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 "print.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  43
  44
  45 /*
  46  *  Default constructor
  47  */
  48
  49 pseries::pseries() : inherited(TINFO_pseries) { }
  50
  51
  52 /*
  53  *  Other ctors
  54  */
  55
  56 /** Construct pseries from a vector of coefficients and powers.
  57  *  expair.rest holds the coefficient, expair.coeff holds the power.
  58  *  The powers must be integers (positive or negative) and in ascending order;
  59  *  the last coefficient can be Order(_ex1) to represent a truncated,
  60  *  non-terminating series.
  61  *
  62  *  @param rel_  expansion variable and point (must hold a relational)
  63  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  64  *  @return newly constructed pseries */
  65 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  66 {
  67         GINAC_ASSERT(is_a<relational>(rel_));
  68         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  69         point = rel_.rhs();
  70         var = rel_.lhs();
  71 }
  72
  73
  74 /*
  75  *  Archiving
  76  */
  77
  78 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  79 {
  80         for (unsigned int i=0; true; ++i) {
  81                 ex rest;
  82                 ex coeff;
  83                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  84                         seq.push_back(expair(rest, coeff));
  85                 else
  86                         break;
  87         }
  88         n.find_ex("var", var, sym_lst);
  89         n.find_ex("point", point, sym_lst);
  90 }
  91
  92 void pseries::archive(archive_node &n) const
  93 {
  94         inherited::archive(n);
  95         epvector::const_iterator i = seq.begin(), iend = seq.end();
  96         while (i != iend) {
  97                 n.add_ex("coeff", i->rest);
  98                 n.add_ex("power", i->coeff);
  99                 ++i;
 100         }
 101         n.add_ex("var", var);
 102         n.add_ex("point", point);
 103 }
 104
 105 DEFAULT_UNARCHIVE(pseries)
 106
 107 //////////
 108 // functions overriding virtual functions from base classes
 109 //////////
 110
 111 void pseries::print(const print_context & c, unsigned level) const
 112 {
 113         if (is_a<print_tree>(c)) {
 114
 115                 c.s << std::string(level, ' ') << class_name()
 116                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 117                     << std::endl;
 118                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 119                 size_t num = seq.size();
 120                 for (size_t i=0; i<num; ++i) {
 121                         seq[i].rest.print(c, level + delta_indent);
 122                         seq[i].coeff.print(c, level + delta_indent);
 123                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 124                 }
 125                 var.print(c, level + delta_indent);
 126                 point.print(c, level + delta_indent);
 127
 128         } else if (is_a<print_python_repr>(c)) {
 129                 c.s << class_name() << "(relational(";
 130                 var.print(c);
 131                 c.s << ',';
 132                 point.print(c);
 133                 c.s << "),[";
 134                 size_t num = seq.size();
 135                 for (size_t i=0; i<num; ++i) {
 136                         if (i)
 137                                 c.s << ',';
 138                         c.s << '(';
 139                         seq[i].rest.print(c);
 140                         c.s << ',';
 141                         seq[i].coeff.print(c);
 142                         c.s << ')';
 143                 }
 144                 c.s << "])";
 145         } else {
 146
 147                 if (precedence() <= level)
 148                         c.s << "(";
 149
 150                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
 151                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
 152
 153                 // objects of type pseries must not have any zero entries, so the
 154                 // trivial (zero) pseries needs a special treatment here:
 155                 if (seq.empty())
 156                         c.s << '0';
 157                 epvector::const_iterator i = seq.begin(), end = seq.end();
 158                 while (i != end) {
 159                         // print a sign, if needed
 160                         if (i != seq.begin())
 161                                 c.s << '+';
 162                         if (!is_order_function(i->rest)) {
 163                                 // print 'rest', i.e. the expansion coefficient
 164                                 if (i->rest.info(info_flags::numeric) &&
 165                                         i->rest.info(info_flags::positive)) {
 166                                         i->rest.print(c);
 167                                 } else {
 168                                         c.s << par_open;
 169                                         i->rest.print(c);
 170                                         c.s << par_close;
 171                                 }
 172                                 // print 'coeff', something like (x-1)^42
 173                                 if (!i->coeff.is_zero()) {
 174                                         if (is_a<print_latex>(c))
 175                                                 c.s << ' ';
 176                                         else
 177                                                 c.s << '*';
 178                                         if (!point.is_zero()) {
 179                                                 c.s << par_open;
 180                                                 (var-point).print(c);
 181                                                 c.s << par_close;
 182                                         } else
 183                                                 var.print(c);
 184                                         if (i->coeff.compare(_ex1)) {
 185                                                 if (is_a<print_python>(c))
 186                                                         c.s << "**";
 187                                                 else
 188                                                         c.s << '^';
 189                                                 if (i->coeff.info(info_flags::negative)) {
 190                                                         c.s << par_open;
 191                                                         i->coeff.print(c);
 192                                                         c.s << par_close;
 193                                                 } else {
 194                                                         if (is_a<print_latex>(c)) {
 195                                                                 c.s << '{';
 196                                                                 i->coeff.print(c);
 197                                                                 c.s << '}';
 198                                                         } else
 199                                                                 i->coeff.print(c);
 200                                                 }
 201                                         }
 202                                 }
 203                         } else
 204                                 Order(power(var-point,i->coeff)).print(c);
 205                         ++i;
 206                 }
 207
 208                 if (precedence() <= level)
 209                         c.s << ")";
 210         }
 211 }
 212
 213 int pseries::compare_same_type(const basic & other) const
 214 {
 215         GINAC_ASSERT(is_a<pseries>(other));
 216         const pseries &o = static_cast<const pseries &>(other);
 217
 218         // first compare the lengths of the series...
 219         if (seq.size()>o.seq.size())
 220                 return 1;
 221         if (seq.size()<o.seq.size())
 222                 return -1;
 223
 224         // ...then the expansion point...
 225         int cmpval = var.compare(o.var);
 226         if (cmpval)
 227                 return cmpval;
 228         cmpval = point.compare(o.point);
 229         if (cmpval)
 230                 return cmpval;
 231
 232         // ...and if that failed the individual elements
 233         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 234         while (it!=seq.end() && o_it!=o.seq.end()) {
 235                 cmpval = it->compare(*o_it);
 236                 if (cmpval)
 237                         return cmpval;
 238                 ++it;
 239                 ++o_it;
 240         }
 241
 242         // so they are equal.
 243         return 0;
 244 }
 245
 246 /** Return the number of operands including a possible order term. */
 247 size_t pseries::nops() const
 248 {
 249         return seq.size();
 250 }
 251
 252 /** Return the ith term in the series when represented as a sum. */
 253 ex pseries::op(size_t i) const
 254 {
 255         if (i >= seq.size())
 256                 throw (std::out_of_range("op() out of range"));
 257
 258         return seq[i].rest * power(var - point, seq[i].coeff);
 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 exmap & m, unsigned options) 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 (m.find(var) != m.end())
 405                 return convert_to_poly(true).subs(m, options);
 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(m, options), it->coeff));
 414                 ++it;
 415         }
 416         return (new pseries(relational(var,point.subs(m, options)), 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.
 436  *  @see ex::diff */
 437 ex pseries::derivative(const symbol & s) const
 438 {
 439         epvector new_seq;
 440         epvector::const_iterator it = seq.begin(), itend = seq.end();
 441
 442         if (s == var) {
 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
 456         } else {
 457
 458                 while (it != itend) {
 459                         if (is_order_function(it->rest)) {
 460                                 new_seq.push_back(*it);
 461                         } else {
 462                                 ex c = it->rest.diff(s);
 463                                 if (!c.is_zero())
 464                                         new_seq.push_back(expair(c, it->coeff));
 465                         }
 466                         ++it;
 467                 }
 468         }
 469
 470         return pseries(relational(var,point), new_seq);
 471 }
 472
 473 ex pseries::convert_to_poly(bool no_order) const
 474 {
 475         ex e;
 476         epvector::const_iterator it = seq.begin(), itend = seq.end();
 477
 478         while (it != itend) {
 479                 if (is_order_function(it->rest)) {
 480                         if (!no_order)
 481                                 e += Order(power(var - point, it->coeff));
 482                 } else
 483                         e += it->rest * power(var - point, it->coeff);
 484                 ++it;
 485         }
 486         return e;
 487 }
 488
 489 bool pseries::is_terminating() const
 490 {
 491         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 492 }
 493
 494
 495 /*
 496  *  Implementations of series expansion
 497  */
 498
 499 /** Default implementation of ex::series(). This performs Taylor expansion.
 500  *  @see ex::series */
 501 ex basic::series(const relational & r, int order, unsigned options) const
 502 {
 503         epvector seq;
 504         numeric fac = 1;
 505         ex deriv = *this;
 506         ex coeff = deriv.subs(r, subs_options::no_pattern);
 507         const symbol &s = ex_to<symbol>(r.lhs());
 508
 509         if (!coeff.is_zero())
 510                 seq.push_back(expair(coeff, _ex0));
 511
 512         int n;
 513         for (n=1; n<order; ++n) {
 514                 fac = fac.mul(n);
 515                 // We need to test for zero in order to see if the series terminates.
 516                 // The problem is that there is no such thing as a perfect test for
 517                 // zero.  Expanding the term occasionally helps a little...
 518                 deriv = deriv.diff(s).expand();
 519                 if (deriv.is_zero())  // Series terminates
 520                         return pseries(r, seq);
 521
 522                 coeff = deriv.subs(r, subs_options::no_pattern);
 523                 if (!coeff.is_zero())
 524                         seq.push_back(expair(fac.inverse() * coeff, n));
 525         }
 526
 527         // Higher-order terms, if present
 528         deriv = deriv.diff(s);
 529         if (!deriv.expand().is_zero())
 530                 seq.push_back(expair(Order(_ex1), n));
 531         return pseries(r, seq);
 532 }
 533
 534
 535 /** Implementation of ex::series() for symbols.
 536  *  @see ex::series */
 537 ex symbol::series(const relational & r, int order, unsigned options) const
 538 {
 539         epvector seq;
 540         const ex point = r.rhs();
 541         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 542
 543         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 544                 if (order > 0 && !point.is_zero())
 545                         seq.push_back(expair(point, _ex0));
 546                 if (order > 1)
 547                         seq.push_back(expair(_ex1, _ex1));
 548                 else
 549                         seq.push_back(expair(Order(_ex1), numeric(order)));
 550         } else
 551                 seq.push_back(expair(*this, _ex0));
 552         return pseries(r, seq);
 553 }
 554
 555
 556 /** Add one series object to another, producing a pseries object that
 557  *  represents the sum.
 558  *
 559  *  @param other  pseries object to add with
 560  *  @return the sum as a pseries */
 561 ex pseries::add_series(const pseries &other) const
 562 {
 563         // Adding two series with different variables or expansion points
 564         // results in an empty (constant) series
 565         if (!is_compatible_to(other)) {
 566                 epvector nul;
 567                 nul.push_back(expair(Order(_ex1), _ex0));
 568                 return pseries(relational(var,point), nul);
 569         }
 570
 571         // Series addition
 572         epvector new_seq;
 573         epvector::const_iterator a = seq.begin();
 574         epvector::const_iterator b = other.seq.begin();
 575         epvector::const_iterator a_end = seq.end();
 576         epvector::const_iterator b_end = other.seq.end();
 577         int pow_a = INT_MAX, pow_b = INT_MAX;
 578         for (;;) {
 579                 // If a is empty, fill up with elements from b and stop
 580                 if (a == a_end) {
 581                         while (b != b_end) {
 582                                 new_seq.push_back(*b);
 583                                 ++b;
 584                         }
 585                         break;
 586                 } else
 587                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 588
 589                 // If b is empty, fill up with elements from a and stop
 590                 if (b == b_end) {
 591                         while (a != a_end) {
 592                                 new_seq.push_back(*a);
 593                                 ++a;
 594                         }
 595                         break;
 596                 } else
 597                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 598
 599                 // a and b are non-empty, compare powers
 600                 if (pow_a < pow_b) {
 601                         // a has lesser power, get coefficient from a
 602                         new_seq.push_back(*a);
 603                         if (is_order_function((*a).rest))
 604                                 break;
 605                         ++a;
 606                 } else if (pow_b < pow_a) {
 607                         // b has lesser power, get coefficient from b
 608                         new_seq.push_back(*b);
 609                         if (is_order_function((*b).rest))
 610                                 break;
 611                         ++b;
 612                 } else {
 613                         // Add coefficient of a and b
 614                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 615                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 616                                 break;  // Order term ends the sequence
 617                         } else {
 618                                 ex sum = (*a).rest + (*b).rest;
 619                                 if (!(sum.is_zero()))
 620                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 621                                 ++a;
 622                                 ++b;
 623                         }
 624                 }
 625         }
 626         return pseries(relational(var,point), new_seq);
 627 }
 628
 629
 630 /** Implementation of ex::series() for sums. This performs series addition when
 631  *  adding pseries objects.
 632  *  @see ex::series */
 633 ex add::series(const relational & r, int order, unsigned options) const
 634 {
 635         ex acc; // Series accumulator
 636
 637         // Get first term from overall_coeff
 638         acc = overall_coeff.series(r, order, options);
 639
 640         // Add remaining terms
 641         epvector::const_iterator it = seq.begin();
 642         epvector::const_iterator itend = seq.end();
 643         for (; it!=itend; ++it) {
 644                 ex op;
 645                 if (is_exactly_a<pseries>(it->rest))
 646                         op = it->rest;
 647                 else
 648                         op = it->rest.series(r, order, options);
 649                 if (!it->coeff.is_equal(_ex1))
 650                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 651
 652                 // Series addition
 653                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 654         }
 655         return acc;
 656 }
 657
 658
 659 /** Multiply a pseries object with a numeric constant, producing a pseries
 660  *  object that represents the product.
 661  *
 662  *  @param other  constant to multiply with
 663  *  @return the product as a pseries */
 664 ex pseries::mul_const(const numeric &other) const
 665 {
 666         epvector new_seq;
 667         new_seq.reserve(seq.size());
 668
 669         epvector::const_iterator it = seq.begin(), itend = seq.end();
 670         while (it != itend) {
 671                 if (!is_order_function(it->rest))
 672                         new_seq.push_back(expair(it->rest * other, it->coeff));
 673                 else
 674                         new_seq.push_back(*it);
 675                 ++it;
 676         }
 677         return pseries(relational(var,point), new_seq);
 678 }
 679
 680
 681 /** Multiply one pseries object to another, producing a pseries object that
 682  *  represents the product.
 683  *
 684  *  @param other  pseries object to multiply with
 685  *  @return the product as a pseries */
 686 ex pseries::mul_series(const pseries &other) const
 687 {
 688         // Multiplying two series with different variables or expansion points
 689         // results in an empty (constant) series
 690         if (!is_compatible_to(other)) {
 691                 epvector nul;
 692                 nul.push_back(expair(Order(_ex1), _ex0));
 693                 return pseries(relational(var,point), nul);
 694         }
 695
 696         // Series multiplication
 697         epvector new_seq;
 698         int a_max = degree(var);
 699         int b_max = other.degree(var);
 700         int a_min = ldegree(var);
 701         int b_min = other.ldegree(var);
 702         int cdeg_min = a_min + b_min;
 703         int cdeg_max = a_max + b_max;
 704
 705         int higher_order_a = INT_MAX;
 706         int higher_order_b = INT_MAX;
 707         if (is_order_function(coeff(var, a_max)))
 708                 higher_order_a = a_max + b_min;
 709         if (is_order_function(other.coeff(var, b_max)))
 710                 higher_order_b = b_max + a_min;
 711         int higher_order_c = std::min(higher_order_a, higher_order_b);
 712         if (cdeg_max >= higher_order_c)
 713                 cdeg_max = higher_order_c - 1;
 714
 715         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 716                 ex co = _ex0;
 717                 // c(i)=a(0)b(i)+...+a(i)b(0)
 718                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 719                         ex a_coeff = coeff(var, i);
 720                         ex b_coeff = other.coeff(var, cdeg-i);
 721                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 722                                 co += a_coeff * b_coeff;
 723                 }
 724                 if (!co.is_zero())
 725                         new_seq.push_back(expair(co, numeric(cdeg)));
 726         }
 727         if (higher_order_c < INT_MAX)
 728                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 729         return pseries(relational(var, point), new_seq);
 730 }
 731
 732
 733 /** Implementation of ex::series() for product. This performs series
 734  *  multiplication when multiplying series.
 735  *  @see ex::series */
 736 ex mul::series(const relational & r, int order, unsigned options) const
 737 {
 738         pseries acc; // Series accumulator
 739
 740         // Multiply with remaining terms
 741         const epvector::const_iterator itbeg = seq.begin();
 742         const epvector::const_iterator itend = seq.end();
 743         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 744                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 745
 746                 // Series multiplication
 747                 if (it==itbeg)
 748                         acc = ex_to<pseries>(op);
 749                 else
 750                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 751         }
 752         return acc.mul_const(ex_to<numeric>(overall_coeff));
 753 }
 754
 755
 756 /** Compute the p-th power of a series.
 757  *
 758  *  @param p  power to compute
 759  *  @param deg  truncation order of series calculation */
 760 ex pseries::power_const(const numeric &p, int deg) const
 761 {
 762         // method:
 763         // (due to Leonhard Euler)
 764         // let A(x) be this series and for the time being let it start with a
 765         // constant (later we'll generalize):
 766         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 767         // We want to compute
 768         //     C(x) = A(x)^p
 769         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 770         // Taking the derivative on both sides and multiplying with A(x) one
 771         // immediately arrives at
 772         //     C'(x)*A(x) = p*C(x)*A'(x)
 773         // Multiplying this out and comparing coefficients we get the recurrence
 774         // formula
 775         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 776         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 777         // which can easily be solved given the starting value c_0 = (a_0)^p.
 778         // For the more general case where the leading coefficient of A(x) is not
 779         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 780         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 781         // then of course x^(p*m) but the recurrence formula still holds.
 782
 783         if (seq.empty()) {
 784                 // as a special case, handle the empty (zero) series honoring the
 785                 // usual power laws such as implemented in power::eval()
 786                 if (p.real().is_zero())
 787                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 788                 else if (p.real().is_negative())
 789                         throw pole_error("pseries::power_const(): division by zero",1);
 790                 else
 791                         return *this;
 792         }
 793
 794         const int ldeg = ldegree(var);
 795         if (!(p*ldeg).is_integer())
 796                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 797
 798         // O(x^n)^(-m) is undefined
 799         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 800                 throw pole_error("pseries::power_const(): division by zero",1);
 801
 802         // Compute coefficients of the powered series
 803         exvector co;
 804         co.reserve(deg);
 805         co.push_back(power(coeff(var, ldeg), p));
 806         bool all_sums_zero = true;
 807         for (int i=1; i<deg; ++i) {
 808                 ex sum = _ex0;
 809                 for (int j=1; j<=i; ++j) {
 810                         ex c = coeff(var, j + ldeg);
 811                         if (is_order_function(c)) {
 812                                 co.push_back(Order(_ex1));
 813                                 break;
 814                         } else
 815                                 sum += (p * j - (i - j)) * co[i - j] * c;
 816                 }
 817                 if (!sum.is_zero())
 818                         all_sums_zero = false;
 819                 co.push_back(sum / coeff(var, ldeg) / i);
 820         }
 821
 822         // Construct new series (of non-zero coefficients)
 823         epvector new_seq;
 824         bool higher_order = false;
 825         for (int i=0; i<deg; ++i) {
 826                 if (!co[i].is_zero())
 827                         new_seq.push_back(expair(co[i], p * ldeg + i));
 828                 if (is_order_function(co[i])) {
 829                         higher_order = true;
 830                         break;
 831                 }
 832         }
 833         if (!higher_order && !all_sums_zero)
 834                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 835         return pseries(relational(var,point), new_seq);
 836 }
 837
 838
 839 /** Return a new pseries object with the powers shifted by deg. */
 840 pseries pseries::shift_exponents(int deg) const
 841 {
 842         epvector newseq = seq;
 843         epvector::iterator i = newseq.begin(), end  = newseq.end();
 844         while (i != end) {
 845                 i->coeff += deg;
 846                 ++i;
 847         }
 848         return pseries(relational(var, point), newseq);
 849 }
 850
 851
 852 /** Implementation of ex::series() for powers. This performs Laurent expansion
 853  *  of reciprocals of series at singularities.
 854  *  @see ex::series */
 855 ex power::series(const relational & r, int order, unsigned options) const
 856 {
 857         // If basis is already a series, just power it
 858         if (is_exactly_a<pseries>(basis))
 859                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 860
 861         // Basis is not a series, may there be a singularity?
 862         bool must_expand_basis = false;
 863         try {
 864                 basis.subs(r, subs_options::no_pattern);
 865         } catch (pole_error) {
 866                 must_expand_basis = true;
 867         }
 868
 869         // Is the expression of type something^(-int)?
 870         if (!must_expand_basis && !exponent.info(info_flags::negint))
 871                 return basic::series(r, order, options);
 872
 873         // Is the expression of type 0^something?
 874         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero())
 875                 return basic::series(r, order, options);
 876
 877         // Singularity encountered, is the basis equal to (var - point)?
 878         if (basis.is_equal(r.lhs() - r.rhs())) {
 879                 epvector new_seq;
 880                 if (ex_to<numeric>(exponent).to_int() < order)
 881                         new_seq.push_back(expair(_ex1, exponent));
 882                 else
 883                         new_seq.push_back(expair(Order(_ex1), exponent));
 884                 return pseries(r, new_seq);
 885         }
 886
 887         // No, expand basis into series
 888         ex e = basis.series(r, order, options);
 889         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 890 }
 891
 892
 893 /** Re-expansion of a pseries object. */
 894 ex pseries::series(const relational & r, int order, unsigned options) const
 895 {
 896         const ex p = r.rhs();
 897         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 898         const symbol &s = ex_to<symbol>(r.lhs());
 899
 900         if (var.is_equal(s) && point.is_equal(p)) {
 901                 if (order > degree(s))
 902                         return *this;
 903                 else {
 904                         epvector new_seq;
 905                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 906                         while (it != itend) {
 907                                 int o = ex_to<numeric>(it->coeff).to_int();
 908                                 if (o >= order) {
 909                                         new_seq.push_back(expair(Order(_ex1), o));
 910                                         break;
 911                                 }
 912                                 new_seq.push_back(*it);
 913                                 ++it;
 914                         }
 915                         return pseries(r, new_seq);
 916                 }
 917         } else
 918                 return convert_to_poly().series(r, order, options);
 919 }
 920
 921
 922 /** Compute the truncated series expansion of an expression.
 923  *  This function returns an expression containing an object of class pseries
 924  *  to represent the series. If the series does not terminate within the given
 925  *  truncation order, the last term of the series will be an order term.
 926  *
 927  *  @param r  expansion relation, lhs holds variable and rhs holds point
 928  *  @param order  truncation order of series calculations
 929  *  @param options  of class series_options
 930  *  @return an expression holding a pseries object */
 931 ex ex::series(const ex & r, int order, unsigned options) const
 932 {
 933         ex e;
 934         relational rel_;
 935
 936         if (is_a<relational>(r))
 937                 rel_ = ex_to<relational>(r);
 938         else if (is_a<symbol>(r))
 939                 rel_ = relational(r,_ex0);
 940         else
 941                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 942
 943         try {
 944                 e = bp->series(rel_, order, options);
 945         } catch (std::exception &x) {
 946                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 947         }
 948         return e;
 949 }
 950
 951 } // namespace GiNaC