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) 1999-2003 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 <iostream>
  25 #include <stdexcept>
  26
  27 #include "pseries.h"
  28 #include "add.h"
  29 #include "inifcns.h" // for Order function
  30 #include "lst.h"
  31 #include "mul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "operators.h"
  35 #include "symbol.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  42
  43
  44 /*
  45  *  Default constructor
  46  */
  47
  48 pseries::pseries() : inherited(TINFO_pseries) { }
  49
  50
  51 /*
  52  *  Other ctors
  53  */
  54
  55 /** Construct pseries from a vector of coefficients and powers.
  56  *  expair.rest holds the coefficient, expair.coeff holds the power.
  57  *  The powers must be integers (positive or negative) and in ascending order;
  58  *  the last coefficient can be Order(_ex1) to represent a truncated,
  59  *  non-terminating series.
  60  *
  61  *  @param rel_  expansion variable and point (must hold a relational)
  62  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  63  *  @return newly constructed pseries */
  64 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  65 {
  66         GINAC_ASSERT(is_a<relational>(rel_));
  67         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  68         point = rel_.rhs();
  69         var = rel_.lhs();
  70 }
  71
  72
  73 /*
  74  *  Archiving
  75  */
  76
  77 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  78 {
  79         for (unsigned int i=0; true; ++i) {
  80                 ex rest;
  81                 ex coeff;
  82                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  83                         seq.push_back(expair(rest, coeff));
  84                 else
  85                         break;
  86         }
  87         n.find_ex("var", var, sym_lst);
  88         n.find_ex("point", point, sym_lst);
  89 }
  90
  91 void pseries::archive(archive_node &n) const
  92 {
  93         inherited::archive(n);
  94         epvector::const_iterator i = seq.begin(), iend = seq.end();
  95         while (i != iend) {
  96                 n.add_ex("coeff", i->rest);
  97                 n.add_ex("power", i->coeff);
  98                 ++i;
  99         }
 100         n.add_ex("var", var);
 101         n.add_ex("point", point);
 102 }
 103
 104 DEFAULT_UNARCHIVE(pseries)
 105
 106 //////////
 107 // functions overriding virtual functions from base classes
 108 //////////
 109
 110 void pseries::print(const print_context & c, unsigned level) const
 111 {
 112         if (is_a<print_tree>(c)) {
 113
 114                 c.s << std::string(level, ' ') << class_name()
 115                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 116                     << std::endl;
 117                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 118                 size_t num = seq.size();
 119                 for (size_t i=0; i<num; ++i) {
 120                         seq[i].rest.print(c, level + delta_indent);
 121                         seq[i].coeff.print(c, level + delta_indent);
 122                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 123                 }
 124                 var.print(c, level + delta_indent);
 125                 point.print(c, level + delta_indent);
 126
 127         } else if (is_a<print_python_repr>(c)) {
 128                 c.s << class_name() << "(relational(";
 129                 var.print(c);
 130                 c.s << ',';
 131                 point.print(c);
 132                 c.s << "),[";
 133                 size_t num = seq.size();
 134                 for (size_t i=0; i<num; ++i) {
 135                         if (i)
 136                                 c.s << ',';
 137                         c.s << '(';
 138                         seq[i].rest.print(c);
 139                         c.s << ',';
 140                         seq[i].coeff.print(c);
 141                         c.s << ')';
 142                 }
 143                 c.s << "])";
 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                                                 if (is_a<print_python>(c))
 185                                                         c.s << "**";
 186                                                 else
 187                                                         c.s << '^';
 188                                                 if (i->coeff.info(info_flags::negative)) {
 189                                                         c.s << par_open;
 190                                                         i->coeff.print(c);
 191                                                         c.s << par_close;
 192                                                 } else {
 193                                                         if (is_a<print_latex>(c)) {
 194                                                                 c.s << '{';
 195                                                                 i->coeff.print(c);
 196                                                                 c.s << '}';
 197                                                         } else
 198                                                                 i->coeff.print(c);
 199                                                 }
 200                                         }
 201                                 }
 202                         } else
 203                                 Order(power(var-point,i->coeff)).print(c);
 204                         ++i;
 205                 }
 206
 207                 if (precedence() <= level)
 208                         c.s << ")";
 209         }
 210 }
 211
 212 int pseries::compare_same_type(const basic & other) const
 213 {
 214         GINAC_ASSERT(is_a<pseries>(other));
 215         const pseries &o = static_cast<const pseries &>(other);
 216
 217         // first compare the lengths of the series...
 218         if (seq.size()>o.seq.size())
 219                 return 1;
 220         if (seq.size()<o.seq.size())
 221                 return -1;
 222
 223         // ...then the expansion point...
 224         int cmpval = var.compare(o.var);
 225         if (cmpval)
 226                 return cmpval;
 227         cmpval = point.compare(o.point);
 228         if (cmpval)
 229                 return cmpval;
 230
 231         // ...and if that failed the individual elements
 232         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 233         while (it!=seq.end() && o_it!=o.seq.end()) {
 234                 cmpval = it->compare(*o_it);
 235                 if (cmpval)
 236                         return cmpval;
 237                 ++it;
 238                 ++o_it;
 239         }
 240
 241         // so they are equal.
 242         return 0;
 243 }
 244
 245 /** Return the number of operands including a possible order term. */
 246 size_t pseries::nops() const
 247 {
 248         return seq.size();
 249 }
 250
 251 /** Return the ith term in the series when represented as a sum. */
 252 ex pseries::op(size_t i) const
 253 {
 254         if (i >= seq.size())
 255                 throw (std::out_of_range("op() out of range"));
 256
 257         return seq[i].rest * power(var - point, seq[i].coeff);
 258 }
 259
 260 /** Return degree of highest power of the series.  This is usually the exponent
 261  *  of the Order term.  If s is not the expansion variable of the series, the
 262  *  series is examined termwise. */
 263 int pseries::degree(const ex &s) const
 264 {
 265         if (var.is_equal(s)) {
 266                 // Return last exponent
 267                 if (seq.size())
 268                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 269                 else
 270                         return 0;
 271         } else {
 272                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 273                 if (it == itend)
 274                         return 0;
 275                 int max_pow = INT_MIN;
 276                 while (it != itend) {
 277                         int pow = it->rest.degree(s);
 278                         if (pow > max_pow)
 279                                 max_pow = pow;
 280                         ++it;
 281                 }
 282                 return max_pow;
 283         }
 284 }
 285
 286 /** Return degree of lowest power of the series.  This is usually the exponent
 287  *  of the leading term.  If s is not the expansion variable of the series, the
 288  *  series is examined termwise.  If s is the expansion variable but the
 289  *  expansion point is not zero the series is not expanded to find the degree.
 290  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 291 int pseries::ldegree(const ex &s) const
 292 {
 293         if (var.is_equal(s)) {
 294                 // Return first exponent
 295                 if (seq.size())
 296                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 297                 else
 298                         return 0;
 299         } else {
 300                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 301                 if (it == itend)
 302                         return 0;
 303                 int min_pow = INT_MAX;
 304                 while (it != itend) {
 305                         int pow = it->rest.ldegree(s);
 306                         if (pow < min_pow)
 307                                 min_pow = pow;
 308                         ++it;
 309                 }
 310                 return min_pow;
 311         }
 312 }
 313
 314 /** Return coefficient of degree n in power series if s is the expansion
 315  *  variable.  If the expansion point is nonzero, by definition the n=1
 316  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 317  *  the expansion took place in the s in the first place).
 318  *  If s is not the expansion variable, an attempt is made to convert the
 319  *  series to a polynomial and return the corresponding coefficient from
 320  *  there. */
 321 ex pseries::coeff(const ex &s, int n) const
 322 {
 323         if (var.is_equal(s)) {
 324                 if (seq.empty())
 325                         return _ex0;
 326
 327                 // Binary search in sequence for given power
 328                 numeric looking_for = numeric(n);
 329                 int lo = 0, hi = seq.size() - 1;
 330                 while (lo <= hi) {
 331                         int mid = (lo + hi) / 2;
 332                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 333                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 334                         switch (cmp) {
 335                                 case -1:
 336                                         lo = mid + 1;
 337                                         break;
 338                                 case 0:
 339                                         return seq[mid].rest;
 340                                 case 1:
 341                                         hi = mid - 1;
 342                                         break;
 343                                 default:
 344                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 345                         }
 346                 }
 347                 return _ex0;
 348         } else
 349                 return convert_to_poly().coeff(s, n);
 350 }
 351
 352 /** Does nothing. */
 353 ex pseries::collect(const ex &s, bool distributed) const
 354 {
 355         return *this;
 356 }
 357
 358 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 359 ex pseries::eval(int level) const
 360 {
 361         if (level == 1)
 362                 return this->hold();
 363
 364         if (level == -max_recursion_level)
 365                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 366
 367         // Construct a new series with evaluated coefficients
 368         epvector new_seq;
 369         new_seq.reserve(seq.size());
 370         epvector::const_iterator it = seq.begin(), itend = seq.end();
 371         while (it != itend) {
 372                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 373                 ++it;
 374         }
 375         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 376 }
 377
 378 /** Evaluate coefficients numerically. */
 379 ex pseries::evalf(int level) const
 380 {
 381         if (level == 1)
 382                 return *this;
 383
 384         if (level == -max_recursion_level)
 385                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 386
 387         // Construct a new series with evaluated coefficients
 388         epvector new_seq;
 389         new_seq.reserve(seq.size());
 390         epvector::const_iterator it = seq.begin(), itend = seq.end();
 391         while (it != itend) {
 392                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 393                 ++it;
 394         }
 395         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 396 }
 397
 398 ex pseries::subs(const exmap & m, unsigned options) const
 399 {
 400         // If expansion variable is being substituted, convert the series to a
 401         // polynomial and do the substitution there because the result might
 402         // no longer be a power series
 403         if (m.find(var) != m.end())
 404                 return convert_to_poly(true).subs(m, options);
 405
 406         // Otherwise construct a new series with substituted coefficients and
 407         // expansion point
 408         epvector newseq;
 409         newseq.reserve(seq.size());
 410         epvector::const_iterator it = seq.begin(), itend = seq.end();
 411         while (it != itend) {
 412                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
 413                 ++it;
 414         }
 415         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
 416 }
 417
 418 /** Implementation of ex::expand() for a power series.  It expands all the
 419  *  terms individually and returns the resulting series as a new pseries. */
 420 ex pseries::expand(unsigned options) const
 421 {
 422         epvector newseq;
 423         epvector::const_iterator i = seq.begin(), end = seq.end();
 424         while (i != end) {
 425                 ex restexp = i->rest.expand();
 426                 if (!restexp.is_zero())
 427                         newseq.push_back(expair(restexp, i->coeff));
 428                 ++i;
 429         }
 430         return (new pseries(relational(var,point), newseq))
 431                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 432 }
 433
 434 /** Implementation of ex::diff() for a power series.
 435  *  @see ex::diff */
 436 ex pseries::derivative(const symbol & s) const
 437 {
 438         epvector new_seq;
 439         epvector::const_iterator it = seq.begin(), itend = seq.end();
 440
 441         if (s == var) {
 442
 443                 // FIXME: coeff might depend on var
 444                 while (it != itend) {
 445                         if (is_order_function(it->rest)) {
 446                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 447                         } else {
 448                                 ex c = it->rest * it->coeff;
 449                                 if (!c.is_zero())
 450                                         new_seq.push_back(expair(c, it->coeff - 1));
 451                         }
 452                         ++it;
 453                 }
 454
 455         } else {
 456
 457                 while (it != itend) {
 458                         if (is_order_function(it->rest)) {
 459                                 new_seq.push_back(*it);
 460                         } else {
 461                                 ex c = it->rest.diff(s);
 462                                 if (!c.is_zero())
 463                                         new_seq.push_back(expair(c, it->coeff));
 464                         }
 465                         ++it;
 466                 }
 467         }
 468
 469         return pseries(relational(var,point), new_seq);
 470 }
 471
 472 ex pseries::convert_to_poly(bool no_order) const
 473 {
 474         ex e;
 475         epvector::const_iterator it = seq.begin(), itend = seq.end();
 476
 477         while (it != itend) {
 478                 if (is_order_function(it->rest)) {
 479                         if (!no_order)
 480                                 e += Order(power(var - point, it->coeff));
 481                 } else
 482                         e += it->rest * power(var - point, it->coeff);
 483                 ++it;
 484         }
 485         return e;
 486 }
 487
 488 bool pseries::is_terminating() const
 489 {
 490         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 491 }
 492
 493
 494 /*
 495  *  Implementations of series expansion
 496  */
 497
 498 /** Default implementation of ex::series(). This performs Taylor expansion.
 499  *  @see ex::series */
 500 ex basic::series(const relational & r, int order, unsigned options) const
 501 {
 502         epvector seq;
 503         numeric fac = 1;
 504         ex deriv = *this;
 505         ex coeff = deriv.subs(r, subs_options::no_pattern);
 506         const symbol &s = ex_to<symbol>(r.lhs());
 507
 508         if (!coeff.is_zero())
 509                 seq.push_back(expair(coeff, _ex0));
 510
 511         int n;
 512         for (n=1; n<order; ++n) {
 513                 fac = fac.mul(n);
 514                 // We need to test for zero in order to see if the series terminates.
 515                 // The problem is that there is no such thing as a perfect test for
 516                 // zero.  Expanding the term occasionally helps a little...
 517                 deriv = deriv.diff(s).expand();
 518                 if (deriv.is_zero())  // Series terminates
 519                         return pseries(r, seq);
 520
 521                 coeff = deriv.subs(r, subs_options::no_pattern);
 522                 if (!coeff.is_zero())
 523                         seq.push_back(expair(fac.inverse() * coeff, n));
 524         }
 525
 526         // Higher-order terms, if present
 527         deriv = deriv.diff(s);
 528         if (!deriv.expand().is_zero())
 529                 seq.push_back(expair(Order(_ex1), n));
 530         return pseries(r, seq);
 531 }
 532
 533
 534 /** Implementation of ex::series() for symbols.
 535  *  @see ex::series */
 536 ex symbol::series(const relational & r, int order, unsigned options) const
 537 {
 538         epvector seq;
 539         const ex point = r.rhs();
 540         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 541
 542         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 543                 if (order > 0 && !point.is_zero())
 544                         seq.push_back(expair(point, _ex0));
 545                 if (order > 1)
 546                         seq.push_back(expair(_ex1, _ex1));
 547                 else
 548                         seq.push_back(expair(Order(_ex1), numeric(order)));
 549         } else
 550                 seq.push_back(expair(*this, _ex0));
 551         return pseries(r, seq);
 552 }
 553
 554
 555 /** Add one series object to another, producing a pseries object that
 556  *  represents the sum.
 557  *
 558  *  @param other  pseries object to add with
 559  *  @return the sum as a pseries */
 560 ex pseries::add_series(const pseries &other) const
 561 {
 562         // Adding two series with different variables or expansion points
 563         // results in an empty (constant) series
 564         if (!is_compatible_to(other)) {
 565                 epvector nul;
 566                 nul.push_back(expair(Order(_ex1), _ex0));
 567                 return pseries(relational(var,point), nul);
 568         }
 569
 570         // Series addition
 571         epvector new_seq;
 572         epvector::const_iterator a = seq.begin();
 573         epvector::const_iterator b = other.seq.begin();
 574         epvector::const_iterator a_end = seq.end();
 575         epvector::const_iterator b_end = other.seq.end();
 576         int pow_a = INT_MAX, pow_b = INT_MAX;
 577         for (;;) {
 578                 // If a is empty, fill up with elements from b and stop
 579                 if (a == a_end) {
 580                         while (b != b_end) {
 581                                 new_seq.push_back(*b);
 582                                 ++b;
 583                         }
 584                         break;
 585                 } else
 586                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 587
 588                 // If b is empty, fill up with elements from a and stop
 589                 if (b == b_end) {
 590                         while (a != a_end) {
 591                                 new_seq.push_back(*a);
 592                                 ++a;
 593                         }
 594                         break;
 595                 } else
 596                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 597
 598                 // a and b are non-empty, compare powers
 599                 if (pow_a < pow_b) {
 600                         // a has lesser power, get coefficient from a
 601                         new_seq.push_back(*a);
 602                         if (is_order_function((*a).rest))
 603                                 break;
 604                         ++a;
 605                 } else if (pow_b < pow_a) {
 606                         // b has lesser power, get coefficient from b
 607                         new_seq.push_back(*b);
 608                         if (is_order_function((*b).rest))
 609                                 break;
 610                         ++b;
 611                 } else {
 612                         // Add coefficient of a and b
 613                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 614                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 615                                 break;  // Order term ends the sequence
 616                         } else {
 617                                 ex sum = (*a).rest + (*b).rest;
 618                                 if (!(sum.is_zero()))
 619                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 620                                 ++a;
 621                                 ++b;
 622                         }
 623                 }
 624         }
 625         return pseries(relational(var,point), new_seq);
 626 }
 627
 628
 629 /** Implementation of ex::series() for sums. This performs series addition when
 630  *  adding pseries objects.
 631  *  @see ex::series */
 632 ex add::series(const relational & r, int order, unsigned options) const
 633 {
 634         ex acc; // Series accumulator
 635
 636         // Get first term from overall_coeff
 637         acc = overall_coeff.series(r, order, options);
 638
 639         // Add remaining terms
 640         epvector::const_iterator it = seq.begin();
 641         epvector::const_iterator itend = seq.end();
 642         for (; it!=itend; ++it) {
 643                 ex op;
 644                 if (is_exactly_a<pseries>(it->rest))
 645                         op = it->rest;
 646                 else
 647                         op = it->rest.series(r, order, options);
 648                 if (!it->coeff.is_equal(_ex1))
 649                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 650
 651                 // Series addition
 652                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 653         }
 654         return acc;
 655 }
 656
 657
 658 /** Multiply a pseries object with a numeric constant, producing a pseries
 659  *  object that represents the product.
 660  *
 661  *  @param other  constant to multiply with
 662  *  @return the product as a pseries */
 663 ex pseries::mul_const(const numeric &other) const
 664 {
 665         epvector new_seq;
 666         new_seq.reserve(seq.size());
 667
 668         epvector::const_iterator it = seq.begin(), itend = seq.end();
 669         while (it != itend) {
 670                 if (!is_order_function(it->rest))
 671                         new_seq.push_back(expair(it->rest * other, it->coeff));
 672                 else
 673                         new_seq.push_back(*it);
 674                 ++it;
 675         }
 676         return pseries(relational(var,point), new_seq);
 677 }
 678
 679
 680 /** Multiply one pseries object to another, producing a pseries object that
 681  *  represents the product.
 682  *
 683  *  @param other  pseries object to multiply with
 684  *  @return the product as a pseries */
 685 ex pseries::mul_series(const pseries &other) const
 686 {
 687         // Multiplying two series with different variables or expansion points
 688         // results in an empty (constant) series
 689         if (!is_compatible_to(other)) {
 690                 epvector nul;
 691                 nul.push_back(expair(Order(_ex1), _ex0));
 692                 return pseries(relational(var,point), nul);
 693         }
 694
 695         // Series multiplication
 696         epvector new_seq;
 697         int a_max = degree(var);
 698         int b_max = other.degree(var);
 699         int a_min = ldegree(var);
 700         int b_min = other.ldegree(var);
 701         int cdeg_min = a_min + b_min;
 702         int cdeg_max = a_max + b_max;
 703
 704         int higher_order_a = INT_MAX;
 705         int higher_order_b = INT_MAX;
 706         if (is_order_function(coeff(var, a_max)))
 707                 higher_order_a = a_max + b_min;
 708         if (is_order_function(other.coeff(var, b_max)))
 709                 higher_order_b = b_max + a_min;
 710         int higher_order_c = std::min(higher_order_a, higher_order_b);
 711         if (cdeg_max >= higher_order_c)
 712                 cdeg_max = higher_order_c - 1;
 713
 714         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 715                 ex co = _ex0;
 716                 // c(i)=a(0)b(i)+...+a(i)b(0)
 717                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 718                         ex a_coeff = coeff(var, i);
 719                         ex b_coeff = other.coeff(var, cdeg-i);
 720                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 721                                 co += a_coeff * b_coeff;
 722                 }
 723                 if (!co.is_zero())
 724                         new_seq.push_back(expair(co, numeric(cdeg)));
 725         }
 726         if (higher_order_c < INT_MAX)
 727                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 728         return pseries(relational(var, point), new_seq);
 729 }
 730
 731
 732 /** Implementation of ex::series() for product. This performs series
 733  *  multiplication when multiplying series.
 734  *  @see ex::series */
 735 ex mul::series(const relational & r, int order, unsigned options) const
 736 {
 737         pseries acc; // Series accumulator
 738
 739         // Multiply with remaining terms
 740         const epvector::const_iterator itbeg = seq.begin();
 741         const epvector::const_iterator itend = seq.end();
 742         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 743                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 744
 745                 // Series multiplication
 746                 if (it==itbeg)
 747                         acc = ex_to<pseries>(op);
 748                 else
 749                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 750         }
 751         return acc.mul_const(ex_to<numeric>(overall_coeff));
 752 }
 753
 754
 755 /** Compute the p-th power of a series.
 756  *
 757  *  @param p  power to compute
 758  *  @param deg  truncation order of series calculation */
 759 ex pseries::power_const(const numeric &p, int deg) const
 760 {
 761         // method:
 762         // (due to Leonhard Euler)
 763         // let A(x) be this series and for the time being let it start with a
 764         // constant (later we'll generalize):
 765         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 766         // We want to compute
 767         //     C(x) = A(x)^p
 768         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 769         // Taking the derivative on both sides and multiplying with A(x) one
 770         // immediately arrives at
 771         //     C'(x)*A(x) = p*C(x)*A'(x)
 772         // Multiplying this out and comparing coefficients we get the recurrence
 773         // formula
 774         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 775         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 776         // which can easily be solved given the starting value c_0 = (a_0)^p.
 777         // For the more general case where the leading coefficient of A(x) is not
 778         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 779         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 780         // then of course x^(p*m) but the recurrence formula still holds.
 781
 782         if (seq.empty()) {
 783                 // as a special case, handle the empty (zero) series honoring the
 784                 // usual power laws such as implemented in power::eval()
 785                 if (p.real().is_zero())
 786                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 787                 else if (p.real().is_negative())
 788                         throw pole_error("pseries::power_const(): division by zero",1);
 789                 else
 790                         return *this;
 791         }
 792
 793         const int ldeg = ldegree(var);
 794         if (!(p*ldeg).is_integer())
 795                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 796
 797         // O(x^n)^(-m) is undefined
 798         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 799                 throw pole_error("pseries::power_const(): division by zero",1);
 800
 801         // Compute coefficients of the powered series
 802         exvector co;
 803         co.reserve(deg);
 804         co.push_back(power(coeff(var, ldeg), p));
 805         bool all_sums_zero = true;
 806         for (int i=1; i<deg; ++i) {
 807                 ex sum = _ex0;
 808                 for (int j=1; j<=i; ++j) {
 809                         ex c = coeff(var, j + ldeg);
 810                         if (is_order_function(c)) {
 811                                 co.push_back(Order(_ex1));
 812                                 break;
 813                         } else
 814                                 sum += (p * j - (i - j)) * co[i - j] * c;
 815                 }
 816                 if (!sum.is_zero())
 817                         all_sums_zero = false;
 818                 co.push_back(sum / coeff(var, ldeg) / i);
 819         }
 820
 821         // Construct new series (of non-zero coefficients)
 822         epvector new_seq;
 823         bool higher_order = false;
 824         for (int i=0; i<deg; ++i) {
 825                 if (!co[i].is_zero())
 826                         new_seq.push_back(expair(co[i], p * ldeg + i));
 827                 if (is_order_function(co[i])) {
 828                         higher_order = true;
 829                         break;
 830                 }
 831         }
 832         if (!higher_order && !all_sums_zero)
 833                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 834         return pseries(relational(var,point), new_seq);
 835 }
 836
 837
 838 /** Return a new pseries object with the powers shifted by deg. */
 839 pseries pseries::shift_exponents(int deg) const
 840 {
 841         epvector newseq = seq;
 842         epvector::iterator i = newseq.begin(), end  = newseq.end();
 843         while (i != end) {
 844                 i->coeff += deg;
 845                 ++i;
 846         }
 847         return pseries(relational(var, point), newseq);
 848 }
 849
 850
 851 /** Implementation of ex::series() for powers. This performs Laurent expansion
 852  *  of reciprocals of series at singularities.
 853  *  @see ex::series */
 854 ex power::series(const relational & r, int order, unsigned options) const
 855 {
 856         // If basis is already a series, just power it
 857         if (is_exactly_a<pseries>(basis))
 858                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 859
 860         // Basis is not a series, may there be a singularity?
 861         bool must_expand_basis = false;
 862         try {
 863                 basis.subs(r, subs_options::no_pattern);
 864         } catch (pole_error) {
 865                 must_expand_basis = true;
 866         }
 867
 868         // Is the expression of type something^(-int)?
 869         if (!must_expand_basis && !exponent.info(info_flags::negint))
 870                 return basic::series(r, order, options);
 871
 872         // Is the expression of type 0^something?
 873         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero())
 874                 return basic::series(r, order, options);
 875
 876         // Singularity encountered, is the basis equal to (var - point)?
 877         if (basis.is_equal(r.lhs() - r.rhs())) {
 878                 epvector new_seq;
 879                 if (ex_to<numeric>(exponent).to_int() < order)
 880                         new_seq.push_back(expair(_ex1, exponent));
 881                 else
 882                         new_seq.push_back(expair(Order(_ex1), exponent));
 883                 return pseries(r, new_seq);
 884         }
 885
 886         // No, expand basis into series
 887         ex e = basis.series(r, order, options);
 888         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 889 }
 890
 891
 892 /** Re-expansion of a pseries object. */
 893 ex pseries::series(const relational & r, int order, unsigned options) const
 894 {
 895         const ex p = r.rhs();
 896         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 897         const symbol &s = ex_to<symbol>(r.lhs());
 898
 899         if (var.is_equal(s) && point.is_equal(p)) {
 900                 if (order > degree(s))
 901                         return *this;
 902                 else {
 903                         epvector new_seq;
 904                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 905                         while (it != itend) {
 906                                 int o = ex_to<numeric>(it->coeff).to_int();
 907                                 if (o >= order) {
 908                                         new_seq.push_back(expair(Order(_ex1), o));
 909                                         break;
 910                                 }
 911                                 new_seq.push_back(*it);
 912                                 ++it;
 913                         }
 914                         return pseries(r, new_seq);
 915                 }
 916         } else
 917                 return convert_to_poly().series(r, order, options);
 918 }
 919
 920
 921 /** Compute the truncated series expansion of an expression.
 922  *  This function returns an expression containing an object of class pseries
 923  *  to represent the series. If the series does not terminate within the given
 924  *  truncation order, the last term of the series will be an order term.
 925  *
 926  *  @param r  expansion relation, lhs holds variable and rhs holds point
 927  *  @param order  truncation order of series calculations
 928  *  @param options  of class series_options
 929  *  @return an expression holding a pseries object */
 930 ex ex::series(const ex & r, int order, unsigned options) const
 931 {
 932         ex e;
 933         relational rel_;
 934
 935         if (is_a<relational>(r))
 936                 rel_ = ex_to<relational>(r);
 937         else if (is_a<symbol>(r))
 938                 rel_ = relational(r,_ex0);
 939         else
 940                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 941
 942         try {
 943                 e = bp->series(rel_, order, options);
 944         } catch (std::exception &x) {
 945                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 946         }
 947         return e;
 948 }
 949
 950 } // namespace GiNaC