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" // for Order function
  29 #include "lst.h"
  30 #include "mul.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "symbol.h"
  34 #include "print.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37 #include "debugmsg.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  42
  43
  44 /*
  45  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  46  */
  47
  48 pseries::pseries() : basic(TINFO_pseries)
  49 {
  50         debugmsg("pseries default ctor", 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 DEFAULT_DESTROY(pseries)
  62
  63
  64 /*
  65  *  Other ctors
  66  */
  67
  68 /** Construct pseries from a vector of coefficients and powers.
  69  *  expair.rest holds the coefficient, expair.coeff holds the power.
  70  *  The powers must be integers (positive or negative) and in ascending order;
  71  *  the last coefficient can be Order(_ex1) to represent a truncated,
  72  *  non-terminating series.
  73  *
  74  *  @param rel_  expansion variable and point (must hold a relational)
  75  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  76  *  @return newly constructed pseries */
  77 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  78 {
  79         debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
  80         GINAC_ASSERT(is_exactly_a<relational>(rel_));
  81         GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
  82         point = rel_.rhs();
  83         var = rel_.lhs();
  84 }
  85
  86
  87 /*
  88  *  Archiving
  89  */
  90
  91 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  92 {
  93         debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
  94         for (unsigned int i=0; true; ++i) {
  95                 ex rest;
  96                 ex coeff;
  97                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  98                         seq.push_back(expair(rest, coeff));
  99                 else
 100                         break;
 101         }
 102         n.find_ex("var", var, sym_lst);
 103         n.find_ex("point", point, sym_lst);
 104 }
 105
 106 void pseries::archive(archive_node &n) const
 107 {
 108         inherited::archive(n);
 109         epvector::const_iterator i = seq.begin(), iend = seq.end();
 110         while (i != iend) {
 111                 n.add_ex("coeff", i->rest);
 112                 n.add_ex("power", i->coeff);
 113                 ++i;
 114         }
 115         n.add_ex("var", var);
 116         n.add_ex("point", point);
 117 }
 118
 119 DEFAULT_UNARCHIVE(pseries)
 120
 121 //////////
 122 // functions overriding virtual functions from base classes
 123 //////////
 124
 125 void pseries::print(const print_context & c, unsigned level) const
 126 {
 127         debugmsg("pseries print", LOGLEVEL_PRINT);
 128
 129         if (is_a<print_tree>(c)) {
 130
 131                 c.s << std::string(level, ' ') << class_name()
 132                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 133                     << std::endl;
 134                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 135                 unsigned num = seq.size();
 136                 for (unsigned i=0; i<num; ++i) {
 137                         seq[i].rest.print(c, level + delta_indent);
 138                         seq[i].coeff.print(c, level + delta_indent);
 139                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 140                 }
 141                 var.print(c, level + delta_indent);
 142                 point.print(c, level + delta_indent);
 143
 144         } else {
 145
 146                 if (precedence() <= level)
 147                         c.s << "(";
 148
 149                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
 150                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
 151
 152                 // objects of type pseries must not have any zero entries, so the
 153                 // trivial (zero) pseries needs a special treatment here:
 154                 if (seq.empty())
 155                         c.s << '0';
 156                 epvector::const_iterator i = seq.begin(), end = seq.end();
 157                 while (i != end) {
 158                         // print a sign, if needed
 159                         if (i != seq.begin())
 160                                 c.s << '+';
 161                         if (!is_order_function(i->rest)) {
 162                                 // print 'rest', i.e. the expansion coefficient
 163                                 if (i->rest.info(info_flags::numeric) &&
 164                                         i->rest.info(info_flags::positive)) {
 165                                         i->rest.print(c);
 166                                 } else {
 167                                         c.s << par_open;
 168                                         i->rest.print(c);
 169                                         c.s << par_close;
 170                                 }
 171                                 // print 'coeff', something like (x-1)^42
 172                                 if (!i->coeff.is_zero()) {
 173                                         if (is_a<print_latex>(c))
 174                                                 c.s << ' ';
 175                                         else
 176                                                 c.s << '*';
 177                                         if (!point.is_zero()) {
 178                                                 c.s << par_open;
 179                                                 (var-point).print(c);
 180                                                 c.s << par_close;
 181                                         } else
 182                                                 var.print(c);
 183                                         if (i->coeff.compare(_ex1)) {
 184                                                 c.s << '^';
 185                                                 if (i->coeff.info(info_flags::negative)) {
 186                                                         c.s << par_open;
 187                                                         i->coeff.print(c);
 188                                                         c.s << par_close;
 189                                                 } else {
 190                                                         if (is_a<print_latex>(c)) {
 191                                                                 c.s << '{';
 192                                                                 i->coeff.print(c);
 193                                                                 c.s << '}';
 194                                                         } else
 195                                                                 i->coeff.print(c);
 196                                                 }
 197                                         }
 198                                 }
 199                         } else
 200                                 Order(power(var-point,i->coeff)).print(c);
 201                         ++i;
 202                 }
 203
 204                 if (precedence() <= level)
 205                         c.s << ")";
 206         }
 207 }
 208
 209 int pseries::compare_same_type(const basic & other) const
 210 {
 211         GINAC_ASSERT(is_a<pseries>(other));
 212         const pseries &o = static_cast<const pseries &>(other);
 213
 214         // first compare the lengths of the series...
 215         if (seq.size()>o.seq.size())
 216                 return 1;
 217         if (seq.size()<o.seq.size())
 218                 return -1;
 219
 220         // ...then the expansion point...
 221         int cmpval = var.compare(o.var);
 222         if (cmpval)
 223                 return cmpval;
 224         cmpval = point.compare(o.point);
 225         if (cmpval)
 226                 return cmpval;
 227
 228         // ...and if that failed the individual elements
 229         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 230         while (it!=seq.end() && o_it!=o.seq.end()) {
 231                 cmpval = it->compare(*o_it);
 232                 if (cmpval)
 233                         return cmpval;
 234                 ++it;
 235                 ++o_it;
 236         }
 237
 238         // so they are equal.
 239         return 0;
 240 }
 241
 242 /** Return the number of operands including a possible order term. */
 243 unsigned pseries::nops(void) const
 244 {
 245         return seq.size();
 246 }
 247
 248 /** Return the ith term in the series when represented as a sum. */
 249 ex pseries::op(int i) const
 250 {
 251         if (i < 0 || unsigned(i) >= seq.size())
 252                 throw (std::out_of_range("op() out of range"));
 253         return seq[i].rest * power(var - point, seq[i].coeff);
 254 }
 255
 256 ex &pseries::let_op(int i)
 257 {
 258         throw (std::logic_error("let_op not defined for pseries"));
 259 }
 260
 261 /** Return degree of highest power of the series.  This is usually the exponent
 262  *  of the Order term.  If s is not the expansion variable of the series, the
 263  *  series is examined termwise. */
 264 int pseries::degree(const ex &s) const
 265 {
 266         if (var.is_equal(s)) {
 267                 // Return last exponent
 268                 if (seq.size())
 269                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 270                 else
 271                         return 0;
 272         } else {
 273                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 274                 if (it == itend)
 275                         return 0;
 276                 int max_pow = INT_MIN;
 277                 while (it != itend) {
 278                         int pow = it->rest.degree(s);
 279                         if (pow > max_pow)
 280                                 max_pow = pow;
 281                         ++it;
 282                 }
 283                 return max_pow;
 284         }
 285 }
 286
 287 /** Return degree of lowest power of the series.  This is usually the exponent
 288  *  of the leading term.  If s is not the expansion variable of the series, the
 289  *  series is examined termwise.  If s is the expansion variable but the
 290  *  expansion point is not zero the series is not expanded to find the degree.
 291  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 292 int pseries::ldegree(const ex &s) const
 293 {
 294         if (var.is_equal(s)) {
 295                 // Return first exponent
 296                 if (seq.size())
 297                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 298                 else
 299                         return 0;
 300         } else {
 301                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 302                 if (it == itend)
 303                         return 0;
 304                 int min_pow = INT_MAX;
 305                 while (it != itend) {
 306                         int pow = it->rest.ldegree(s);
 307                         if (pow < min_pow)
 308                                 min_pow = pow;
 309                         ++it;
 310                 }
 311                 return min_pow;
 312         }
 313 }
 314
 315 /** Return coefficient of degree n in power series if s is the expansion
 316  *  variable.  If the expansion point is nonzero, by definition the n=1
 317  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 318  *  the expansion took place in the s in the first place).
 319  *  If s is not the expansion variable, an attempt is made to convert the
 320  *  series to a polynomial and return the corresponding coefficient from
 321  *  there. */
 322 ex pseries::coeff(const ex &s, int n) const
 323 {
 324         if (var.is_equal(s)) {
 325                 if (seq.empty())
 326                         return _ex0;
 327
 328                 // Binary search in sequence for given power
 329                 numeric looking_for = numeric(n);
 330                 int lo = 0, hi = seq.size() - 1;
 331                 while (lo <= hi) {
 332                         int mid = (lo + hi) / 2;
 333                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 334                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 335                         switch (cmp) {
 336                                 case -1:
 337                                         lo = mid + 1;
 338                                         break;
 339                                 case 0:
 340                                         return seq[mid].rest;
 341                                 case 1:
 342                                         hi = mid - 1;
 343                                         break;
 344                                 default:
 345                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 346                         }
 347                 }
 348                 return _ex0;
 349         } else
 350                 return convert_to_poly().coeff(s, n);
 351 }
 352
 353 /** Does nothing. */
 354 ex pseries::collect(const ex &s, bool distributed) const
 355 {
 356         return *this;
 357 }
 358
 359 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 360 ex pseries::eval(int level) const
 361 {
 362         if (level == 1)
 363                 return this->hold();
 364
 365         if (level == -max_recursion_level)
 366                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 367
 368         // Construct a new series with evaluated coefficients
 369         epvector new_seq;
 370         new_seq.reserve(seq.size());
 371         epvector::const_iterator it = seq.begin(), itend = seq.end();
 372         while (it != itend) {
 373                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 374                 ++it;
 375         }
 376         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 377 }
 378
 379 /** Evaluate coefficients numerically. */
 380 ex pseries::evalf(int level) const
 381 {
 382         if (level == 1)
 383                 return *this;
 384
 385         if (level == -max_recursion_level)
 386                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 387
 388         // Construct a new series with evaluated coefficients
 389         epvector new_seq;
 390         new_seq.reserve(seq.size());
 391         epvector::const_iterator it = seq.begin(), itend = seq.end();
 392         while (it != itend) {
 393                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 394                 ++it;
 395         }
 396         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 397 }
 398
 399 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
 400 {
 401         // If expansion variable is being substituted, convert the series to a
 402         // polynomial and do the substitution there because the result might
 403         // no longer be a power series
 404         if (ls.has(var))
 405                 return convert_to_poly(true).subs(ls, lr, no_pattern);
 406
 407         // Otherwise construct a new series with substituted coefficients and
 408         // expansion point
 409         epvector newseq;
 410         newseq.reserve(seq.size());
 411         epvector::const_iterator it = seq.begin(), itend = seq.end();
 412         while (it != itend) {
 413                 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
 414                 ++it;
 415         }
 416         return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
 417 }
 418
 419 /** Implementation of ex::expand() for a power series.  It expands all the
 420  *  terms individually and returns the resulting series as a new pseries. */
 421 ex pseries::expand(unsigned options) const
 422 {
 423         epvector newseq;
 424         epvector::const_iterator i = seq.begin(), end = seq.end();
 425         while (i != end) {
 426                 ex restexp = i->rest.expand();
 427                 if (!restexp.is_zero())
 428                         newseq.push_back(expair(restexp, i->coeff));
 429                 ++i;
 430         }
 431         return (new pseries(relational(var,point), newseq))
 432                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 433 }
 434
 435 /** Implementation of ex::diff() for a power series.  It treats the series as a
 436  *  polynomial.
 437  *  @see ex::diff */
 438 ex pseries::derivative(const symbol & s) const
 439 {
 440         if (s == var) {
 441                 epvector new_seq;
 442                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 443
 444                 // FIXME: coeff might depend on var
 445                 while (it != itend) {
 446                         if (is_order_function(it->rest)) {
 447                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 448                         } else {
 449                                 ex c = it->rest * it->coeff;
 450                                 if (!c.is_zero())
 451                                         new_seq.push_back(expair(c, it->coeff - 1));
 452                         }
 453                         ++it;
 454                 }
 455                 return pseries(relational(var,point), new_seq);
 456         } else {
 457                 return *this;
 458         }
 459 }
 460
 461 ex pseries::convert_to_poly(bool no_order) const
 462 {
 463         ex e;
 464         epvector::const_iterator it = seq.begin(), itend = seq.end();
 465
 466         while (it != itend) {
 467                 if (is_order_function(it->rest)) {
 468                         if (!no_order)
 469                                 e += Order(power(var - point, it->coeff));
 470                 } else
 471                         e += it->rest * power(var - point, it->coeff);
 472                 ++it;
 473         }
 474         return e;
 475 }
 476
 477 bool pseries::is_terminating(void) const
 478 {
 479         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 480 }
 481
 482
 483 /*
 484  *  Implementations of series expansion
 485  */
 486
 487 /** Default implementation of ex::series(). This performs Taylor expansion.
 488  *  @see ex::series */
 489 ex basic::series(const relational & r, int order, unsigned options) const
 490 {
 491         epvector seq;
 492         numeric fac = 1;
 493         ex deriv = *this;
 494         ex coeff = deriv.subs(r);
 495         const symbol &s = ex_to<symbol>(r.lhs());
 496
 497         if (!coeff.is_zero())
 498                 seq.push_back(expair(coeff, _ex0));
 499
 500         int n;
 501         for (n=1; n<order; ++n) {
 502                 fac = fac.mul(n);
 503                 // We need to test for zero in order to see if the series terminates.
 504                 // The problem is that there is no such thing as a perfect test for
 505                 // zero.  Expanding the term occasionally helps a little...
 506                 deriv = deriv.diff(s).expand();
 507                 if (deriv.is_zero())  // Series terminates
 508                         return pseries(r, seq);
 509
 510                 coeff = deriv.subs(r);
 511                 if (!coeff.is_zero())
 512                         seq.push_back(expair(fac.inverse() * coeff, n));
 513         }
 514
 515         // Higher-order terms, if present
 516         deriv = deriv.diff(s);
 517         if (!deriv.expand().is_zero())
 518                 seq.push_back(expair(Order(_ex1), n));
 519         return pseries(r, seq);
 520 }
 521
 522
 523 /** Implementation of ex::series() for symbols.
 524  *  @see ex::series */
 525 ex symbol::series(const relational & r, int order, unsigned options) const
 526 {
 527         epvector seq;
 528         const ex point = r.rhs();
 529         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 530
 531         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 532                 if (order > 0 && !point.is_zero())
 533                         seq.push_back(expair(point, _ex0));
 534                 if (order > 1)
 535                         seq.push_back(expair(_ex1, _ex1));
 536                 else
 537                         seq.push_back(expair(Order(_ex1), numeric(order)));
 538         } else
 539                 seq.push_back(expair(*this, _ex0));
 540         return pseries(r, seq);
 541 }
 542
 543
 544 /** Add one series object to another, producing a pseries object that
 545  *  represents the sum.
 546  *
 547  *  @param other  pseries object to add with
 548  *  @return the sum as a pseries */
 549 ex pseries::add_series(const pseries &other) const
 550 {
 551         // Adding two series with different variables or expansion points
 552         // results in an empty (constant) series
 553         if (!is_compatible_to(other)) {
 554                 epvector nul;
 555                 nul.push_back(expair(Order(_ex1), _ex0));
 556                 return pseries(relational(var,point), nul);
 557         }
 558
 559         // Series addition
 560         epvector new_seq;
 561         epvector::const_iterator a = seq.begin();
 562         epvector::const_iterator b = other.seq.begin();
 563         epvector::const_iterator a_end = seq.end();
 564         epvector::const_iterator b_end = other.seq.end();
 565         int pow_a = INT_MAX, pow_b = INT_MAX;
 566         for (;;) {
 567                 // If a is empty, fill up with elements from b and stop
 568                 if (a == a_end) {
 569                         while (b != b_end) {
 570                                 new_seq.push_back(*b);
 571                                 ++b;
 572                         }
 573                         break;
 574                 } else
 575                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 576
 577                 // If b is empty, fill up with elements from a and stop
 578                 if (b == b_end) {
 579                         while (a != a_end) {
 580                                 new_seq.push_back(*a);
 581                                 ++a;
 582                         }
 583                         break;
 584                 } else
 585                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 586
 587                 // a and b are non-empty, compare powers
 588                 if (pow_a < pow_b) {
 589                         // a has lesser power, get coefficient from a
 590                         new_seq.push_back(*a);
 591                         if (is_order_function((*a).rest))
 592                                 break;
 593                         ++a;
 594                 } else if (pow_b < pow_a) {
 595                         // b has lesser power, get coefficient from b
 596                         new_seq.push_back(*b);
 597                         if (is_order_function((*b).rest))
 598                                 break;
 599                         ++b;
 600                 } else {
 601                         // Add coefficient of a and b
 602                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 603                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 604                                 break;  // Order term ends the sequence
 605                         } else {
 606                                 ex sum = (*a).rest + (*b).rest;
 607                                 if (!(sum.is_zero()))
 608                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 609                                 ++a;
 610                                 ++b;
 611                         }
 612                 }
 613         }
 614         return pseries(relational(var,point), new_seq);
 615 }
 616
 617
 618 /** Implementation of ex::series() for sums. This performs series addition when
 619  *  adding pseries objects.
 620  *  @see ex::series */
 621 ex add::series(const relational & r, int order, unsigned options) const
 622 {
 623         ex acc; // Series accumulator
 624
 625         // Get first term from overall_coeff
 626         acc = overall_coeff.series(r, order, options);
 627
 628         // Add remaining terms
 629         epvector::const_iterator it = seq.begin();
 630         epvector::const_iterator itend = seq.end();
 631         for (; it!=itend; ++it) {
 632                 ex op;
 633                 if (is_ex_exactly_of_type(it->rest, pseries))
 634                         op = it->rest;
 635                 else
 636                         op = it->rest.series(r, order, options);
 637                 if (!it->coeff.is_equal(_ex1))
 638                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 639
 640                 // Series addition
 641                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 642         }
 643         return acc;
 644 }
 645
 646
 647 /** Multiply a pseries object with a numeric constant, producing a pseries
 648  *  object that represents the product.
 649  *
 650  *  @param other  constant to multiply with
 651  *  @return the product as a pseries */
 652 ex pseries::mul_const(const numeric &other) const
 653 {
 654         epvector new_seq;
 655         new_seq.reserve(seq.size());
 656
 657         epvector::const_iterator it = seq.begin(), itend = seq.end();
 658         while (it != itend) {
 659                 if (!is_order_function(it->rest))
 660                         new_seq.push_back(expair(it->rest * other, it->coeff));
 661                 else
 662                         new_seq.push_back(*it);
 663                 ++it;
 664         }
 665         return pseries(relational(var,point), new_seq);
 666 }
 667
 668
 669 /** Multiply one pseries object to another, producing a pseries object that
 670  *  represents the product.
 671  *
 672  *  @param other  pseries object to multiply with
 673  *  @return the product as a pseries */
 674 ex pseries::mul_series(const pseries &other) const
 675 {
 676         // Multiplying two series with different variables or expansion points
 677         // results in an empty (constant) series
 678         if (!is_compatible_to(other)) {
 679                 epvector nul;
 680                 nul.push_back(expair(Order(_ex1), _ex0));
 681                 return pseries(relational(var,point), nul);
 682         }
 683
 684         // Series multiplication
 685         epvector new_seq;
 686         int a_max = degree(var);
 687         int b_max = other.degree(var);
 688         int a_min = ldegree(var);
 689         int b_min = other.ldegree(var);
 690         int cdeg_min = a_min + b_min;
 691         int cdeg_max = a_max + b_max;
 692
 693         int higher_order_a = INT_MAX;
 694         int higher_order_b = INT_MAX;
 695         if (is_order_function(coeff(var, a_max)))
 696                 higher_order_a = a_max + b_min;
 697         if (is_order_function(other.coeff(var, b_max)))
 698                 higher_order_b = b_max + a_min;
 699         int higher_order_c = std::min(higher_order_a, higher_order_b);
 700         if (cdeg_max >= higher_order_c)
 701                 cdeg_max = higher_order_c - 1;
 702
 703         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 704                 ex co = _ex0;
 705                 // c(i)=a(0)b(i)+...+a(i)b(0)
 706                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 707                         ex a_coeff = coeff(var, i);
 708                         ex b_coeff = other.coeff(var, cdeg-i);
 709                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 710                                 co += a_coeff * b_coeff;
 711                 }
 712                 if (!co.is_zero())
 713                         new_seq.push_back(expair(co, numeric(cdeg)));
 714         }
 715         if (higher_order_c < INT_MAX)
 716                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 717         return pseries(relational(var, point), new_seq);
 718 }
 719
 720
 721 /** Implementation of ex::series() for product. This performs series
 722  *  multiplication when multiplying series.
 723  *  @see ex::series */
 724 ex mul::series(const relational & r, int order, unsigned options) const
 725 {
 726         pseries acc; // Series accumulator
 727
 728         // Multiply with remaining terms
 729         const epvector::const_iterator itbeg = seq.begin();
 730         const epvector::const_iterator itend = seq.end();
 731         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 732                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 733
 734                 // Series multiplication
 735                 if (it==itbeg)
 736                         acc = ex_to<pseries>(op);
 737                 else
 738                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 739         }
 740         return acc.mul_const(ex_to<numeric>(overall_coeff));
 741 }
 742
 743
 744 /** Compute the p-th power of a series.
 745  *
 746  *  @param p  power to compute
 747  *  @param deg  truncation order of series calculation */
 748 ex pseries::power_const(const numeric &p, int deg) const
 749 {
 750         // method:
 751         // (due to Leonhard Euler)
 752         // let A(x) be this series and for the time being let it start with a
 753         // constant (later we'll generalize):
 754         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 755         // We want to compute
 756         //     C(x) = A(x)^p
 757         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 758         // Taking the derivative on both sides and multiplying with A(x) one
 759         // immediately arrives at
 760         //     C'(x)*A(x) = p*C(x)*A'(x)
 761         // Multiplying this out and comparing coefficients we get the recurrence
 762         // formula
 763         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 764         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 765         // which can easily be solved given the starting value c_0 = (a_0)^p.
 766         // For the more general case where the leading coefficient of A(x) is not
 767         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 768         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 769         // then of course x^(p*m) but the recurrence formula still holds.
 770
 771         if (seq.empty()) {
 772                 // as a special case, handle the empty (zero) series honoring the
 773                 // usual power laws such as implemented in power::eval()
 774                 if (p.real().is_zero())
 775                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 776                 else if (p.real().is_negative())
 777                         throw pole_error("pseries::power_const(): division by zero",1);
 778                 else
 779                         return *this;
 780         }
 781
 782         const int ldeg = ldegree(var);
 783         if (!(p*ldeg).is_integer())
 784                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 785
 786         // O(x^n)^(-m) is undefined
 787         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 788                 throw pole_error("pseries::power_const(): division by zero",1);
 789
 790         // Compute coefficients of the powered series
 791         exvector co;
 792         co.reserve(deg);
 793         co.push_back(power(coeff(var, ldeg), p));
 794         bool all_sums_zero = true;
 795         for (int i=1; i<deg; ++i) {
 796                 ex sum = _ex0;
 797                 for (int j=1; j<=i; ++j) {
 798                         ex c = coeff(var, j + ldeg);
 799                         if (is_order_function(c)) {
 800                                 co.push_back(Order(_ex1));
 801                                 break;
 802                         } else
 803                                 sum += (p * j - (i - j)) * co[i - j] * c;
 804                 }
 805                 if (!sum.is_zero())
 806                         all_sums_zero = false;
 807                 co.push_back(sum / coeff(var, ldeg) / i);
 808         }
 809
 810         // Construct new series (of non-zero coefficients)
 811         epvector new_seq;
 812         bool higher_order = false;
 813         for (int i=0; i<deg; ++i) {
 814                 if (!co[i].is_zero())
 815                         new_seq.push_back(expair(co[i], p * ldeg + i));
 816                 if (is_order_function(co[i])) {
 817                         higher_order = true;
 818                         break;
 819                 }
 820         }
 821         if (!higher_order && !all_sums_zero)
 822                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 823         return pseries(relational(var,point), new_seq);
 824 }
 825
 826
 827 /** Return a new pseries object with the powers shifted by deg. */
 828 pseries pseries::shift_exponents(int deg) const
 829 {
 830         epvector newseq = seq;
 831         epvector::iterator i = newseq.begin(), end  = newseq.end();
 832         while (i != end) {
 833                 i->coeff += deg;
 834                 ++i;
 835         }
 836         return pseries(relational(var, point), newseq);
 837 }
 838
 839
 840 /** Implementation of ex::series() for powers. This performs Laurent expansion
 841  *  of reciprocals of series at singularities.
 842  *  @see ex::series */
 843 ex power::series(const relational & r, int order, unsigned options) const
 844 {
 845         // If basis is already a series, just power it
 846         if (is_ex_exactly_of_type(basis, pseries))
 847                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 848
 849         // Basis is not a series, may there be a singularity?
 850         bool must_expand_basis = false;
 851         try {
 852                 basis.subs(r);
 853         } catch (pole_error) {
 854                 must_expand_basis = true;
 855         }
 856
 857         // Is the expression of type something^(-int)?
 858         if (!must_expand_basis && !exponent.info(info_flags::negint))
 859                 return basic::series(r, order, options);
 860
 861         // Is the expression of type 0^something?
 862         if (!must_expand_basis && !basis.subs(r).is_zero())
 863                 return basic::series(r, order, options);
 864
 865         // Singularity encountered, is the basis equal to (var - point)?
 866         if (basis.is_equal(r.lhs() - r.rhs())) {
 867                 epvector new_seq;
 868                 if (ex_to<numeric>(exponent).to_int() < order)
 869                         new_seq.push_back(expair(_ex1, exponent));
 870                 else
 871                         new_seq.push_back(expair(Order(_ex1), exponent));
 872                 return pseries(r, new_seq);
 873         }
 874
 875         // No, expand basis into series
 876         ex e = basis.series(r, order, options);
 877         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 878 }
 879
 880
 881 /** Re-expansion of a pseries object. */
 882 ex pseries::series(const relational & r, int order, unsigned options) const
 883 {
 884         const ex p = r.rhs();
 885         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 886         const symbol &s = ex_to<symbol>(r.lhs());
 887
 888         if (var.is_equal(s) && point.is_equal(p)) {
 889                 if (order > degree(s))
 890                         return *this;
 891                 else {
 892                         epvector new_seq;
 893                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 894                         while (it != itend) {
 895                                 int o = ex_to<numeric>(it->coeff).to_int();
 896                                 if (o >= order) {
 897                                         new_seq.push_back(expair(Order(_ex1), o));
 898                                         break;
 899                                 }
 900                                 new_seq.push_back(*it);
 901                                 ++it;
 902                         }
 903                         return pseries(r, new_seq);
 904                 }
 905         } else
 906                 return convert_to_poly().series(r, order, options);
 907 }
 908
 909
 910 /** Compute the truncated series expansion of an expression.
 911  *  This function returns an expression containing an object of class pseries
 912  *  to represent the series. If the series does not terminate within the given
 913  *  truncation order, the last term of the series will be an order term.
 914  *
 915  *  @param r  expansion relation, lhs holds variable and rhs holds point
 916  *  @param order  truncation order of series calculations
 917  *  @param options  of class series_options
 918  *  @return an expression holding a pseries object */
 919 ex ex::series(const ex & r, int order, unsigned options) const
 920 {
 921         GINAC_ASSERT(bp!=0);
 922         ex e;
 923         relational rel_;
 924
 925         if (is_ex_exactly_of_type(r,relational))
 926                 rel_ = ex_to<relational>(r);
 927         else if (is_ex_exactly_of_type(r,symbol))
 928                 rel_ = relational(r,_ex0);
 929         else
 930                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 931
 932         try {
 933                 e = bp->series(rel_, order, options);
 934         } catch (std::exception &x) {
 935                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 936         }
 937         return e;
 938 }
 939
 940 } // namespace GiNaC