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) 1999-2001 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 void pseries::copy(const pseries &other)
  54 {
  55         inherited::copy(other);
  56         seq = other.seq;
  57         var = other.var;
  58         point = other.point;
  59 }
  60
  61 void pseries::destroy(bool call_parent)
  62 {
  63         if (call_parent)
  64                 inherited::destroy(call_parent);
  65 }
  66
  67
  68 /*
  69  *  Other constructors
  70  */
  71
  72 /** Construct pseries from a vector of coefficients and powers.
  73  *  expair.rest holds the coefficient, expair.coeff holds the power.
  74  *  The powers must be integers (positive or negative) and in ascending order;
  75  *  the last coefficient can be Order(_ex1()) to represent a truncated,
  76  *  non-terminating series.
  77  *
  78  *  @param rel_  expansion variable and point (must hold a relational)
  79  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  80  *  @return newly constructed pseries */
  81 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  82 {
  83         debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
  84         GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
  85         GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
  86         point = rel_.rhs();
  87         var = *static_cast<symbol *>(rel_.lhs().bp);
  88 }
  89
  90
  91 /*
  92  *  Archiving
  93  */
  94
  95 /** Construct object from archive_node. */
  96 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  97 {
  98         debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
  99         for (unsigned int i=0; true; ++i) {
 100                 ex rest;
 101                 ex coeff;
 102                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
 103                         seq.push_back(expair(rest, coeff));
 104                 else
 105                         break;
 106         }
 107         n.find_ex("var", var, sym_lst);
 108         n.find_ex("point", point, sym_lst);
 109 }
 110
 111 /** Unarchive the object. */
 112 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
 113 {
 114         return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
 115 }
 116
 117 /** Archive the object. */
 118 void pseries::archive(archive_node &n) const
 119 {
 120         inherited::archive(n);
 121         epvector::const_iterator i = seq.begin(), iend = seq.end();
 122         while (i != iend) {
 123                 n.add_ex("coeff", i->rest);
 124                 n.add_ex("power", i->coeff);
 125                 ++i;
 126         }
 127         n.add_ex("var", var);
 128         n.add_ex("point", point);
 129 }
 130
 131 //////////
 132 // functions overriding virtual functions from bases classes
 133 //////////
 134
 135 basic *pseries::duplicate() const
 136 {
 137         debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
 138         return new pseries(*this);
 139 }
 140
 141 void pseries::print(std::ostream &os, unsigned upper_precedence) const
 142 {
 143         debugmsg("pseries print", LOGLEVEL_PRINT);
 144         if (precedence<=upper_precedence) os << "(";
 145         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 146                 // omit zero terms
 147                 if (i->rest.is_zero())
 148                         continue;
 149                 // print a sign, if needed
 150                 if (i!=seq.begin())
 151                         os << '+';
 152                 if (!is_order_function(i->rest)) {
 153                         // print 'rest', i.e. the expansion coefficient
 154                         if (i->rest.info(info_flags::numeric) &&
 155                                 i->rest.info(info_flags::positive)) {
 156                                 os << i->rest;
 157                         } else
 158                                 os << "(" << i->rest << ')';
 159                         // print 'coeff', something like (x-1)^42
 160                         if (!i->coeff.is_zero()) {
 161                                 os << '*';
 162                                 if (!point.is_zero())
 163                                         os << '(' << var-point << ')';
 164                                 else
 165                                         os << var;
 166                                 if (i->coeff.compare(_ex1())) {
 167                                         os << '^';
 168                                         if (i->coeff.info(info_flags::negative))
 169                                                 os << '(' << i->coeff << ')';
 170                                         else
 171                                                 os << i->coeff;
 172                                 }
 173                         }
 174                 } else {
 175                         os << Order(power(var-point,i->coeff));
 176                 }
 177         }
 178         if (precedence<=upper_precedence) os << ")";
 179 }
 180
 181
 182 void pseries::printraw(std::ostream &os) const
 183 {
 184         debugmsg("pseries printraw", LOGLEVEL_PRINT);
 185         os << "pseries(" << var << ";" << point << ";";
 186         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 187                 os << "(" << (*i).rest << "," << (*i).coeff << "),";
 188         }
 189         os << ")";
 190 }
 191
 192
 193 void pseries::printtree(std::ostream & os, unsigned indent) const
 194 {
 195         debugmsg("pseries printtree",LOGLEVEL_PRINT);
 196         os << std::string(indent,' ') << "pseries "
 197            << ", hash=" << hashvalue
 198            << " (0x" << std::hex << hashvalue << std::dec << ")"
 199            << ", flags=" << flags << std::endl;
 200         for (unsigned i=0; i<seq.size(); ++i) {
 201                 seq[i].rest.printtree(os,indent+delta_indent);
 202                 seq[i].coeff.printtree(os,indent+delta_indent);
 203                 if (i!=seq.size()-1)
 204                         os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
 205         }
 206         var.printtree(os, indent+delta_indent);
 207         point.printtree(os, indent+delta_indent);
 208 }
 209
 210 /** Return the number of operands including a possible order term. */
 211 unsigned pseries::nops(void) const
 212 {
 213         return seq.size();
 214 }
 215
 216
 217 /** Return the ith term in the series when represented as a sum. */
 218 ex pseries::op(int i) const
 219 {
 220         if (i < 0 || unsigned(i) >= seq.size())
 221                 throw (std::out_of_range("op() out of range"));
 222         return seq[i].rest * power(var - point, seq[i].coeff);
 223 }
 224
 225
 226 ex &pseries::let_op(int i)
 227 {
 228         throw (std::logic_error("let_op not defined for pseries"));
 229 }
 230
 231
 232 /** Return degree of highest power of the series.  This is usually the exponent
 233  *  of the Order term.  If s is not the expansion variable of the series, the
 234  *  series is examined termwise. */
 235 int pseries::degree(const symbol &s) const
 236 {
 237         if (var.is_equal(s)) {
 238                 // Return last exponent
 239                 if (seq.size())
 240                         return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 241                 else
 242                         return 0;
 243         } else {
 244                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 245                 if (it == itend)
 246                         return 0;
 247                 int max_pow = INT_MIN;
 248                 while (it != itend) {
 249                         int pow = it->rest.degree(s);
 250                         if (pow > max_pow)
 251                                 max_pow = pow;
 252                         ++it;
 253                 }
 254                 return max_pow;
 255         }
 256 }
 257
 258 /** Return degree of lowest power of the series.  This is usually the exponent
 259  *  of the leading term.  If s is not the expansion variable of the series, the
 260  *  series is examined termwise.  If s is the expansion variable but the
 261  *  expansion point is not zero the series is not expanded to find the degree.
 262  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 263 int pseries::ldegree(const symbol &s) const
 264 {
 265         if (var.is_equal(s)) {
 266                 // Return first exponent
 267                 if (seq.size())
 268                         return ex_to_numeric((*(seq.begin())).coeff).to_int();
 269                 else
 270                         return 0;
 271         } else {
 272                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 273                 if (it == itend)
 274                         return 0;
 275                 int min_pow = INT_MAX;
 276                 while (it != itend) {
 277                         int pow = it->rest.ldegree(s);
 278                         if (pow < min_pow)
 279                                 min_pow = pow;
 280                         ++it;
 281                 }
 282                 return min_pow;
 283         }
 284 }
 285
 286 ex pseries::coeff(const symbol &s, int n) const
 287 {
 288         if (var.is_equal(s)) {
 289                 if (seq.size() == 0)
 290                         return _ex0();
 291
 292                 // Binary search in sequence for given power
 293                 numeric looking_for = numeric(n);
 294                 int lo = 0, hi = seq.size() - 1;
 295                 while (lo <= hi) {
 296                         int mid = (lo + hi) / 2;
 297                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 298                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 299                         switch (cmp) {
 300                                 case -1:
 301                                         lo = mid + 1;
 302                                         break;
 303                                 case 0:
 304                                         return seq[mid].rest;
 305                                 case 1:
 306                                         hi = mid - 1;
 307                                         break;
 308                                 default:
 309                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 310                         }
 311                 }
 312                 return _ex0();
 313         } else
 314                 return convert_to_poly().coeff(s, n);
 315 }
 316
 317
 318 ex pseries::collect(const symbol &s) const
 319 {
 320         return *this;
 321 }
 322
 323
 324 /** Evaluate coefficients. */
 325 ex pseries::eval(int level) const
 326 {
 327         if (level == 1)
 328                 return this->hold();
 329
 330         if (level == -max_recursion_level)
 331                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 332
 333         // Construct a new series with evaluated coefficients
 334         epvector new_seq;
 335         new_seq.reserve(seq.size());
 336         epvector::const_iterator it = seq.begin(), itend = seq.end();
 337         while (it != itend) {
 338                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 339                 ++it;
 340         }
 341         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 342 }
 343
 344
 345 /** Evaluate coefficients numerically. */
 346 ex pseries::evalf(int level) const
 347 {
 348         if (level == 1)
 349                 return *this;
 350
 351         if (level == -max_recursion_level)
 352                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 353
 354         // Construct a new series with evaluated coefficients
 355         epvector new_seq;
 356         new_seq.reserve(seq.size());
 357         epvector::const_iterator it = seq.begin(), itend = seq.end();
 358         while (it != itend) {
 359                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 360                 ++it;
 361         }
 362         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 363 }
 364
 365
 366 ex pseries::subs(const lst & ls, const lst & lr) const
 367 {
 368         // If expansion variable is being substituted, convert the series to a
 369         // polynomial and do the substitution there because the result might
 370         // no longer be a power series
 371         if (ls.has(var))
 372                 return convert_to_poly(true).subs(ls, lr);
 373
 374         // Otherwise construct a new series with substituted coefficients and
 375         // expansion point
 376         epvector newseq;
 377         newseq.reserve(seq.size());
 378         epvector::const_iterator it = seq.begin(), itend = seq.end();
 379         while (it != itend) {
 380                 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 381                 ++it;
 382         }
 383         return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
 384 }
 385
 386
 387 /** Implementation of ex::expand() for a power series.  It expands all the
 388  *  terms individually and returns the resulting series as a new pseries.
 389  *  @see ex::diff */
 390 ex pseries::expand(unsigned options) const
 391 {
 392         epvector newseq;
 393         newseq.reserve(seq.size());
 394         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 395                 newseq.push_back(expair(i->rest.expand(), i->coeff));
 396         return (new pseries(relational(var,point), newseq))
 397                 ->setflag(status_flags::dynallocated | status_flags::expanded);
 398 }
 399
 400
 401 /** Implementation of ex::diff() for a power series.  It treats the series as a
 402  *  polynomial.
 403  *  @see ex::diff */
 404 ex pseries::derivative(const symbol & s) const
 405 {
 406         if (s == var) {
 407                 epvector new_seq;
 408                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 409
 410                 // FIXME: coeff might depend on var
 411                 while (it != itend) {
 412                         if (is_order_function(it->rest)) {
 413                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 414                         } else {
 415                                 ex c = it->rest * it->coeff;
 416                                 if (!c.is_zero())
 417                                         new_seq.push_back(expair(c, it->coeff - 1));
 418                         }
 419                         ++it;
 420                 }
 421                 return pseries(relational(var,point), new_seq);
 422         } else {
 423                 return *this;
 424         }
 425 }
 426
 427
 428 /*
 429  *  Construct ordinary polynomial out of series
 430  */
 431
 432 /** Convert a pseries object to an ordinary polynomial.
 433  *
 434  *  @param no_order flag: discard higher order terms */
 435 ex pseries::convert_to_poly(bool no_order) const
 436 {
 437         ex e;
 438         epvector::const_iterator it = seq.begin(), itend = seq.end();
 439
 440         while (it != itend) {
 441                 if (is_order_function(it->rest)) {
 442                         if (!no_order)
 443                                 e += Order(power(var - point, it->coeff));
 444                 } else
 445                         e += it->rest * power(var - point, it->coeff);
 446                 ++it;
 447         }
 448         return e;
 449 }
 450
 451 /** Returns true if there is no order term, i.e. the series terminates and
 452  *  false otherwise. */
 453 bool pseries::is_terminating(void) const
 454 {
 455         return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
 456 }
 457
 458
 459 /*
 460  *  Implementation of series expansion
 461  */
 462
 463 /** Default implementation of ex::series(). This performs Taylor expansion.
 464  *  @see ex::series */
 465 ex basic::series(const relational & r, int order, unsigned options) const
 466 {
 467         epvector seq;
 468         numeric fac(1);
 469         ex deriv = *this;
 470         ex coeff = deriv.subs(r);
 471         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 472
 473         if (!coeff.is_zero())
 474                 seq.push_back(expair(coeff, numeric(0)));
 475
 476         int n;
 477         for (n=1; n<order; ++n) {
 478                 fac = fac.mul(numeric(n));
 479                 deriv = deriv.diff(*s).expand();
 480                 if (deriv.is_zero()) {
 481                         // Series terminates
 482                         return pseries(r, seq);
 483                 }
 484                 coeff = deriv.subs(r);
 485                 if (!coeff.is_zero())
 486                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
 487         }
 488
 489         // Higher-order terms, if present
 490         deriv = deriv.diff(*s);
 491         if (!deriv.expand().is_zero())
 492                 seq.push_back(expair(Order(_ex1()), numeric(n)));
 493         return pseries(r, seq);
 494 }
 495
 496
 497 /** Implementation of ex::series() for symbols.
 498  *  @see ex::series */
 499 ex symbol::series(const relational & r, int order, unsigned options) const
 500 {
 501         epvector seq;
 502         const ex point = r.rhs();
 503         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 504         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 505
 506         if (this->is_equal(*s)) {
 507                 if (order > 0 && !point.is_zero())
 508                         seq.push_back(expair(point, _ex0()));
 509                 if (order > 1)
 510                         seq.push_back(expair(_ex1(), _ex1()));
 511                 else
 512                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 513         } else
 514                 seq.push_back(expair(*this, _ex0()));
 515         return pseries(r, seq);
 516 }
 517
 518
 519 /** Add one series object to another, producing a pseries object that
 520  *  represents the sum.
 521  *
 522  *  @param other  pseries object to add with
 523  *  @return the sum as a pseries */
 524 ex pseries::add_series(const pseries &other) const
 525 {
 526         // Adding two series with different variables or expansion points
 527         // results in an empty (constant) series
 528         if (!is_compatible_to(other)) {
 529                 epvector nul;
 530                 nul.push_back(expair(Order(_ex1()), _ex0()));
 531                 return pseries(relational(var,point), nul);
 532         }
 533
 534         // Series addition
 535         epvector new_seq;
 536         epvector::const_iterator a = seq.begin();
 537         epvector::const_iterator b = other.seq.begin();
 538         epvector::const_iterator a_end = seq.end();
 539         epvector::const_iterator b_end = other.seq.end();
 540         int pow_a = INT_MAX, pow_b = INT_MAX;
 541         for (;;) {
 542                 // If a is empty, fill up with elements from b and stop
 543                 if (a == a_end) {
 544                         while (b != b_end) {
 545                                 new_seq.push_back(*b);
 546                                 ++b;
 547                         }
 548                         break;
 549                 } else
 550                         pow_a = ex_to_numeric((*a).coeff).to_int();
 551
 552                 // If b is empty, fill up with elements from a and stop
 553                 if (b == b_end) {
 554                         while (a != a_end) {
 555                                 new_seq.push_back(*a);
 556                                 ++a;
 557                         }
 558                         break;
 559                 } else
 560                         pow_b = ex_to_numeric((*b).coeff).to_int();
 561
 562                 // a and b are non-empty, compare powers
 563                 if (pow_a < pow_b) {
 564                         // a has lesser power, get coefficient from a
 565                         new_seq.push_back(*a);
 566                         if (is_order_function((*a).rest))
 567                                 break;
 568                         ++a;
 569                 } else if (pow_b < pow_a) {
 570                         // b has lesser power, get coefficient from b
 571                         new_seq.push_back(*b);
 572                         if (is_order_function((*b).rest))
 573                                 break;
 574                         ++b;
 575                 } else {
 576                         // Add coefficient of a and b
 577                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 578                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 579                                 break;  // Order term ends the sequence
 580                         } else {
 581                                 ex sum = (*a).rest + (*b).rest;
 582                                 if (!(sum.is_zero()))
 583                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 584                                 ++a;
 585                                 ++b;
 586                         }
 587                 }
 588         }
 589         return pseries(relational(var,point), new_seq);
 590 }
 591
 592
 593 /** Implementation of ex::series() for sums. This performs series addition when
 594  *  adding pseries objects.
 595  *  @see ex::series */
 596 ex add::series(const relational & r, int order, unsigned options) const
 597 {
 598         ex acc; // Series accumulator
 599
 600         // Get first term from overall_coeff
 601         acc = overall_coeff.series(r, order, options);
 602
 603         // Add remaining terms
 604         epvector::const_iterator it = seq.begin();
 605         epvector::const_iterator itend = seq.end();
 606         for (; it!=itend; ++it) {
 607                 ex op;
 608                 if (is_ex_exactly_of_type(it->rest, pseries))
 609                         op = it->rest;
 610                 else
 611                         op = it->rest.series(r, order, options);
 612                 if (!it->coeff.is_equal(_ex1()))
 613                         op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 614
 615                 // Series addition
 616                 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 617         }
 618         return acc;
 619 }
 620
 621
 622 /** Multiply a pseries object with a numeric constant, producing a pseries
 623  *  object that represents the product.
 624  *
 625  *  @param other  constant to multiply with
 626  *  @return the product as a pseries */
 627 ex pseries::mul_const(const numeric &other) const
 628 {
 629         epvector new_seq;
 630         new_seq.reserve(seq.size());
 631
 632         epvector::const_iterator it = seq.begin(), itend = seq.end();
 633         while (it != itend) {
 634                 if (!is_order_function(it->rest))
 635                         new_seq.push_back(expair(it->rest * other, it->coeff));
 636                 else
 637                         new_seq.push_back(*it);
 638                 ++it;
 639         }
 640         return pseries(relational(var,point), new_seq);
 641 }
 642
 643
 644 /** Multiply one pseries object to another, producing a pseries object that
 645  *  represents the product.
 646  *
 647  *  @param other  pseries object to multiply with
 648  *  @return the product as a pseries */
 649 ex pseries::mul_series(const pseries &other) const
 650 {
 651         // Multiplying two series with different variables or expansion points
 652         // results in an empty (constant) series
 653         if (!is_compatible_to(other)) {
 654                 epvector nul;
 655                 nul.push_back(expair(Order(_ex1()), _ex0()));
 656                 return pseries(relational(var,point), nul);
 657         }
 658
 659         // Series multiplication
 660         epvector new_seq;
 661
 662         const symbol *s = static_cast<symbol *>(var.bp);
 663         int a_max = degree(*s);
 664         int b_max = other.degree(*s);
 665         int a_min = ldegree(*s);
 666         int b_min = other.ldegree(*s);
 667         int cdeg_min = a_min + b_min;
 668         int cdeg_max = a_max + b_max;
 669
 670         int higher_order_a = INT_MAX;
 671         int higher_order_b = INT_MAX;
 672         if (is_order_function(coeff(*s, a_max)))
 673                 higher_order_a = a_max + b_min;
 674         if (is_order_function(other.coeff(*s, b_max)))
 675                 higher_order_b = b_max + a_min;
 676         int higher_order_c = std::min(higher_order_a, higher_order_b);
 677         if (cdeg_max >= higher_order_c)
 678                 cdeg_max = higher_order_c - 1;
 679
 680         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 681                 ex co = _ex0();
 682                 // c(i)=a(0)b(i)+...+a(i)b(0)
 683                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 684                         ex a_coeff = coeff(*s, i);
 685                         ex b_coeff = other.coeff(*s, cdeg-i);
 686                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 687                                 co += a_coeff * b_coeff;
 688                 }
 689                 if (!co.is_zero())
 690                         new_seq.push_back(expair(co, numeric(cdeg)));
 691         }
 692         if (higher_order_c < INT_MAX)
 693                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 694         return pseries(relational(var,point), new_seq);
 695 }
 696
 697
 698 /** Implementation of ex::series() for product. This performs series
 699  *  multiplication when multiplying series.
 700  *  @see ex::series */
 701 ex mul::series(const relational & r, int order, unsigned options) const
 702 {
 703         ex acc; // Series accumulator
 704
 705         // Get first term from overall_coeff
 706         acc = overall_coeff.series(r, order, options);
 707
 708         // Multiply with remaining terms
 709         epvector::const_iterator it = seq.begin();
 710         epvector::const_iterator itend = seq.end();
 711         for (; it!=itend; ++it) {
 712                 ex op = it->rest;
 713                 if (op.info(info_flags::numeric)) {
 714                         // series * const (special case, faster)
 715                         ex f = power(op, it->coeff);
 716                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 717                         continue;
 718                 } else if (!is_ex_exactly_of_type(op, pseries))
 719                         op = op.series(r, order, options);
 720                 if (!it->coeff.is_equal(_ex1()))
 721                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 722
 723                 // Series multiplication
 724                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 725         }
 726         return acc;
 727 }
 728
 729
 730 /** Compute the p-th power of a series.
 731  *
 732  *  @param p  power to compute
 733  *  @param deg  truncation order of series calculation */
 734 ex pseries::power_const(const numeric &p, int deg) const
 735 {
 736         // method:
 737         // let A(x) be this series and for the time being let it start with a
 738         // constant (later we'll generalize):
 739         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 740         // We want to compute
 741         //     C(x) = A(x)^p
 742         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 743         // Taking the derivative on both sides and multiplying with A(x) one
 744         // immediately arrives at
 745         //     C'(x)*A(x) = p*C(x)*A'(x)
 746         // Multiplying this out and comparing coefficients we get the recurrence
 747         // formula
 748         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 749         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 750         // which can easily be solved given the starting value c_0 = (a_0)^p.
 751         // For the more general case where the leading coefficient of A(x) is not
 752         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 753         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 754         // then of course x^(p*m) but the recurrence formula still holds.
 755         const symbol *s = static_cast<symbol *>(var.bp);
 756         int ldeg = ldegree(*s);
 757
 758         // Compute coefficients of the powered series
 759         exvector co;
 760         co.reserve(deg);
 761         co.push_back(power(coeff(*s, ldeg), p));
 762         bool all_sums_zero = true;
 763         for (int i=1; i<deg; ++i) {
 764                 ex sum = _ex0();
 765                 for (int j=1; j<=i; ++j) {
 766                         ex c = coeff(*s, j + ldeg);
 767                         if (is_order_function(c)) {
 768                                 co.push_back(Order(_ex1()));
 769                                 break;
 770                         } else
 771                                 sum += (p * j - (i - j)) * co[i - j] * c;
 772                 }
 773                 if (!sum.is_zero())
 774                         all_sums_zero = false;
 775                 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
 776         }
 777
 778         // Construct new series (of non-zero coefficients)
 779         epvector new_seq;
 780         bool higher_order = false;
 781         for (int i=0; i<deg; ++i) {
 782                 if (!co[i].is_zero())
 783                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 784                 if (is_order_function(co[i])) {
 785                         higher_order = true;
 786                         break;
 787                 }
 788         }
 789         if (!higher_order && !all_sums_zero)
 790                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 791         return pseries(relational(var,point), new_seq);
 792 }
 793
 794
 795 /** Return a new pseries object with the powers shifted by deg. */
 796 pseries pseries::shift_exponents(int deg) const
 797 {
 798         epvector newseq(seq);
 799         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
 800                 i->coeff = i->coeff + deg;
 801         return pseries(relational(var, point), newseq);
 802 }
 803
 804
 805 /** Implementation of ex::series() for powers. This performs Laurent expansion
 806  *  of reciprocals of series at singularities.
 807  *  @see ex::series */
 808 ex power::series(const relational & r, int order, unsigned options) const
 809 {
 810         ex e;
 811         if (!is_ex_exactly_of_type(basis, pseries)) {
 812                 // Basis is not a series, may there be a singularity?
 813                 bool must_expand_basis = false;
 814                 try {
 815                         basis.subs(r);
 816                 } catch (pole_error) {
 817                         must_expand_basis = true;
 818                 }
 819
 820                 // Is the expression of type something^(-int)?
 821                 if (!must_expand_basis && !exponent.info(info_flags::negint))
 822                         return basic::series(r, order, options);
 823
 824                 // Is the expression of type 0^something?
 825                 if (!must_expand_basis && !basis.subs(r).is_zero())
 826                         return basic::series(r, order, options);
 827
 828                 // Singularity encountered, expand basis into series
 829                 e = basis.series(r, order, options);
 830         } else {
 831                 // Basis is a series
 832                 e = basis;
 833         }
 834
 835         // Power e
 836         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 837 }
 838
 839
 840 /** Re-expansion of a pseries object. */
 841 ex pseries::series(const relational & r, int order, unsigned options) const
 842 {
 843         const ex p = r.rhs();
 844         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 845         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 846
 847         if (var.is_equal(*s) && point.is_equal(p)) {
 848                 if (order > degree(*s))
 849                         return *this;
 850                 else {
 851                         epvector new_seq;
 852                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 853                         while (it != itend) {
 854                                 int o = ex_to_numeric(it->coeff).to_int();
 855                                 if (o >= order) {
 856                                         new_seq.push_back(expair(Order(_ex1()), o));
 857                                         break;
 858                                 }
 859                                 new_seq.push_back(*it);
 860                                 ++it;
 861                         }
 862                         return pseries(r, new_seq);
 863                 }
 864         } else
 865                 return convert_to_poly().series(r, order, options);
 866 }
 867
 868
 869 /** Compute the truncated series expansion of an expression.
 870  *  This function returns an expression containing an object of class pseries
 871  *  to represent the series. If the series does not terminate within the given
 872  *  truncation order, the last term of the series will be an order term.
 873  *
 874  *  @param r  expansion relation, lhs holds variable and rhs holds point
 875  *  @param order  truncation order of series calculations
 876  *  @param options  of class series_options
 877  *  @return an expression holding a pseries object */
 878 ex ex::series(const ex & r, int order, unsigned options) const
 879 {
 880         GINAC_ASSERT(bp!=0);
 881         ex e;
 882         relational rel_;
 883
 884         if (is_ex_exactly_of_type(r,relational))
 885                 rel_ = ex_to_relational(r);
 886         else if (is_ex_exactly_of_type(r,symbol))
 887                 rel_ = relational(r,_ex0());
 888         else
 889                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 890
 891         try {
 892                 e = bp->series(rel_, order, options);
 893         } catch (std::exception &x) {
 894                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 895         }
 896         return e;
 897 }
 898
 899 //////////
 900 // static member variables
 901 //////////
 902
 903 // protected
 904
 905 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
 906
 907 #ifndef NO_NAMESPACE_GINAC
 908 } // namespace GiNaC
 909 #endif // ndef NO_NAMESPACE_GINAC