seq.push_back(*it);
 791                 ++it;
 792         }
 793         return pseries(relational(var,point), new_seq);
 794 }
 795
 796
 797 /** Multiply one pseries object to another, producing a pseries object that
 798  *  represents the product.
 799  *
 800  *  @param other  pseries object to multiply with
 801  *  @return the product as a pseries */
 802 ex pseries::mul_series(const pseries &other) const
 803 {
 804         // Multiplying two series with different variables or expansion points
 805         // results in an empty (constant) series
 806         if (!is_compatible_to(other)) {
 807                 epvector nul;
 808                 nul.push_back(expair(Order(_ex1), _ex0));
 809                 return pseries(relational(var,point), nul);
 810         }
 811
 812         if (seq.empty() || other.seq.empty()) {
 813                 return (new pseries(var==point, epvector()))
 814                        ->setflag(status_flags::dynallocated);
 815         }
 816
 817         // Series multiplication
 818         epvector new_seq;
 819         int a_max = degree(var);
 820         int b_max = other.degree(var);
 821         int a_min = ldegree(var);
 822         int b_min = other.ldegree(var);
 823         int cdeg_min = a_min + b_min;
 824         int cdeg_max = a_max + b_max;
 825
 826         int higher_order_a = INT_MAX;
 827         int higher_order_b = INT_MAX;
 828         if (is_order_function(coeff(var, a_max)))
 829                 higher_order_a = a_max + b_min;
 830         if (is_order_function(other.coeff(var, b_max)))
 831                 higher_order_b = b_max + a_min;
 832         int higher_order_c = std::min(higher_order_a, higher_order_b);
 833         if (cdeg_max >= higher_order_c)
 834                 cdeg_max = higher_order_c - 1;
 835
 836         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 837                 ex co = _ex0;
 838                 // c(i)=a(0)b(i)+...+a(i)b(0)
 839                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 840                         ex a_coeff = coeff(var, i);
 841                         ex b_coeff = other.coeff(var, cdeg-i);
 842                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 843                                 co += a_coeff * b_coeff;
 844                 }
 845                 if (!co.is_zero())
 846                         new_seq.push_back(expair(co, numeric(cdeg)));
 847         }
 848         if (higher_order_c < INT_MAX)
 849                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 850         return pseries(relational(var, point), new_seq);
 851 }
 852
 853
 854 /** Implementation of ex::series() for product. This performs series
 855  *  multiplication when multiplying series.
 856  *  @see ex::series */
 857 ex mul::series(const relational & r, int order, unsigned options) const
 858 {
 859         pseries acc; // Series accumulator
 860
 861         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 862         const ex& sym = r.lhs();
 863
 864         // holds ldegrees of the series of individual factors
 865         std::vector<int> ldegrees;
 866
 867         // find minimal degrees
 868         const epvector::const_iterator itbeg = seq.begin();
 869         const epvector::const_iterator itend = seq.end();
 870         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 871
 872                 ex expon = it->coeff;
 873                 int factor = 1;
 874                 ex buf;
 875                 if (expon.info(info_flags::integer)) {
 876                         buf = it->rest;
 877                         factor = ex_to<numeric>(expon).to_int();
 878                 } else {
 879                         buf = recombine_pair_to_ex(*it);
 880                 }
 881
 882                 int real_ldegree = 0;
 883                 try {
 884                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 885                 } catch (std::runtime_error) {}
 886
 887                 if (real_ldegree == 0) {
 888                         int orderloop = 0;
 889                         do {
 890                                 orderloop++;
 891                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 892                         } while (real_ldegree == orderloop);
 893                 }
 894
 895                 ldegrees.push_back(factor * real_ldegree);
 896         }
 897
 898         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 899
 900         if (degsum >= order) {
 901                 epvector epv;
 902                 epv.push_back(expair(Order(_ex1), order));
 903                 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
 904         }
 905
 906         // Multiply with remaining terms
 907         std::vector<int>::const_iterator itd = ldegrees.begin();
 908         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
 909
 910                 // do series expansion with adjusted order
 911                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 912
 913                 // Series multiplication
 914                 if (it == itbeg)
 915                         acc = ex_to<pseries>(op);
 916                 else
 917                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 918         }
 919
 920         return acc.mul_const(ex_to<numeric>(overall_coeff));
 921 }
 922
 923
 924 /** Compute the p-th power of a series.
 925  *
 926  *  @param p  power to compute
 927  *  @param deg  truncation order of series calculation */
 928 ex pseries::power_const(const numeric &p, int deg) const
 929 {
 930         // method:
 931         // (due to Leonhard Euler)
 932         // let A(x) be this series and for the time being let it start with a
 933         // constant (later we'll generalize):
 934         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 935         // We want to compute
 936         //     C(x) = A(x)^p
 937         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 938         // Taking the derivative on both sides and multiplying with A(x) one
 939         // immediately arrives at
 940         //     C'(x)*A(x) = p*C(x)*A'(x)
 941         // Multiplying this out and comparing coefficients we get the recurrence
 942         // formula
 943         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 944         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 945         // which can easily be solved given the starting value c_0 = (a_0)^p.
 946         // For the more general case where the leading coefficient of A(x) is not
 947         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 948         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 949         // then of course x^(p*m) but the recurrence formula still holds.
 950
 951         if (seq.empty()) {
 952                 // as a special case, handle the empty (zero) series honoring the
 953                 // usual power laws such as implemented in power::eval()
 954                 if (p.real().is_zero())
 955                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 956                 else if (p.real().is_negative())
 957                         throw pole_error("pseries::power_const(): division by zero",1);
 958                 else
 959                         return *this;
 960         }
 961
 962         const int ldeg = ldegree(var);
 963         if (!(p*ldeg).is_integer())
 964                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 965
 966         // adjust number of coefficients
 967         int numcoeff = deg - (p*ldeg).to_int();
 968         if (numcoeff <= 0) {
 969                 epvector epv;
 970                 epv.reserve(1);
 971                 epv.push_back(expair(Order(_ex1), deg));
 972                 return (new pseries(relational(var,point), epv))
 973                        ->setflag(status_flags::dynallocated);
 974         }
 975
 976         // O(x^n)^(-m) is undefined
 977         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 978                 throw pole_error("pseries::power_const(): division by zero",1);
 979
 980         // Compute coefficients of the powered series
 981         exvector co;
 982         co.reserve(numcoeff);
 983         co.push_back(power(coeff(var, ldeg), p));
 984         for (int i=1; i<numcoeff; ++i) {
 985                 ex sum = _ex0;
 986                 for (int j=1; j<=i; ++j) {
 987                         ex c = coeff(var, j + ldeg);
 988                         if (is_order_function(c)) {
 989                                 co.push_back(Order(_ex1));
 990                                 break;
 991                         } else
 992                                 sum += (p * j - (i - j)) * co[i - j] * c;
 993                 }
 994                 co.push_back(sum / coeff(var, ldeg) / i);
 995         }
 996
 997         // Construct new series (of non-zero coefficients)
 998         epvector new_seq;
 999         bool higher_order = false;
1000         for (int i=0; i<numcoeff; ++i) {
1001                 if (!co[i].is_zero())
1002                         new_seq.push_back(expair(co[i], p * ldeg + i));
1003                 if (is_order_function(co[i])) {
1004                         higher_order = true;
1005                         break;
1006                 }
1007         }
1008         if (!higher_order)
1009                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1010
1011         return pseries(relational(var,point), new_seq);
1012 }
1013
1014
1015 /** Return a new pseries object with the powers shifted by deg. */
1016 pseries pseries::shift_exponents(int deg) const
1017 {
1018         epvector newseq = seq;
1019         epvector::iterator i = newseq.begin(), end  = newseq.end();
1020         while (i != end) {
1021                 i->coeff += deg;
1022                 ++i;
1023         }
1024         return pseries(relational(var, point), newseq);
1025 }
1026
1027
1028 /** Implementation of ex::series() for powers. This performs Laurent expansion
1029  *  of reciprocals of series at singularities.
1030  *  @see ex::series */
1031 ex power::series(const relational & r, int order, unsigned options) const
1032 {
1033         // If basis is already a series, just power it
1034         if (is_exactly_a<pseries>(basis))
1035                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1036
1037         // Basis is not a series, may there be a singularity?
1038         bool must_expand_basis = false;
1039         try {
1040                 basis.subs(r, subs_options::no_pattern);
1041         } catch (pole_error) {
1042                 must_expand_basis = true;
1043         }
1044
1045         // Is the expression of type something^(-int)?
1046         if (!must_expand_basis && !exponent.info(info_flags::negint)
1047          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1048                 return basic::series(r, order, options);
1049
1050         // Is the expression of type 0^something?
1051         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1052          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1053                 return basic::series(r, order, options);
1054
1055         // Singularity encountered, is the basis equal to (var - point)?
1056         if (basis.is_equal(r.lhs() - r.rhs())) {
1057                 epvector new_seq;
1058                 if (ex_to<numeric>(exponent).to_int() < order)
1059                         new_seq.push_back(expair(_ex1, exponent));
1060                 else
1061                         new_seq.push_back(expair(Order(_ex1), exponent));
1062                 return pseries(r, new_seq);
1063         }
1064
1065         // No, expand basis into series
1066
1067         numeric numexp;
1068         if (is_a<numeric>(exponent)) {
1069                 numexp = ex_to<numeric>(exponent);
1070         } else {
1071                 numexp = 0;
1072         }
1073         const ex& sym = r.lhs();
1074         // find existing minimal degree
1075         int real_ldegree = basis.expand().ldegree(sym-r.rhs());
1076         if (real_ldegree == 0) {
1077                 int orderloop = 0;
1078                 do {
1079                         orderloop++;
1080                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1081                 } while (real_ldegree == orderloop);
1082         }
1083
1084         if (!(real_ldegree*numexp).is_integer())
1085                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1086         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1087
1088         ex result;
1089         try {
1090                 result = ex_to<pseries>(e).power_const(numexp, order);
1091         } catch (pole_error) {
1092                 epvector ser;
1093                 ser.push_back(expair(Order(_ex1), order));
1094                 result = pseries(r, ser);
1095         }
1096
1097         return result;
1098 }
1099
1100
1101 /** Re-expansion of a pseries object. */
1102 ex pseries::series(const relational & r, int order, unsigned options) const
1103 {
1104         const ex p = r.rhs();
1105         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1106         const symbol &s = ex_to<symbol>(r.lhs());
1107
1108         if (var.is_equal(s) && point.is_equal(p)) {
1109                 if (order > degree(s))
1110                         return *this;
1111                 else {
1112                         epvector new_seq;
1113                         epvector::const_iterator it = seq.begin(), itend = seq.end();
1114                         while (it != itend) {
1115                                 int o = ex_to<numeric>(it->coeff).to_int();
1116                                 if (o >= order) {
1117                                         new_seq.push_back(expair(Order(_ex1), o));
1118                                         break;
1119                                 }
1120                                 new_seq.push_back(*it);
1121                                 ++it;
1122                         }
1123                         return pseries(r, new_seq);
1124                 }
1125         } else
1126                 return convert_to_poly().series(r, order, options);
1127 }
1128
1129 ex integral::series(const relational & r, int order, unsigned options) const
1130 {
1131         if (x.subs(r) != x)
1132                 throw std::logic_error("Cannot series expand wrt dummy variable");
1133
1134         // Expanding integrant with r substituted taken in boundaries.
1135         ex fseries = f.series(r, order, options);
1136         epvector fexpansion;
1137         fexpansion.reserve(fseries.nops());
1138         for (size_t i=0; i<fseries.nops(); ++i) {
1139                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1140                 currcoeff = (currcoeff == Order(_ex1))
1141                         ? currcoeff
1142                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1143                 if (currcoeff != 0)
1144                         fexpansion.push_back(
1145                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1146         }
1147
1148         // Expanding lower boundary
1149         ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1150         ex aseries = (a-a.subs(r)).series(r, order, options);
1151         fseries = f.series(x == (a.subs(r)), order, options);
1152         for (size_t i=0; i<fseries.nops(); ++i) {
1153                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1154                 if (is_order_function(currcoeff))
1155                         break;
1156                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1157                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1158                 currcoeff = currcoeff.series(r, orderforf);
1159                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1160                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1161                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1162                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1163         }
1164
1165         // Expanding upper boundary
1166         ex bseries = (b-b.subs(r)).series(r, order, options);
1167         fseries = f.series(x == (b.subs(r)), order, options);
1168         for (size_t i=0; i<fseries.nops(); ++i) {
1169                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1170                 if (is_order_function(currcoeff))
1171                         break;
1172                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1173                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1174                 currcoeff = currcoeff.series(r, orderforf);
1175                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1176                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1177                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1178                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1179         }
1180
1181         return result;
1182 }
1183
1184
1185 /** Compute the truncated series expansion of an expression.
1186  *  This function returns an expression containing an object of class pseries
1187  *  to represent the series. If the series does not terminate within the given
1188  *  truncation order, the last term of the series will be an order term.
1189  *
1190  *  @param r  expansion relation, lhs holds variable and rhs holds point
1191  *  @param order  truncation order of series calculations
1192  *  @param options  of class series_options
1193  *  @return an expression holding a pseries object */
1194 ex ex::series(const ex & r, int order, unsigned options) const
1195 {
1196         ex e;
1197         relational rel_;
1198
1199         if (is_a<relational>(r))
1200                 rel_ = ex_to<relational>(r);
1201         else if (is_a<symbol>(r))
1202                 rel_ = relational(r,_ex0);
1203         else
1204                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1205
1206         try {
1207                 e = bp->series(rel_, order, options);
1208         } catch (std::exception &x) {
1209                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1210         }
1211         return e;
1212 }
1213
1214 } // namespace GiNaC