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_) : basic(TINFO_pseries), seq(ops_)
 104 {
 105         debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
 106         GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
 107         GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
 108         point = rel_.rhs();
 109         var = *static_cast<symbol *>(rel_.lhs().bp);
 110 }
 111
 112
 113 /*
 114  *  Archiving
 115  */
 116
 117 /** Construct object from archive_node. */
 118 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 119 {
 120         debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
 121         for (unsigned int i=0; true; ++i) {
 122                 ex rest;
 123                 ex coeff;
 124                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
 125                         seq.push_back(expair(rest, coeff));
 126                 else
 127                         break;
 128         }
 129         n.find_ex("var", var, sym_lst);
 130         n.find_ex("point", point, sym_lst);
 131 }
 132
 133 /** Unarchive the object. */
 134 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
 135 {
 136         return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
 137 }
 138
 139 /** Archive the object. */
 140 void pseries::archive(archive_node &n) const
 141 {
 142         inherited::archive(n);
 143         epvector::const_iterator i = seq.begin(), iend = seq.end();
 144         while (i != iend) {
 145                 n.add_ex("coeff", i->rest);
 146                 n.add_ex("power", i->coeff);
 147                 ++i;
 148         }
 149         n.add_ex("var", var);
 150         n.add_ex("point", point);
 151 }
 152
 153 //////////
 154 // functions overriding virtual functions from bases classes
 155 //////////
 156
 157 basic *pseries::duplicate() const
 158 {
 159         debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
 160         return new pseries(*this);
 161 }
 162
 163 void pseries::print(std::ostream &os, unsigned upper_precedence) const
 164 {
 165         debugmsg("pseries print", LOGLEVEL_PRINT);
 166         if (precedence<=upper_precedence) os << "(";
 167         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 168                 // omit zero terms
 169                 if (i->rest.is_zero())
 170                         continue;
 171                 // print a sign, if needed
 172                 if (i!=seq.begin())
 173                         os << '+';
 174                 if (!is_order_function(i->rest)) {
 175                         // print 'rest', i.e. the expansion coefficient
 176                         if (i->rest.info(info_flags::numeric) &&
 177                                 i->rest.info(info_flags::positive)) {
 178                                 os << i->rest;
 179                         } else
 180                                 os << "(" << i->rest << ')';
 181                         // print 'coeff', something like (x-1)^42
 182                         if (!i->coeff.is_zero()) {
 183                                 os << '*';
 184                                 if (!point.is_zero())
 185                                         os << '(' << var-point << ')';
 186                                 else
 187                                         os << var;
 188                                 if (i->coeff.compare(_ex1())) {
 189                                         os << '^';
 190                                         if (i->coeff.info(info_flags::negative))
 191                                                 os << '(' << i->coeff << ')';
 192                                         else
 193                                                 os << i->coeff;
 194                                 }
 195                         }
 196                 } else {
 197                         os << Order(power(var-point,i->coeff));
 198                 }
 199         }
 200         if (precedence<=upper_precedence) os << ")";
 201 }
 202
 203
 204 void pseries::printraw(std::ostream &os) const
 205 {
 206         debugmsg("pseries printraw", LOGLEVEL_PRINT);
 207         os << "pseries(" << var << ";" << point << ";";
 208         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 209                 os << "(" << (*i).rest << "," << (*i).coeff << "),";
 210         }
 211         os << ")";
 212 }
 213
 214
 215 void pseries::printtree(std::ostream & os, unsigned indent) const
 216 {
 217         debugmsg("pseries printtree",LOGLEVEL_PRINT);
 218         os << std::string(indent,' ') << "pseries "
 219            << ", hash=" << hashvalue
 220            << " (0x" << std::hex << hashvalue << std::dec << ")"
 221            << ", flags=" << flags << std::endl;
 222         for (unsigned i=0; i<seq.size(); ++i) {
 223                 seq[i].rest.printtree(os,indent+delta_indent);
 224                 seq[i].coeff.printtree(os,indent+delta_indent);
 225                 if (i!=seq.size()-1)
 226                         os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
 227         }
 228         var.printtree(os, indent+delta_indent);
 229         point.printtree(os, indent+delta_indent);
 230 }
 231
 232 /** Return the number of operands including a possible order term. */
 233 unsigned pseries::nops(void) const
 234 {
 235         return seq.size();
 236 }
 237
 238
 239 /** Return the ith term in the series when represented as a sum. */
 240 ex pseries::op(int i) const
 241 {
 242         if (i < 0 || unsigned(i) >= seq.size())
 243                 throw (std::out_of_range("op() out of range"));
 244         return seq[i].rest * power(var - point, seq[i].coeff);
 245 }
 246
 247
 248 ex &pseries::let_op(int i)
 249 {
 250         throw (std::logic_error("let_op not defined for pseries"));
 251 }
 252
 253
 254 /** Return degree of highest power of the series.  This is usually the exponent
 255  *  of the Order term.  If s is not the expansion variable of the series, the
 256  *  series is examined termwise. */
 257 int pseries::degree(const symbol &s) const
 258 {
 259         if (var.is_equal(s)) {
 260                 // Return last exponent
 261                 if (seq.size())
 262                         return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 263                 else
 264                         return 0;
 265         } else {
 266                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 267                 if (it == itend)
 268                         return 0;
 269                 int max_pow = INT_MIN;
 270                 while (it != itend) {
 271                         int pow = it->rest.degree(s);
 272                         if (pow > max_pow)
 273                                 max_pow = pow;
 274                         ++it;
 275                 }
 276                 return max_pow;
 277         }
 278 }
 279
 280 /** Return degree of lowest power of the series.  This is usually the exponent
 281  *  of the leading term.  If s is not the expansion variable of the series, the
 282  *  series is examined termwise.  If s is the expansion variable but the
 283  *  expansion point is not zero the series is not expanded to find the degree.
 284  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 285 int pseries::ldegree(const symbol &s) const
 286 {
 287         if (var.is_equal(s)) {
 288                 // Return first exponent
 289                 if (seq.size())
 290                         return ex_to_numeric((*(seq.begin())).coeff).to_int();
 291                 else
 292                         return 0;
 293         } else {
 294                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 295                 if (it == itend)
 296                         return 0;
 297                 int min_pow = INT_MAX;
 298                 while (it != itend) {
 299                         int pow = it->rest.ldegree(s);
 300                         if (pow < min_pow)
 301                                 min_pow = pow;
 302                         ++it;
 303                 }
 304                 return min_pow;
 305         }
 306 }
 307
 308 ex pseries::coeff(const symbol &s, int n) const
 309 {
 310         if (var.is_equal(s)) {
 311                 if (seq.size() == 0)
 312                         return _ex0();
 313
 314                 // Binary search in sequence for given power
 315                 numeric looking_for = numeric(n);
 316                 int lo = 0, hi = seq.size() - 1;
 317                 while (lo <= hi) {
 318                         int mid = (lo + hi) / 2;
 319                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 320                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 321                         switch (cmp) {
 322                                 case -1:
 323                                         lo = mid + 1;
 324                                         break;
 325                                 case 0:
 326                                         return seq[mid].rest;
 327                                 case 1:
 328                                         hi = mid - 1;
 329                                         break;
 330                                 default:
 331                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 332                         }
 333                 }
 334                 return _ex0();
 335         } else
 336                 return convert_to_poly().coeff(s, n);
 337 }
 338
 339
 340 ex pseries::collect(const symbol &s) const
 341 {
 342         return *this;
 343 }
 344
 345
 346 /** Evaluate coefficients. */
 347 ex pseries::eval(int level) const
 348 {
 349         if (level == 1)
 350                 return this->hold();
 351
 352         if (level == -max_recursion_level)
 353                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 354
 355         // Construct a new series with evaluated coefficients
 356         epvector new_seq;
 357         new_seq.reserve(seq.size());
 358         epvector::const_iterator it = seq.begin(), itend = seq.end();
 359         while (it != itend) {
 360                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 361                 ++it;
 362         }
 363         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 364 }
 365
 366
 367 /** Evaluate coefficients numerically. */
 368 ex pseries::evalf(int level) const
 369 {
 370         if (level == 1)
 371                 return *this;
 372
 373         if (level == -max_recursion_level)
 374                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 375
 376         // Construct a new series with evaluated coefficients
 377         epvector new_seq;
 378         new_seq.reserve(seq.size());
 379         epvector::const_iterator it = seq.begin(), itend = seq.end();
 380         while (it != itend) {
 381                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 382                 ++it;
 383         }
 384         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 385 }
 386
 387
 388 ex pseries::subs(const lst & ls, const lst & lr) const
 389 {
 390         // If expansion variable is being substituted, convert the series to a
 391         // polynomial and do the substitution there because the result might
 392         // no longer be a power series
 393         if (ls.has(var))
 394                 return convert_to_poly(true).subs(ls, lr);
 395
 396         // Otherwise construct a new series with substituted coefficients and
 397         // expansion point
 398         epvector newseq;
 399         newseq.reserve(seq.size());
 400         epvector::const_iterator it = seq.begin(), itend = seq.end();
 401         while (it != itend) {
 402                 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 403                 ++it;
 404         }
 405         return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
 406 }
 407
 408
 409 /** Implementation of ex::expand() for a power series.  It expands all the
 410  *  terms individually and returns the resulting series as a new pseries.
 411  *  @see ex::diff */
 412 ex pseries::expand(unsigned options) const
 413 {
 414         epvector newseq;
 415         newseq.reserve(seq.size());
 416         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 417                 newseq.push_back(expair(i->rest.expand(), i->coeff));
 418         return (new pseries(relational(var,point), newseq))
 419                 ->setflag(status_flags::dynallocated | status_flags::expanded);
 420 }
 421
 422
 423 /** Implementation of ex::diff() for a power series.  It treats the series as a
 424  *  polynomial.
 425  *  @see ex::diff */
 426 ex pseries::derivative(const symbol & s) const
 427 {
 428         if (s == var) {
 429                 epvector new_seq;
 430                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 431
 432                 // FIXME: coeff might depend on var
 433                 while (it != itend) {
 434                         if (is_order_function(it->rest)) {
 435                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 436                         } else {
 437                                 ex c = it->rest * it->coeff;
 438                                 if (!c.is_zero())
 439                                         new_seq.push_back(expair(c, it->coeff - 1));
 440                         }
 441                         ++it;
 442                 }
 443                 return pseries(relational(var,point), new_seq);
 444         } else {
 445                 return *this;
 446         }
 447 }
 448
 449
 450 /*
 451  *  Construct ordinary polynomial out of series
 452  */
 453
 454 /** Convert a pseries object to an ordinary polynomial.
 455  *
 456  *  @param no_order flag: discard higher order terms */
 457 ex pseries::convert_to_poly(bool no_order) const
 458 {
 459         ex e;
 460         epvector::const_iterator it = seq.begin(), itend = seq.end();
 461
 462         while (it != itend) {
 463                 if (is_order_function(it->rest)) {
 464                         if (!no_order)
 465                                 e += Order(power(var - point, it->coeff));
 466                 } else
 467                         e += it->rest * power(var - point, it->coeff);
 468                 ++it;
 469         }
 470         return e;
 471 }
 472
 473 /** Returns true if there is no order term, i.e. the series terminates and
 474  *  false otherwise. */
 475 bool pseries::is_terminating(void) const
 476 {
 477         return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
 478 }
 479
 480
 481 /*
 482  *  Implementation of series expansion
 483  */
 484
 485 /** Default implementation of ex::series(). This performs Taylor expansion.
 486  *  @see ex::series */
 487 ex basic::series(const relational & r, int order, unsigned options) const
 488 {
 489         epvector seq;
 490         numeric fac(1);
 491         ex deriv = *this;
 492         ex coeff = deriv.subs(r);
 493         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 494
 495         if (!coeff.is_zero())
 496                 seq.push_back(expair(coeff, numeric(0)));
 497
 498         int n;
 499         for (n=1; n<order; ++n) {
 500                 fac = fac.mul(numeric(n));
 501                 deriv = deriv.diff(*s).expand();
 502                 if (deriv.is_zero()) {
 503                         // Series terminates
 504                         return pseries(r, seq);
 505                 }
 506                 coeff = deriv.subs(r);
 507                 if (!coeff.is_zero())
 508                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
 509         }
 510
 511         // Higher-order terms, if present
 512         deriv = deriv.diff(*s);
 513         if (!deriv.expand().is_zero())
 514                 seq.push_back(expair(Order(_ex1()), numeric(n)));
 515         return pseries(r, seq);
 516 }
 517
 518
 519 /** Implementation of ex::series() for symbols.
 520  *  @see ex::series */
 521 ex symbol::series(const relational & r, int order, unsigned options) const
 522 {
 523         epvector seq;
 524         const ex point = r.rhs();
 525         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 526         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 527
 528         if (this->is_equal(*s)) {
 529                 if (order > 0 && !point.is_zero())
 530                         seq.push_back(expair(point, _ex0()));
 531                 if (order > 1)
 532                         seq.push_back(expair(_ex1(), _ex1()));
 533                 else
 534                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 535         } else
 536                 seq.push_back(expair(*this, _ex0()));
 537         return pseries(r, seq);
 538 }
 539
 540
 541 /** Add one series object to another, producing a pseries object that
 542  *  represents the sum.
 543  *
 544  *  @param other  pseries object to add with
 545  *  @return the sum as a pseries */
 546 ex pseries::add_series(const pseries &other) const
 547 {
 548         // Adding two series with different variables or expansion points
 549         // results in an empty (constant) series
 550         if (!is_compatible_to(other)) {
 551                 epvector nul;
 552                 nul.push_back(expair(Order(_ex1()), _ex0()));
 553                 return pseries(relational(var,point), nul);
 554         }
 555
 556         // Series addition
 557         epvector new_seq;
 558         epvector::const_iterator a = seq.begin();
 559         epvector::const_iterator b = other.seq.begin();
 560         epvector::const_iterator a_end = seq.end();
 561         epvector::const_iterator b_end = other.seq.end();
 562         int pow_a = INT_MAX, pow_b = INT_MAX;
 563         for (;;) {
 564                 // If a is empty, fill up with elements from b and stop
 565                 if (a == a_end) {
 566                         while (b != b_end) {
 567                                 new_seq.push_back(*b);
 568                                 ++b;
 569                         }
 570                         break;
 571                 } else
 572                         pow_a = ex_to_numeric((*a).coeff).to_int();
 573
 574                 // If b is empty, fill up with elements from a and stop
 575                 if (b == b_end) {
 576                         while (a != a_end) {
 577                                 new_seq.push_back(*a);
 578                                 ++a;
 579                         }
 580                         break;
 581                 } else
 582                         pow_b = ex_to_numeric((*b).coeff).to_int();
 583
 584                 // a and b are non-empty, compare powers
 585                 if (pow_a < pow_b) {
 586                         // a has lesser power, get coefficient from a
 587                         new_seq.push_back(*a);
 588                         if (is_order_function((*a).rest))
 589                                 break;
 590                         ++a;
 591                 } else if (pow_b < pow_a) {
 592                         // b has lesser power, get coefficient from b
 593                         new_seq.push_back(*b);
 594                         if (is_order_function((*b).rest))
 595                                 break;
 596                         ++b;
 597                 } else {
 598                         // Add coefficient of a and b
 599                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 600                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 601                                 break;  // Order term ends the sequence
 602                         } else {
 603                                 ex sum = (*a).rest + (*b).rest;
 604                                 if (!(sum.is_zero()))
 605                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 606                                 ++a;
 607                                 ++b;
 608                         }
 609                 }
 610         }
 611         return pseries(relational(var,point), new_seq);
 612 }
 613
 614
 615 /** Implementation of ex::series() for sums. This performs series addition when
 616  *  adding pseries objects.
 617  *  @see ex::series */
 618 ex add::series(const relational & r, int order, unsigned options) const
 619 {
 620         ex acc; // Series accumulator
 621
 622         // Get first term from overall_coeff
 623         acc = overall_coeff.series(r, order, options);
 624
 625         // Add remaining terms
 626         epvector::const_iterator it = seq.begin();
 627         epvector::const_iterator itend = seq.end();
 628         for (; it!=itend; ++it) {
 629                 ex op;
 630                 if (is_ex_exactly_of_type(it->rest, pseries))
 631                         op = it->rest;
 632                 else
 633                         op = it->rest.series(r, order, options);
 634                 if (!it->coeff.is_equal(_ex1()))
 635                         op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 636
 637                 // Series addition
 638                 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 639         }
 640         return acc;
 641 }
 642
 643
 644 /** Multiply a pseries object with a numeric constant, producing a pseries
 645  *  object that represents the product.
 646  *
 647  *  @param other  constant to multiply with
 648  *  @return the product as a pseries */
 649 ex pseries::mul_const(const numeric &other) const
 650 {
 651         epvector new_seq;
 652         new_seq.reserve(seq.size());
 653
 654         epvector::const_iterator it = seq.begin(), itend = seq.end();
 655         while (it != itend) {
 656                 if (!is_order_function(it->rest))
 657                         new_seq.push_back(expair(it->rest * other, it->coeff));
 658                 else
 659                         new_seq.push_back(*it);
 660                 ++it;
 661         }
 662         return pseries(relational(var,point), new_seq);
 663 }
 664
 665
 666 /** Multiply one pseries object to another, producing a pseries object that
 667  *  represents the product.
 668  *
 669  *  @param other  pseries object to multiply with
 670  *  @return the product as a pseries */
 671 ex pseries::mul_series(const pseries &other) const
 672 {
 673         // Multiplying two series with different variables or expansion points
 674         // results in an empty (constant) series
 675         if (!is_compatible_to(other)) {
 676                 epvector nul;
 677                 nul.push_back(expair(Order(_ex1()), _ex0()));
 678                 return pseries(relational(var,point), nul);
 679         }
 680
 681         // Series multiplication
 682         epvector new_seq;
 683
 684         const symbol *s = static_cast<symbol *>(var.bp);
 685         int a_max = degree(*s);
 686         int b_max = other.degree(*s);
 687         int a_min = ldegree(*s);
 688         int b_min = other.ldegree(*s);
 689         int cdeg_min = a_min + b_min;
 690         int cdeg_max = a_max + b_max;
 691
 692         int higher_order_a = INT_MAX;
 693         int higher_order_b = INT_MAX;
 694         if (is_order_function(coeff(*s, a_max)))
 695                 higher_order_a = a_max + b_min;
 696         if (is_order_function(other.coeff(*s, b_max)))
 697                 higher_order_b = b_max + a_min;
 698         int higher_order_c = std::min(higher_order_a, higher_order_b);
 699         if (cdeg_max >= higher_order_c)
 700                 cdeg_max = higher_order_c - 1;
 701
 702         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 703                 ex co = _ex0();
 704                 // c(i)=a(0)b(i)+...+a(i)b(0)
 705                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 706                         ex a_coeff = coeff(*s, i);
 707                         ex b_coeff = other.coeff(*s, cdeg-i);
 708                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 709                                 co += a_coeff * b_coeff;
 710                 }
 711                 if (!co.is_zero())
 712                         new_seq.push_back(expair(co, numeric(cdeg)));
 713         }
 714         if (higher_order_c < INT_MAX)
 715                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 716         return pseries(relational(var,point), new_seq);
 717 }
 718
 719
 720 /** Implementation of ex::series() for product. This performs series
 721  *  multiplication when multiplying series.
 722  *  @see ex::series */
 723 ex mul::series(const relational & r, int order, unsigned options) const
 724 {
 725         ex acc; // Series accumulator
 726
 727         // Get first term from overall_coeff
 728         acc = overall_coeff.series(r, order, options);
 729
 730         // Multiply with remaining terms
 731         epvector::const_iterator it = seq.begin();
 732         epvector::const_iterator itend = seq.end();
 733         for (; it!=itend; ++it) {
 734                 ex op = it->rest;
 735                 if (op.info(info_flags::numeric)) {
 736                         // series * const (special case, faster)
 737                         ex f = power(op, it->coeff);
 738                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 739                         continue;
 740                 } else if (!is_ex_exactly_of_type(op, pseries))
 741                         op = op.series(r, order, options);
 742                 if (!it->coeff.is_equal(_ex1()))
 743                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 744
 745                 // Series multiplication
 746                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 747         }
 748         return acc;
 749 }
 750
 751
 752 /** Compute the p-th power of a series.
 753  *
 754  *  @param p  power to compute
 755  *  @param deg  truncation order of series calculation */
 756 ex pseries::power_const(const numeric &p, int deg) const
 757 {
 758         int i;
 759         const symbol *s = static_cast<symbol *>(var.bp);
 760         int ldeg = ldegree(*s);
 761
 762         // Calculate coefficients of powered series
 763         exvector co;
 764         co.reserve(deg);
 765         ex co0;
 766         co.push_back(co0 = power(coeff(*s, ldeg), p));
 767         bool all_sums_zero = true;
 768         for (i=1; i<deg; ++i) {
 769                 ex sum = _ex0();
 770                 for (int j=1; j<=i; ++j) {
 771                         ex c = coeff(*s, j + ldeg);
 772                         if (is_order_function(c)) {
 773                                 co.push_back(Order(_ex1()));
 774                                 break;
 775                         } else
 776                                 sum += (p * j - (i - j)) * co[i - j] * c;
 777                 }
 778                 if (!sum.is_zero())
 779                         all_sums_zero = false;
 780                 co.push_back(co0 * sum / numeric(i));
 781         }
 782
 783         // Construct new series (of non-zero coefficients)
 784         epvector new_seq;
 785         bool higher_order = false;
 786         for (i=0; i<deg; ++i) {
 787                 if (!co[i].is_zero())
 788                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 789                 if (is_order_function(co[i])) {
 790                         higher_order = true;
 791                         break;
 792                 }
 793         }
 794         if (!higher_order && !all_sums_zero)
 795                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 796         return pseries(relational(var,point), new_seq);
 797 }
 798
 799
 800 /** Return a new pseries object with the powers shifted by deg. */
 801 pseries pseries::shift_exponents(int deg) const
 802 {
 803         epvector newseq(seq);
 804         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
 805                 i->coeff = i->coeff + deg;
 806         return pseries(relational(var, point), newseq);
 807 }
 808
 809
 810 /** Implementation of ex::series() for powers. This performs Laurent expansion
 811  *  of reciprocals of series at singularities.
 812  *  @see ex::series */
 813 ex power::series(const relational & r, int order, unsigned options) const
 814 {
 815         ex e;
 816         if (!is_ex_exactly_of_type(basis, pseries)) {
 817                 // Basis is not a series, may there be a singulary?
 818                 if (!exponent.info(info_flags::negint))
 819                         return basic::series(r, order, options);
 820
 821                 // Expression is of type something^(-int), check for singularity
 822                 if (!basis.subs(r).is_zero())
 823                         return basic::series(r, order, options);
 824
 825                 // Singularity encountered, expand basis into series
 826                 e = basis.series(r, order, options);
 827         } else {
 828                 // Basis is a series
 829                 e = basis;
 830         }
 831
 832         // Power e
 833         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 834 }
 835
 836
 837 /** Re-expansion of a pseries object. */
 838 ex pseries::series(const relational & r, int order, unsigned options) const
 839 {
 840         const ex p = r.rhs();
 841         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 842         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 843
 844         if (var.is_equal(*s) && point.is_equal(p)) {
 845                 if (order > degree(*s))
 846                         return *this;
 847                 else {
 848                         epvector new_seq;
 849                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 850                         while (it != itend) {
 851                                 int o = ex_to_numeric(it->coeff).to_int();
 852                                 if (o >= order) {
 853                                         new_seq.push_back(expair(Order(_ex1()), o));
 854                                         break;
 855                                 }
 856                                 new_seq.push_back(*it);
 857                                 ++it;
 858                         }
 859                         return pseries(r, new_seq);
 860                 }
 861         } else
 862                 return convert_to_poly().series(r, order, options);
 863 }
 864
 865
 866 /** Compute the truncated series expansion of an expression.
 867  *  This function returns an expression containing an object of class pseries
 868  *  to represent the series. If the series does not terminate within the given
 869  *  truncation order, the last term of the series will be an order term.
 870  *
 871  *  @param r  expansion relation, lhs holds variable and rhs holds point
 872  *  @param order  truncation order of series calculations
 873  *  @param options  of class series_options
 874  *  @return an expression holding a pseries object */
 875 ex ex::series(const ex & r, int order, unsigned options) const
 876 {
 877         GINAC_ASSERT(bp!=0);
 878         ex e;
 879         relational rel_;
 880
 881         if (is_ex_exactly_of_type(r,relational))
 882                 rel_ = ex_to_relational(r);
 883         else if (is_ex_exactly_of_type(r,symbol))
 884                 rel_ = relational(r,_ex0());
 885         else
 886                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 887
 888         try {
 889                 e = bp->series(rel_, order, options);
 890         } catch (std::exception &x) {
 891                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 892         }
 893         return e;
 894 }
 895
 896 //////////
 897 // static member variables
 898 //////////
 899
 900 // protected
 901
 902 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
 903
 904 //////////
 905 // global constants
 906 //////////
 907
 908 const pseries some_pseries;
 909 const std::type_info & typeid_pseries = typeid(some_pseries);
 910
 911 #ifndef NO_NAMESPACE_GINAC
 912 } // namespace GiNaC
 913 #endif // ndef NO_NAMESPACE_GINAC