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