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) 1999-2000 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  22  */
  23
  24 #include <stdexcept>
  25
  26 #include "pseries.h"
  27 #include "add.h"
  28 #include "inifcns.h"
  29 #include "lst.h"
  30 #include "mul.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "symbol.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36 #include "debugmsg.h"
  37
  38 #ifndef NO_NAMESPACE_GINAC
  39 namespace GiNaC {
  40 #endif // ndef NO_NAMESPACE_GINAC
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  43
  44 /*
  45  *  Default constructor, destructor, copy constructor, assignment operator and helpers
  46  */
  47
  48 pseries::pseries() : basic(TINFO_pseries)
  49 {
  50     debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
  51 }
  52
  53 pseries::~pseries()
  54 {
  55     debugmsg("pseries destructor", LOGLEVEL_DESTRUCT);
  56     destroy(false);
  57 }
  58
  59 pseries::pseries(const pseries &other)
  60 {
  61     debugmsg("pseries copy constructor", LOGLEVEL_CONSTRUCT);
  62     copy(other);
  63 }
  64
  65 const pseries &pseries::operator=(const pseries & other)
  66 {
  67     debugmsg("pseries operator=", LOGLEVEL_ASSIGNMENT);
  68     if (this != &other) {
  69         destroy(true);
  70         copy(other);
  71     }
  72     return *this;
  73 }
  74
  75 void pseries::copy(const pseries &other)
  76 {
  77     inherited::copy(other);
  78     seq = other.seq;
  79     var = other.var;
  80     point = other.point;
  81 }
  82
  83 void pseries::destroy(bool call_parent)
  84 {
  85     if (call_parent)
  86         inherited::destroy(call_parent);
  87 }
  88
  89
  90 /*
  91  *  Other constructors
  92  */
  93
  94 /** Construct pseries from a vector of coefficients and powers.
  95  *  expair.rest holds the coefficient, expair.coeff holds the power.
  96  *  The powers must be integers (positive or negative) and in ascending order;
  97  *  the last coefficient can be Order(_ex1()) to represent a truncated,
  98  *  non-terminating series.
  99  *
 100  *  @param rel__  expansion variable and point (must hold a relational)
 101  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
 102  *  @return newly constructed pseries */
 103 pseries::pseries(const ex &rel_, const epvector &ops_)
 104     : basic(TINFO_pseries), seq(ops_)
 105 {
 106     debugmsg("pseries constructor from rel,epvector", LOGLEVEL_CONSTRUCT);
 107     GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
 108     GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
 109     point = rel_.rhs();
 110     var = *static_cast<symbol *>(rel_.lhs().bp);
 111 }
 112
 113
 114 /*
 115  *  Archiving
 116  */
 117
 118 /** Construct object from archive_node. */
 119 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 120 {
 121     debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
 122     for (unsigned int i=0; true; i++) {
 123         ex rest;
 124         ex coeff;
 125         if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
 126             seq.push_back(expair(rest, coeff));
 127         else
 128             break;
 129     }
 130     n.find_ex("var", var, sym_lst);
 131     n.find_ex("point", point, sym_lst);
 132 }
 133
 134 /** Unarchive the object. */
 135 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
 136 {
 137     return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
 138 }
 139
 140 /** Archive the object. */
 141 void pseries::archive(archive_node &n) const
 142 {
 143     inherited::archive(n);
 144     epvector::const_iterator i = seq.begin(), iend = seq.end();
 145     while (i != iend) {
 146         n.add_ex("coeff", i->rest);
 147         n.add_ex("power", i->coeff);
 148         i++;
 149     }
 150     n.add_ex("var", var);
 151     n.add_ex("point", point);
 152 }
 153
 154
 155 /*
 156  *  Functions overriding virtual functions from base classes
 157  */
 158
 159 basic *pseries::duplicate() const
 160 {
 161     debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
 162     return new pseries(*this);
 163 }
 164
 165 void pseries::print(ostream &os, unsigned upper_precedence) const
 166 {
 167     debugmsg("pseries print", LOGLEVEL_PRINT);
 168     convert_to_poly().print(os, upper_precedence);
 169 }
 170
 171 void pseries::printraw(ostream &os) const
 172 {
 173         debugmsg("pseries printraw", LOGLEVEL_PRINT);
 174         os << "pseries(" << var << ";" << point << ";";
 175         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); i++) {
 176                 os << "(" << (*i).rest << "," << (*i).coeff << "),";
 177         }
 178         os << ")";
 179 }
 180
 181 unsigned pseries::nops(void) const
 182 {
 183     return seq.size();
 184 }
 185
 186 ex pseries::op(int i) const
 187 {
 188     if (i < 0 || unsigned(i) >= seq.size())
 189         throw (std::out_of_range("op() out of range"));
 190     return seq[i].rest * power(var - point, seq[i].coeff);
 191 }
 192
 193 ex &pseries::let_op(int i)
 194 {
 195     throw (std::logic_error("let_op not defined for pseries"));
 196 }
 197
 198 int pseries::degree(const symbol &s) const
 199 {
 200     if (var.is_equal(s)) {
 201         // Return last exponent
 202         if (seq.size())
 203             return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 204         else
 205             return 0;
 206     } else {
 207         epvector::const_iterator it = seq.begin(), itend = seq.end();
 208         if (it == itend)
 209             return 0;
 210         int max_pow = INT_MIN;
 211         while (it != itend) {
 212             int pow = it->rest.degree(s);
 213             if (pow > max_pow)
 214                 max_pow = pow;
 215             it++;
 216         }
 217         return max_pow;
 218     }
 219 }
 220
 221 int pseries::ldegree(const symbol &s) const
 222 {
 223     if (var.is_equal(s)) {
 224         // Return first exponent
 225         if (seq.size())
 226             return ex_to_numeric((*(seq.begin())).coeff).to_int();
 227         else
 228             return 0;
 229     } else {
 230         epvector::const_iterator it = seq.begin(), itend = seq.end();
 231         if (it == itend)
 232             return 0;
 233         int min_pow = INT_MAX;
 234         while (it != itend) {
 235             int pow = it->rest.ldegree(s);
 236             if (pow < min_pow)
 237                 min_pow = pow;
 238             it++;
 239         }
 240         return min_pow;
 241     }
 242 }
 243
 244 ex pseries::coeff(const symbol &s, int n) const
 245 {
 246     if (var.is_equal(s)) {
 247                 if (seq.size() == 0)
 248                         return _ex0();
 249
 250                 // Binary search in sequence for given power
 251                 numeric looking_for = numeric(n);
 252                 int lo = 0, hi = seq.size() - 1;
 253                 while (lo <= hi) {
 254                         int mid = (lo + hi) / 2;
 255                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 256                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 257                         switch (cmp) {
 258                                 case -1:
 259                                         lo = mid + 1;
 260                                         break;
 261                                 case 0:
 262                                         return seq[mid].rest;
 263                                 case 1:
 264                                         hi = mid - 1;
 265                                         break;
 266                                 default:
 267                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 268                         }
 269                 }
 270                 return _ex0();
 271     } else
 272         return convert_to_poly().coeff(s, n);
 273 }
 274
 275 ex pseries::collect(const symbol &s) const
 276 {
 277         return *this;
 278 }
 279
 280 /** Evaluate coefficients. */
 281 ex pseries::eval(int level) const
 282 {
 283     if (level == 1)
 284         return this->hold();
 285
 286         if (level == -max_recursion_level)
 287                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 288
 289     // Construct a new series with evaluated coefficients
 290     epvector new_seq;
 291     new_seq.reserve(seq.size());
 292     epvector::const_iterator it = seq.begin(), itend = seq.end();
 293     while (it != itend) {
 294         new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 295         it++;
 296     }
 297     return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 298 }
 299
 300 /** Evaluate coefficients numerically. */
 301 ex pseries::evalf(int level) const
 302 {
 303         if (level == 1)
 304                 return *this;
 305
 306         if (level == -max_recursion_level)
 307                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 308
 309     // Construct a new series with evaluated coefficients
 310     epvector new_seq;
 311     new_seq.reserve(seq.size());
 312     epvector::const_iterator it = seq.begin(), itend = seq.end();
 313     while (it != itend) {
 314         new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 315         it++;
 316     }
 317     return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 318 }
 319
 320 ex pseries::subs(const lst & ls, const lst & lr) const
 321 {
 322         // If expansion variable is being substituted, convert the series to a
 323         // polynomial and do the substitution there because the result might
 324         // no longer be a power series
 325         if (ls.has(var))
 326                 return convert_to_poly(true).subs(ls, lr);
 327
 328         // Otherwise construct a new series with substituted coefficients and
 329         // expansion point
 330         epvector new_seq;
 331         new_seq.reserve(seq.size());
 332         epvector::const_iterator it = seq.begin(), itend = seq.end();
 333         while (it != itend) {
 334                 new_seq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 335                 it++;
 336         }
 337     return (new pseries(relational(var,point.subs(ls, lr)), new_seq))->setflag(status_flags::dynallocated);
 338 }
 339
 340 /** Implementation of ex::diff() for a power series.  It treats the series as a
 341  *  polynomial.
 342  *  @see ex::diff */
 343 ex pseries::derivative(const symbol & s) const
 344 {
 345     if (s == var) {
 346         epvector new_seq;
 347         epvector::const_iterator it = seq.begin(), itend = seq.end();
 348
 349         // FIXME: coeff might depend on var
 350         while (it != itend) {
 351             if (is_order_function(it->rest)) {
 352                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 353             } else {
 354                 ex c = it->rest * it->coeff;
 355                 if (!c.is_zero())
 356                     new_seq.push_back(expair(c, it->coeff - 1));
 357             }
 358             it++;
 359         }
 360         return pseries(relational(var,point), new_seq);
 361     } else {
 362         return *this;
 363     }
 364 }
 365
 366
 367 /*
 368  *  Construct ordinary polynomial out of series
 369  */
 370
 371 /** Convert a pseries object to an ordinary polynomial.
 372  *
 373  *  @param no_order flag: discard higher order terms */
 374 ex pseries::convert_to_poly(bool no_order) const
 375 {
 376     ex e;
 377     epvector::const_iterator it = seq.begin(), itend = seq.end();
 378
 379     while (it != itend) {
 380         if (is_order_function(it->rest)) {
 381             if (!no_order)
 382                 e += Order(power(var - point, it->coeff));
 383         } else
 384             e += it->rest * power(var - point, it->coeff);
 385         it++;
 386     }
 387     return e;
 388 }
 389
 390
 391 /*
 392  *  Implementation of series expansion
 393  */
 394
 395 /** Default implementation of ex::series(). This performs Taylor expansion.
 396  *  @see ex::series */
 397 ex basic::series(const relational & r, int order) const
 398 {
 399     epvector seq;
 400     numeric fac(1);
 401     ex deriv = *this;
 402     ex coeff = deriv.subs(r);
 403     const symbol *s = static_cast<symbol *>(r.lhs().bp);
 404
 405     if (!coeff.is_zero())
 406         seq.push_back(expair(coeff, numeric(0)));
 407
 408     int n;
 409     for (n=1; n<order; n++) {
 410         fac = fac.mul(numeric(n));
 411         deriv = deriv.diff(*s).expand();
 412         if (deriv.is_zero()) {
 413             // Series terminates
 414             return pseries(r, seq);
 415         }
 416         coeff = fac.inverse() * deriv.subs(r);
 417         if (!coeff.is_zero())
 418             seq.push_back(expair(coeff, numeric(n)));
 419     }
 420
 421     // Higher-order terms, if present
 422     deriv = deriv.diff(*s);
 423     if (!deriv.is_zero())
 424         seq.push_back(expair(Order(_ex1()), numeric(n)));
 425     return pseries(r, seq);
 426 }
 427
 428
 429 /** Implementation of ex::series() for symbols.
 430  *  @see ex::series */
 431 ex symbol::series(const relational & r, int order) const
 432 {
 433         epvector seq;
 434     const ex point = r.rhs();
 435     GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 436     const symbol *s = static_cast<symbol *>(r.lhs().bp);
 437
 438         if (this->is_equal(*s)) {
 439                 if (order > 0 && !point.is_zero())
 440                         seq.push_back(expair(point, _ex0()));
 441                 if (order > 1)
 442                         seq.push_back(expair(_ex1(), _ex1()));
 443                 else
 444                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 445         } else
 446                 seq.push_back(expair(*this, _ex0()));
 447         return pseries(r, seq);
 448 }
 449
 450
 451 /** Add one series object to another, producing a pseries object that
 452  *  represents the sum.
 453  *
 454  *  @param other  pseries object to add with
 455  *  @return the sum as a pseries */
 456 ex pseries::add_series(const pseries &other) const
 457 {
 458     // Adding two series with different variables or expansion points
 459     // results in an empty (constant) series
 460     if (!is_compatible_to(other)) {
 461         epvector nul;
 462         nul.push_back(expair(Order(_ex1()), _ex0()));
 463         return pseries(relational(var,point), nul);
 464     }
 465
 466     // Series addition
 467     epvector new_seq;
 468     epvector::const_iterator a = seq.begin();
 469     epvector::const_iterator b = other.seq.begin();
 470     epvector::const_iterator a_end = seq.end();
 471     epvector::const_iterator b_end = other.seq.end();
 472     int pow_a = INT_MAX, pow_b = INT_MAX;
 473     for (;;) {
 474         // If a is empty, fill up with elements from b and stop
 475         if (a == a_end) {
 476             while (b != b_end) {
 477                 new_seq.push_back(*b);
 478                 b++;
 479             }
 480             break;
 481         } else
 482             pow_a = ex_to_numeric((*a).coeff).to_int();
 483
 484         // If b is empty, fill up with elements from a and stop
 485         if (b == b_end) {
 486             while (a != a_end) {
 487                 new_seq.push_back(*a);
 488                 a++;
 489             }
 490             break;
 491         } else
 492             pow_b = ex_to_numeric((*b).coeff).to_int();
 493
 494         // a and b are non-empty, compare powers
 495         if (pow_a < pow_b) {
 496             // a has lesser power, get coefficient from a
 497             new_seq.push_back(*a);
 498             if (is_order_function((*a).rest))
 499                 break;
 500             a++;
 501         } else if (pow_b < pow_a) {
 502             // b has lesser power, get coefficient from b
 503             new_seq.push_back(*b);
 504             if (is_order_function((*b).rest))
 505                 break;
 506             b++;
 507         } else {
 508             // Add coefficient of a and b
 509             if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 510                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 511                 break;  // Order term ends the sequence
 512             } else {
 513                 ex sum = (*a).rest + (*b).rest;
 514                 if (!(sum.is_zero()))
 515                     new_seq.push_back(expair(sum, numeric(pow_a)));
 516                 a++;
 517                 b++;
 518             }
 519         }
 520     }
 521     return pseries(relational(var,point), new_seq);
 522 }
 523
 524
 525 /** Implementation of ex::series() for sums. This performs series addition when
 526  *  adding pseries objects.
 527  *  @see ex::series */
 528 ex add::series(const relational & r, int order) const
 529 {
 530     ex acc; // Series accumulator
 531
 532     // Get first term from overall_coeff
 533     acc = overall_coeff.series(r, order);
 534
 535     // Add remaining terms
 536     epvector::const_iterator it = seq.begin();
 537     epvector::const_iterator itend = seq.end();
 538     for (; it!=itend; it++) {
 539         ex op;
 540         if (is_ex_exactly_of_type(it->rest, pseries))
 541             op = it->rest;
 542         else
 543             op = it->rest.series(r, order);
 544         if (!it->coeff.is_equal(_ex1()))
 545             op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 546
 547         // Series addition
 548         acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 549     }
 550     return acc;
 551 }
 552
 553
 554 /** Multiply a pseries object with a numeric constant, producing a pseries
 555  *  object that represents the product.
 556  *
 557  *  @param other  constant to multiply with
 558  *  @return the product as a pseries */
 559 ex pseries::mul_const(const numeric &other) const
 560 {
 561     epvector new_seq;
 562     new_seq.reserve(seq.size());
 563
 564     epvector::const_iterator it = seq.begin(), itend = seq.end();
 565     while (it != itend) {
 566         if (!is_order_function(it->rest))
 567             new_seq.push_back(expair(it->rest * other, it->coeff));
 568         else
 569             new_seq.push_back(*it);
 570         it++;
 571     }
 572     return pseries(relational(var,point), new_seq);
 573 }
 574
 575
 576 /** Multiply one pseries object to another, producing a pseries object that
 577  *  represents the product.
 578  *
 579  *  @param other  pseries object to multiply with
 580  *  @return the product as a pseries */
 581 ex pseries::mul_series(const pseries &other) const
 582 {
 583     // Multiplying two series with different variables or expansion points
 584     // results in an empty (constant) series
 585     if (!is_compatible_to(other)) {
 586         epvector nul;
 587         nul.push_back(expair(Order(_ex1()), _ex0()));
 588         return pseries(relational(var,point), nul);
 589     }
 590
 591     // Series multiplication
 592     epvector new_seq;
 593
 594     const symbol *s = static_cast<symbol *>(var.bp);
 595     int a_max = degree(*s);
 596     int b_max = other.degree(*s);
 597     int a_min = ldegree(*s);
 598     int b_min = other.ldegree(*s);
 599     int cdeg_min = a_min + b_min;
 600     int cdeg_max = a_max + b_max;
 601
 602     int higher_order_a = INT_MAX;
 603     int higher_order_b = INT_MAX;
 604     if (is_order_function(coeff(*s, a_max)))
 605         higher_order_a = a_max + b_min;
 606     if (is_order_function(other.coeff(*s, b_max)))
 607         higher_order_b = b_max + a_min;
 608     int higher_order_c = min(higher_order_a, higher_order_b);
 609     if (cdeg_max >= higher_order_c)
 610         cdeg_max = higher_order_c - 1;
 611
 612     for (int cdeg=cdeg_min; cdeg<=cdeg_max; cdeg++) {
 613         ex co = _ex0();
 614         // c(i)=a(0)b(i)+...+a(i)b(0)
 615         for (int i=a_min; cdeg-i>=b_min; i++) {
 616             ex a_coeff = coeff(*s, i);
 617             ex b_coeff = other.coeff(*s, cdeg-i);
 618             if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 619                 co += coeff(*s, i) * other.coeff(*s, cdeg-i);
 620         }
 621         if (!co.is_zero())
 622             new_seq.push_back(expair(co, numeric(cdeg)));
 623     }
 624     if (higher_order_c < INT_MAX)
 625         new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 626     return pseries(relational(var,point), new_seq);
 627 }
 628
 629
 630 /** Implementation of ex::series() for product. This performs series
 631  *  multiplication when multiplying series.
 632  *  @see ex::series */
 633 ex mul::series(const relational & r, int order) const
 634 {
 635     ex acc; // Series accumulator
 636
 637     // Get first term from overall_coeff
 638     acc = overall_coeff.series(r, order);
 639
 640     // Multiply with remaining terms
 641     epvector::const_iterator it = seq.begin();
 642     epvector::const_iterator itend = seq.end();
 643     for (; it!=itend; it++) {
 644         ex op = it->rest;
 645         if (op.info(info_flags::numeric)) {
 646             // series * const (special case, faster)
 647             ex f = power(op, it->coeff);
 648             acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 649             continue;
 650         } else if (!is_ex_exactly_of_type(op, pseries))
 651             op = op.series(r, order);
 652         if (!it->coeff.is_equal(_ex1()))
 653             op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 654
 655         // Series multiplication
 656         acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 657     }
 658     return acc;
 659 }
 660
 661
 662 /** Compute the p-th power of a series.
 663  *
 664  *  @param p  power to compute
 665  *  @param deg  truncation order of series calculation */
 666 ex pseries::power_const(const numeric &p, int deg) const
 667 {
 668     int i;
 669     const symbol *s = static_cast<symbol *>(var.bp);
 670     int ldeg = ldegree(*s);
 671
 672     // Calculate coefficients of powered series
 673     exvector co;
 674     co.reserve(deg);
 675     ex co0;
 676     co.push_back(co0 = power(coeff(*s, ldeg), p));
 677     bool all_sums_zero = true;
 678     for (i=1; i<deg; i++) {
 679         ex sum = _ex0();
 680         for (int j=1; j<=i; j++) {
 681             ex c = coeff(*s, j + ldeg);
 682             if (is_order_function(c)) {
 683                 co.push_back(Order(_ex1()));
 684                 break;
 685             } else
 686                 sum += (p * j - (i - j)) * co[i - j] * c;
 687         }
 688         if (!sum.is_zero())
 689             all_sums_zero = false;
 690         co.push_back(co0 * sum / numeric(i));
 691     }
 692
 693     // Construct new series (of non-zero coefficients)
 694     epvector new_seq;
 695     bool higher_order = false;
 696     for (i=0; i<deg; i++) {
 697         if (!co[i].is_zero())
 698             new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 699         if (is_order_function(co[i])) {
 700             higher_order = true;
 701             break;
 702         }
 703     }
 704     if (!higher_order && !all_sums_zero)
 705         new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 706     return pseries(relational(var,point), new_seq);
 707 }
 708
 709
 710 /** Implementation of ex::series() for powers. This performs Laurent expansion
 711  *  of reciprocals of series at singularities.
 712  *  @see ex::series */
 713 ex power::series(const relational & r, int order) const
 714 {
 715     ex e;
 716     if (!is_ex_exactly_of_type(basis, pseries)) {
 717         // Basis is not a series, may there be a singulary?
 718         if (!exponent.info(info_flags::negint))
 719             return basic::series(r, order);
 720
 721         // Expression is of type something^(-int), check for singularity
 722         if (!basis.subs(r).is_zero())
 723             return basic::series(r, order);
 724
 725         // Singularity encountered, expand basis into series
 726         e = basis.series(r, order);
 727     } else {
 728         // Basis is a series
 729         e = basis;
 730     }
 731
 732     // Power e
 733     return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 734 }
 735
 736
 737 /** Re-expansion of a pseries object. */
 738 ex pseries::series(const relational & r, int order) const
 739 {
 740     const ex p = r.rhs();
 741     GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 742     const symbol *s = static_cast<symbol *>(r.lhs().bp);
 743
 744         if (var.is_equal(*s) && point.is_equal(p)) {
 745                 if (order > degree(*s))
 746                         return *this;
 747                 else {
 748                 epvector new_seq;
 749                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 750                         while (it != itend) {
 751                                 int o = ex_to_numeric(it->coeff).to_int();
 752                                 if (o >= order) {
 753                                         new_seq.push_back(expair(Order(_ex1()), o));
 754                                         break;
 755                                 }
 756                                 new_seq.push_back(*it);
 757                                 it++;
 758                         }
 759                         return pseries(r, new_seq);
 760                 }
 761         } else
 762                 return convert_to_poly().series(r, order);
 763 }
 764
 765
 766 /** Compute the truncated series expansion of an expression.
 767  *  This function returns an expression containing an object of class pseries
 768  *  to represent the series. If the series does not terminate within the given
 769  *  truncation order, the last term of the series will be an order term.
 770  *
 771  *  @param r  expansion relation, lhs holds variable and rhs holds point
 772  *  @param order  truncation order of series calculations
 773  *  @return an expression holding a pseries object */
 774 ex ex::series(const ex & r, int order) const
 775 {
 776     GINAC_ASSERT(bp!=0);
 777     ex e;
 778     relational rel_;
 779
 780     if (is_ex_exactly_of_type(r,relational))
 781         rel_ = ex_to_relational(r);
 782     else if (is_ex_exactly_of_type(r,symbol))
 783         rel_ = relational(r,_ex0());
 784     else
 785         throw (std::logic_error("ex::series(): expansion point has unknown type"));
 786
 787     try {
 788         e = bp->series(rel_, order);
 789     } catch (exception &x) {
 790         throw (std::logic_error(string("unable to compute series (") + x.what() + ")"));
 791     }
 792     return e;
 793 }
 794
 795
 796 // Global constants
 797 const pseries some_pseries;
 798 const type_info & typeid_pseries = typeid(some_pseries);
 799
 800 #ifndef NO_NAMESPACE_GINAC
 801 } // namespace GiNaC
 802 #endif // ndef NO_NAMESPACE_GINAC