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 void pseries::printtree(ostream & os, unsigned indent) const
 182 {
 183     debugmsg("pseries printtree",LOGLEVEL_PRINT);
 184     os << string(indent,' ') << "pseries "
 185        << ", hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
 186        << ", flags=" << flags << endl;
 187     for (unsigned i=0; i<seq.size(); ++i) {
 188         seq[i].rest.printtree(os,indent+delta_indent);
 189         seq[i].coeff.printtree(os,indent+delta_indent);
 190         if (i!=seq.size()-1) {
 191             os << string(indent+delta_indent,' ') << "-----" << endl;
 192         }
 193     }
 194     var.printtree(os, indent+delta_indent);
 195     point.printtree(os, indent+delta_indent);
 196 }
 197
 198 unsigned pseries::nops(void) const
 199 {
 200     return seq.size();
 201 }
 202
 203 ex pseries::op(int i) const
 204 {
 205     if (i < 0 || unsigned(i) >= seq.size())
 206         throw (std::out_of_range("op() out of range"));
 207     return seq[i].rest * power(var - point, seq[i].coeff);
 208 }
 209
 210 ex &pseries::let_op(int i)
 211 {
 212     throw (std::logic_error("let_op not defined for pseries"));
 213 }
 214
 215 int pseries::degree(const symbol &s) const
 216 {
 217     if (var.is_equal(s)) {
 218         // Return last exponent
 219         if (seq.size())
 220             return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 221         else
 222             return 0;
 223     } else {
 224         epvector::const_iterator it = seq.begin(), itend = seq.end();
 225         if (it == itend)
 226             return 0;
 227         int max_pow = INT_MIN;
 228         while (it != itend) {
 229             int pow = it->rest.degree(s);
 230             if (pow > max_pow)
 231                 max_pow = pow;
 232             it++;
 233         }
 234         return max_pow;
 235     }
 236 }
 237
 238 int pseries::ldegree(const symbol &s) const
 239 {
 240     if (var.is_equal(s)) {
 241         // Return first exponent
 242         if (seq.size())
 243             return ex_to_numeric((*(seq.begin())).coeff).to_int();
 244         else
 245             return 0;
 246     } else {
 247         epvector::const_iterator it = seq.begin(), itend = seq.end();
 248         if (it == itend)
 249             return 0;
 250         int min_pow = INT_MAX;
 251         while (it != itend) {
 252             int pow = it->rest.ldegree(s);
 253             if (pow < min_pow)
 254                 min_pow = pow;
 255             it++;
 256         }
 257         return min_pow;
 258     }
 259 }
 260
 261 ex pseries::coeff(const symbol &s, int n) const
 262 {
 263     if (var.is_equal(s)) {
 264                 if (seq.size() == 0)
 265                         return _ex0();
 266
 267                 // Binary search in sequence for given power
 268                 numeric looking_for = numeric(n);
 269                 int lo = 0, hi = seq.size() - 1;
 270                 while (lo <= hi) {
 271                         int mid = (lo + hi) / 2;
 272                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 273                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 274                         switch (cmp) {
 275                                 case -1:
 276                                         lo = mid + 1;
 277                                         break;
 278                                 case 0:
 279                                         return seq[mid].rest;
 280                                 case 1:
 281                                         hi = mid - 1;
 282                                         break;
 283                                 default:
 284                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 285                         }
 286                 }
 287                 return _ex0();
 288     } else
 289         return convert_to_poly().coeff(s, n);
 290 }
 291
 292 ex pseries::collect(const symbol &s) const
 293 {
 294         return *this;
 295 }
 296
 297 /** Evaluate coefficients. */
 298 ex pseries::eval(int level) const
 299 {
 300     if (level == 1)
 301         return this->hold();
 302
 303         if (level == -max_recursion_level)
 304                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 305
 306     // Construct a new series with evaluated coefficients
 307     epvector new_seq;
 308     new_seq.reserve(seq.size());
 309     epvector::const_iterator it = seq.begin(), itend = seq.end();
 310     while (it != itend) {
 311         new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 312         it++;
 313     }
 314     return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 315 }
 316
 317 /** Evaluate coefficients numerically. */
 318 ex pseries::evalf(int level) const
 319 {
 320         if (level == 1)
 321                 return *this;
 322
 323         if (level == -max_recursion_level)
 324                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 325
 326     // Construct a new series with evaluated coefficients
 327     epvector new_seq;
 328     new_seq.reserve(seq.size());
 329     epvector::const_iterator it = seq.begin(), itend = seq.end();
 330     while (it != itend) {
 331         new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 332         it++;
 333     }
 334     return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 335 }
 336
 337 ex pseries::subs(const lst & ls, const lst & lr) const
 338 {
 339         // If expansion variable is being substituted, convert the series to a
 340         // polynomial and do the substitution there because the result might
 341         // no longer be a power series
 342         if (ls.has(var))
 343                 return convert_to_poly(true).subs(ls, lr);
 344
 345         // Otherwise construct a new series with substituted coefficients and
 346         // expansion point
 347         epvector new_seq;
 348         new_seq.reserve(seq.size());
 349         epvector::const_iterator it = seq.begin(), itend = seq.end();
 350         while (it != itend) {
 351                 new_seq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 352                 it++;
 353         }
 354     return (new pseries(relational(var,point.subs(ls, lr)), new_seq))->setflag(status_flags::dynallocated);
 355 }
 356
 357 /** Implementation of ex::diff() for a power series.  It treats the series as a
 358  *  polynomial.
 359  *  @see ex::diff */
 360 ex pseries::derivative(const symbol & s) const
 361 {
 362     if (s == var) {
 363         epvector new_seq;
 364         epvector::const_iterator it = seq.begin(), itend = seq.end();
 365
 366         // FIXME: coeff might depend on var
 367         while (it != itend) {
 368             if (is_order_function(it->rest)) {
 369                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 370             } else {
 371                 ex c = it->rest * it->coeff;
 372                 if (!c.is_zero())
 373                     new_seq.push_back(expair(c, it->coeff - 1));
 374             }
 375             it++;
 376         }
 377         return pseries(relational(var,point), new_seq);
 378     } else {
 379         return *this;
 380     }
 381 }
 382
 383
 384 /*
 385  *  Construct ordinary polynomial out of series
 386  */
 387
 388 /** Convert a pseries object to an ordinary polynomial.
 389  *
 390  *  @param no_order flag: discard higher order terms */
 391 ex pseries::convert_to_poly(bool no_order) const
 392 {
 393     ex e;
 394     epvector::const_iterator it = seq.begin(), itend = seq.end();
 395
 396     while (it != itend) {
 397         if (is_order_function(it->rest)) {
 398             if (!no_order)
 399                 e += Order(power(var - point, it->coeff));
 400         } else
 401             e += it->rest * power(var - point, it->coeff);
 402         it++;
 403     }
 404     return e;
 405 }
 406
 407
 408 /*
 409  *  Implementation of series expansion
 410  */
 411
 412 /** Default implementation of ex::series(). This performs Taylor expansion.
 413  *  @see ex::series */
 414 ex basic::series(const relational & r, int order) const
 415 {
 416     epvector seq;
 417     numeric fac(1);
 418     ex deriv = *this;
 419     ex coeff = deriv.subs(r);
 420     const symbol *s = static_cast<symbol *>(r.lhs().bp);
 421
 422     if (!coeff.is_zero())
 423         seq.push_back(expair(coeff, numeric(0)));
 424
 425     int n;
 426     for (n=1; n<order; n++) {
 427         fac = fac.mul(numeric(n));
 428         deriv = deriv.diff(*s).expand();
 429         if (deriv.is_zero()) {
 430             // Series terminates
 431             return pseries(r, seq);
 432         }
 433         coeff = fac.inverse() * deriv.subs(r);
 434         if (!coeff.is_zero())
 435             seq.push_back(expair(coeff, numeric(n)));
 436     }
 437
 438     // Higher-order terms, if present
 439     deriv = deriv.diff(*s);
 440     if (!deriv.is_zero())
 441         seq.push_back(expair(Order(_ex1()), numeric(n)));
 442     return pseries(r, seq);
 443 }
 444
 445
 446 /** Implementation of ex::series() for symbols.
 447  *  @see ex::series */
 448 ex symbol::series(const relational & r, int order) const
 449 {
 450         epvector seq;
 451     const ex point = r.rhs();
 452     GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 453     const symbol *s = static_cast<symbol *>(r.lhs().bp);
 454
 455         if (this->is_equal(*s)) {
 456                 if (order > 0 && !point.is_zero())
 457                         seq.push_back(expair(point, _ex0()));
 458                 if (order > 1)
 459                         seq.push_back(expair(_ex1(), _ex1()));
 460                 else
 461                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 462         } else
 463                 seq.push_back(expair(*this, _ex0()));
 464         return pseries(r, seq);
 465 }
 466
 467
 468 /** Add one series object to another, producing a pseries object that
 469  *  represents the sum.
 470  *
 471  *  @param other  pseries object to add with
 472  *  @return the sum as a pseries */
 473 ex pseries::add_series(const pseries &other) const
 474 {
 475     // Adding two series with different variables or expansion points
 476     // results in an empty (constant) series
 477     if (!is_compatible_to(other)) {
 478         epvector nul;
 479         nul.push_back(expair(Order(_ex1()), _ex0()));
 480         return pseries(relational(var,point), nul);
 481     }
 482
 483     // Series addition
 484     epvector new_seq;
 485     epvector::const_iterator a = seq.begin();
 486     epvector::const_iterator b = other.seq.begin();
 487     epvector::const_iterator a_end = seq.end();
 488     epvector::const_iterator b_end = other.seq.end();
 489     int pow_a = INT_MAX, pow_b = INT_MAX;
 490     for (;;) {
 491         // If a is empty, fill up with elements from b and stop
 492         if (a == a_end) {
 493             while (b != b_end) {
 494                 new_seq.push_back(*b);
 495                 b++;
 496             }
 497             break;
 498         } else
 499             pow_a = ex_to_numeric((*a).coeff).to_int();
 500
 501         // If b is empty, fill up with elements from a and stop
 502         if (b == b_end) {
 503             while (a != a_end) {
 504                 new_seq.push_back(*a);
 505                 a++;
 506             }
 507             break;
 508         } else
 509             pow_b = ex_to_numeric((*b).coeff).to_int();
 510
 511         // a and b are non-empty, compare powers
 512         if (pow_a < pow_b) {
 513             // a has lesser power, get coefficient from a
 514             new_seq.push_back(*a);
 515             if (is_order_function((*a).rest))
 516                 break;
 517             a++;
 518         } else if (pow_b < pow_a) {
 519             // b has lesser power, get coefficient from b
 520             new_seq.push_back(*b);
 521             if (is_order_function((*b).rest))
 522                 break;
 523             b++;
 524         } else {
 525             // Add coefficient of a and b
 526             if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 527                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 528                 break;  // Order term ends the sequence
 529             } else {
 530                 ex sum = (*a).rest + (*b).rest;
 531                 if (!(sum.is_zero()))
 532                     new_seq.push_back(expair(sum, numeric(pow_a)));
 533                 a++;
 534                 b++;
 535             }
 536         }
 537     }
 538     return pseries(relational(var,point), new_seq);
 539 }
 540
 541
 542 /** Implementation of ex::series() for sums. This performs series addition when
 543  *  adding pseries objects.
 544  *  @see ex::series */
 545 ex add::series(const relational & r, int order) const
 546 {
 547     ex acc; // Series accumulator
 548
 549     // Get first term from overall_coeff
 550     acc = overall_coeff.series(r, order);
 551
 552     // Add remaining terms
 553     epvector::const_iterator it = seq.begin();
 554     epvector::const_iterator itend = seq.end();
 555     for (; it!=itend; it++) {
 556         ex op;
 557         if (is_ex_exactly_of_type(it->rest, pseries))
 558             op = it->rest;
 559         else
 560             op = it->rest.series(r, order);
 561         if (!it->coeff.is_equal(_ex1()))
 562             op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 563
 564         // Series addition
 565         acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 566     }
 567     return acc;
 568 }
 569
 570
 571 /** Multiply a pseries object with a numeric constant, producing a pseries
 572  *  object that represents the product.
 573  *
 574  *  @param other  constant to multiply with
 575  *  @return the product as a pseries */
 576 ex pseries::mul_const(const numeric &other) const
 577 {
 578     epvector new_seq;
 579     new_seq.reserve(seq.size());
 580
 581     epvector::const_iterator it = seq.begin(), itend = seq.end();
 582     while (it != itend) {
 583         if (!is_order_function(it->rest))
 584             new_seq.push_back(expair(it->rest * other, it->coeff));
 585         else
 586             new_seq.push_back(*it);
 587         it++;
 588     }
 589     return pseries(relational(var,point), new_seq);
 590 }
 591
 592
 593 /** Multiply one pseries object to another, producing a pseries object that
 594  *  represents the product.
 595  *
 596  *  @param other  pseries object to multiply with
 597  *  @return the product as a pseries */
 598 ex pseries::mul_series(const pseries &other) const
 599 {
 600     // Multiplying two series with different variables or expansion points
 601     // results in an empty (constant) series
 602     if (!is_compatible_to(other)) {
 603         epvector nul;
 604         nul.push_back(expair(Order(_ex1()), _ex0()));
 605         return pseries(relational(var,point), nul);
 606     }
 607
 608     // Series multiplication
 609     epvector new_seq;
 610
 611     const symbol *s = static_cast<symbol *>(var.bp);
 612     int a_max = degree(*s);
 613     int b_max = other.degree(*s);
 614     int a_min = ldegree(*s);
 615     int b_min = other.ldegree(*s);
 616     int cdeg_min = a_min + b_min;
 617     int cdeg_max = a_max + b_max;
 618
 619     int higher_order_a = INT_MAX;
 620     int higher_order_b = INT_MAX;
 621     if (is_order_function(coeff(*s, a_max)))
 622         higher_order_a = a_max + b_min;
 623     if (is_order_function(other.coeff(*s, b_max)))
 624         higher_order_b = b_max + a_min;
 625     int higher_order_c = min(higher_order_a, higher_order_b);
 626     if (cdeg_max >= higher_order_c)
 627         cdeg_max = higher_order_c - 1;
 628
 629     for (int cdeg=cdeg_min; cdeg<=cdeg_max; cdeg++) {
 630         ex co = _ex0();
 631         // c(i)=a(0)b(i)+...+a(i)b(0)
 632         for (int i=a_min; cdeg-i>=b_min; i++) {
 633             ex a_coeff = coeff(*s, i);
 634             ex b_coeff = other.coeff(*s, cdeg-i);
 635             if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 636                 co += coeff(*s, i) * other.coeff(*s, cdeg-i);
 637         }
 638         if (!co.is_zero())
 639             new_seq.push_back(expair(co, numeric(cdeg)));
 640     }
 641     if (higher_order_c < INT_MAX)
 642         new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 643     return pseries(relational(var,point), new_seq);
 644 }
 645
 646
 647 /** Implementation of ex::series() for product. This performs series
 648  *  multiplication when multiplying series.
 649  *  @see ex::series */
 650 ex mul::series(const relational & r, int order) const
 651 {
 652     ex acc; // Series accumulator
 653
 654     // Get first term from overall_coeff
 655     acc = overall_coeff.series(r, order);
 656
 657     // Multiply with remaining terms
 658     epvector::const_iterator it = seq.begin();
 659     epvector::const_iterator itend = seq.end();
 660     for (; it!=itend; it++) {
 661         ex op = it->rest;
 662         if (op.info(info_flags::numeric)) {
 663             // series * const (special case, faster)
 664             ex f = power(op, it->coeff);
 665             acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 666             continue;
 667         } else if (!is_ex_exactly_of_type(op, pseries))
 668             op = op.series(r, order);
 669         if (!it->coeff.is_equal(_ex1()))
 670             op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 671
 672         // Series multiplication
 673         acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 674     }
 675     return acc;
 676 }
 677
 678
 679 /** Compute the p-th power of a series.
 680  *
 681  *  @param p  power to compute
 682  *  @param deg  truncation order of series calculation */
 683 ex pseries::power_const(const numeric &p, int deg) const
 684 {
 685     int i;
 686     const symbol *s = static_cast<symbol *>(var.bp);
 687     int ldeg = ldegree(*s);
 688
 689     // Calculate coefficients of powered series
 690     exvector co;
 691     co.reserve(deg);
 692     ex co0;
 693     co.push_back(co0 = power(coeff(*s, ldeg), p));
 694     bool all_sums_zero = true;
 695     for (i=1; i<deg; i++) {
 696         ex sum = _ex0();
 697         for (int j=1; j<=i; j++) {
 698             ex c = coeff(*s, j + ldeg);
 699             if (is_order_function(c)) {
 700                 co.push_back(Order(_ex1()));
 701                 break;
 702             } else
 703                 sum += (p * j - (i - j)) * co[i - j] * c;
 704         }
 705         if (!sum.is_zero())
 706             all_sums_zero = false;
 707         co.push_back(co0 * sum / numeric(i));
 708     }
 709
 710     // Construct new series (of non-zero coefficients)
 711     epvector new_seq;
 712     bool higher_order = false;
 713     for (i=0; i<deg; i++) {
 714         if (!co[i].is_zero())
 715             new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 716         if (is_order_function(co[i])) {
 717             higher_order = true;
 718             break;
 719         }
 720     }
 721     if (!higher_order && !all_sums_zero)
 722         new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 723     return pseries(relational(var,point), new_seq);
 724 }
 725
 726
 727 /** Implementation of ex::series() for powers. This performs Laurent expansion
 728  *  of reciprocals of series at singularities.
 729  *  @see ex::series */
 730 ex power::series(const relational & r, int order) const
 731 {
 732     ex e;
 733     if (!is_ex_exactly_of_type(basis, pseries)) {
 734         // Basis is not a series, may there be a singulary?
 735         if (!exponent.info(info_flags::negint))
 736             return basic::series(r, order);
 737
 738         // Expression is of type something^(-int), check for singularity
 739         if (!basis.subs(r).is_zero())
 740             return basic::series(r, order);
 741
 742         // Singularity encountered, expand basis into series
 743         e = basis.series(r, order);
 744     } else {
 745         // Basis is a series
 746         e = basis;
 747     }
 748
 749     // Power e
 750     return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 751 }
 752
 753
 754 /** Re-expansion of a pseries object. */
 755 ex pseries::series(const relational & r, int order) const
 756 {
 757     const ex p = r.rhs();
 758     GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 759     const symbol *s = static_cast<symbol *>(r.lhs().bp);
 760
 761         if (var.is_equal(*s) && point.is_equal(p)) {
 762                 if (order > degree(*s))
 763                         return *this;
 764                 else {
 765                 epvector new_seq;
 766                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 767                         while (it != itend) {
 768                                 int o = ex_to_numeric(it->coeff).to_int();
 769                                 if (o >= order) {
 770                                         new_seq.push_back(expair(Order(_ex1()), o));
 771                                         break;
 772                                 }
 773                                 new_seq.push_back(*it);
 774                                 it++;
 775                         }
 776                         return pseries(r, new_seq);
 777                 }
 778         } else
 779                 return convert_to_poly().series(r, order);
 780 }
 781
 782
 783 /** Compute the truncated series expansion of an expression.
 784  *  This function returns an expression containing an object of class pseries
 785  *  to represent the series. If the series does not terminate within the given
 786  *  truncation order, the last term of the series will be an order term.
 787  *
 788  *  @param r  expansion relation, lhs holds variable and rhs holds point
 789  *  @param order  truncation order of series calculations
 790  *  @return an expression holding a pseries object */
 791 ex ex::series(const ex & r, int order) const
 792 {
 793     GINAC_ASSERT(bp!=0);
 794     ex e;
 795     relational rel_;
 796
 797     if (is_ex_exactly_of_type(r,relational))
 798         rel_ = ex_to_relational(r);
 799     else if (is_ex_exactly_of_type(r,symbol))
 800         rel_ = relational(r,_ex0());
 801     else
 802         throw (std::logic_error("ex::series(): expansion point has unknown type"));
 803
 804     try {
 805         e = bp->series(rel_, order);
 806     } catch (exception &x) {
 807         throw (std::logic_error(string("unable to compute series (") + x.what() + ")"));
 808     }
 809     return e;
 810 }
 811
 812
 813 // Global constants
 814 const pseries some_pseries;
 815 const type_info & typeid_pseries = typeid(some_pseries);
 816
 817 #ifndef NO_NAMESPACE_GINAC
 818 } // namespace GiNaC
 819 #endif // ndef NO_NAMESPACE_GINAC