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