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 /** Return the ith term in the series when represented as a sum. */
 234 ex pseries::op(int i) const
 235 {
 236         if (i < 0 || unsigned(i) >= seq.size())
 237                 throw (std::out_of_range("op() out of range"));
 238         return seq[i].rest * power(var - point, seq[i].coeff);
 239 }
 240
 241 ex &pseries::let_op(int i)
 242 {
 243         throw (std::logic_error("let_op not defined for pseries"));
 244 }
 245
 246 /** Return degree of highest power of the series.  This is usually the exponent
 247  *  of the Order term.  If s is not the expansion variable of the series, the
 248  *  series is examined termwise. */
 249 int pseries::degree(const ex &s) const
 250 {
 251         if (var.is_equal(s)) {
 252                 // Return last exponent
 253                 if (seq.size())
 254                         return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 255                 else
 256                         return 0;
 257         } else {
 258                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 259                 if (it == itend)
 260                         return 0;
 261                 int max_pow = INT_MIN;
 262                 while (it != itend) {
 263                         int pow = it->rest.degree(s);
 264                         if (pow > max_pow)
 265                                 max_pow = pow;
 266                         ++it;
 267                 }
 268                 return max_pow;
 269         }
 270 }
 271
 272 /** Return degree of lowest power of the series.  This is usually the exponent
 273  *  of the leading term.  If s is not the expansion variable of the series, the
 274  *  series is examined termwise.  If s is the expansion variable but the
 275  *  expansion point is not zero the series is not expanded to find the degree.
 276  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 277 int pseries::ldegree(const ex &s) const
 278 {
 279         if (var.is_equal(s)) {
 280                 // Return first exponent
 281                 if (seq.size())
 282                         return ex_to_numeric((*(seq.begin())).coeff).to_int();
 283                 else
 284                         return 0;
 285         } else {
 286                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 287                 if (it == itend)
 288                         return 0;
 289                 int min_pow = INT_MAX;
 290                 while (it != itend) {
 291                         int pow = it->rest.ldegree(s);
 292                         if (pow < min_pow)
 293                                 min_pow = pow;
 294                         ++it;
 295                 }
 296                 return min_pow;
 297         }
 298 }
 299
 300 /** Return coefficient of degree n in power series if s is the expansion
 301  *  variable.  If the expansion point is nonzero, by definition the n=1
 302  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 303  *  the expansion took place in the s in the first place).
 304  *  If s is not the expansion variable, an attempt is made to convert the
 305  *  series to a polynomial and return the corresponding coefficient from
 306  *  there. */
 307 ex pseries::coeff(const ex &s, int n) const
 308 {
 309         if (var.is_equal(s)) {
 310                 if (seq.size() == 0)
 311                         return _ex0();
 312
 313                 // Binary search in sequence for given power
 314                 numeric looking_for = numeric(n);
 315                 int lo = 0, hi = seq.size() - 1;
 316                 while (lo <= hi) {
 317                         int mid = (lo + hi) / 2;
 318                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 319                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 320                         switch (cmp) {
 321                                 case -1:
 322                                         lo = mid + 1;
 323                                         break;
 324                                 case 0:
 325                                         return seq[mid].rest;
 326                                 case 1:
 327                                         hi = mid - 1;
 328                                         break;
 329                                 default:
 330                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 331                         }
 332                 }
 333                 return _ex0();
 334         } else
 335                 return convert_to_poly().coeff(s, n);
 336 }
 337
 338 /** Does nothing. */
 339 ex pseries::collect(const ex &s) const
 340 {
 341         return *this;
 342 }
 343
 344 /** Evaluate coefficients. */
 345 ex pseries::eval(int level) const
 346 {
 347         if (level == 1)
 348                 return this->hold();
 349
 350         if (level == -max_recursion_level)
 351                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 352
 353         // Construct a new series with evaluated coefficients
 354         epvector new_seq;
 355         new_seq.reserve(seq.size());
 356         epvector::const_iterator it = seq.begin(), itend = seq.end();
 357         while (it != itend) {
 358                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 359                 ++it;
 360         }
 361         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 362 }
 363
 364 /** Evaluate coefficients numerically. */
 365 ex pseries::evalf(int level) const
 366 {
 367         if (level == 1)
 368                 return *this;
 369
 370         if (level == -max_recursion_level)
 371                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 372
 373         // Construct a new series with evaluated coefficients
 374         epvector new_seq;
 375         new_seq.reserve(seq.size());
 376         epvector::const_iterator it = seq.begin(), itend = seq.end();
 377         while (it != itend) {
 378                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 379                 ++it;
 380         }
 381         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 382 }
 383
 384 ex pseries::subs(const lst & ls, const lst & lr) const
 385 {
 386         // If expansion variable is being substituted, convert the series to a
 387         // polynomial and do the substitution there because the result might
 388         // no longer be a power series
 389         if (ls.has(var))
 390                 return convert_to_poly(true).subs(ls, lr);
 391
 392         // Otherwise construct a new series with substituted coefficients and
 393         // expansion point
 394         epvector newseq;
 395         newseq.reserve(seq.size());
 396         epvector::const_iterator it = seq.begin(), itend = seq.end();
 397         while (it != itend) {
 398                 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 399                 ++it;
 400         }
 401         return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
 402 }
 403
 404 /** Implementation of ex::expand() for a power series.  It expands all the
 405  *  terms individually and returns the resulting series as a new pseries. */
 406 ex pseries::expand(unsigned options) const
 407 {
 408         epvector newseq;
 409         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 410                 ex restexp = i->rest.expand();
 411                 if (!restexp.is_zero())
 412                         newseq.push_back(expair(restexp, i->coeff));
 413         }
 414         return (new pseries(relational(var,point), newseq))
 415                 ->setflag(status_flags::dynallocated | status_flags::expanded);
 416 }
 417
 418 /** Implementation of ex::diff() for a power series.  It treats the series as a
 419  *  polynomial.
 420  *  @see ex::diff */
 421 ex pseries::derivative(const symbol & s) const
 422 {
 423         if (s == var) {
 424                 epvector new_seq;
 425                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 426
 427                 // FIXME: coeff might depend on var
 428                 while (it != itend) {
 429                         if (is_order_function(it->rest)) {
 430                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 431                         } else {
 432                                 ex c = it->rest * it->coeff;
 433                                 if (!c.is_zero())
 434                                         new_seq.push_back(expair(c, it->coeff - 1));
 435                         }
 436                         ++it;
 437                 }
 438                 return pseries(relational(var,point), new_seq);
 439         } else {
 440                 return *this;
 441         }
 442 }
 443
 444 ex pseries::convert_to_poly(bool no_order) const
 445 {
 446         ex e;
 447         epvector::const_iterator it = seq.begin(), itend = seq.end();
 448
 449         while (it != itend) {
 450                 if (is_order_function(it->rest)) {
 451                         if (!no_order)
 452                                 e += Order(power(var - point, it->coeff));
 453                 } else
 454                         e += it->rest * power(var - point, it->coeff);
 455                 ++it;
 456         }
 457         return e;
 458 }
 459
 460 bool pseries::is_terminating(void) const
 461 {
 462         return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
 463 }
 464
 465
 466 /*
 467  *  Implementations of series expansion
 468  */
 469
 470 /** Default implementation of ex::series(). This performs Taylor expansion.
 471  *  @see ex::series */
 472 ex basic::series(const relational & r, int order, unsigned options) const
 473 {
 474         epvector seq;
 475         numeric fac(1);
 476         ex deriv = *this;
 477         ex coeff = deriv.subs(r);
 478         const symbol &s = static_cast<symbol &>(*r.lhs().bp);
 479
 480         if (!coeff.is_zero())
 481                 seq.push_back(expair(coeff, numeric(0)));
 482
 483         int n;
 484         for (n=1; n<order; ++n) {
 485                 fac = fac.mul(numeric(n));
 486                 deriv = deriv.diff(s).expand();
 487                 if (deriv.is_zero()) {
 488                         // Series terminates
 489                         return pseries(r, seq);
 490                 }
 491                 coeff = deriv.subs(r);
 492                 if (!coeff.is_zero())
 493                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
 494         }
 495
 496         // Higher-order terms, if present
 497         deriv = deriv.diff(s);
 498         if (!deriv.expand().is_zero())
 499                 seq.push_back(expair(Order(_ex1()), numeric(n)));
 500         return pseries(r, seq);
 501 }
 502
 503
 504 /** Implementation of ex::series() for symbols.
 505  *  @see ex::series */
 506 ex symbol::series(const relational & r, int order, unsigned options) const
 507 {
 508         epvector seq;
 509         const ex point = r.rhs();
 510         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 511         ex s = r.lhs();
 512
 513         if (this->is_equal(*s.bp)) {
 514                 if (order > 0 && !point.is_zero())
 515                         seq.push_back(expair(point, _ex0()));
 516                 if (order > 1)
 517                         seq.push_back(expair(_ex1(), _ex1()));
 518                 else
 519                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 520         } else
 521                 seq.push_back(expair(*this, _ex0()));
 522         return pseries(r, seq);
 523 }
 524
 525
 526 /** Add one series object to another, producing a pseries object that
 527  *  represents the sum.
 528  *
 529  *  @param other  pseries object to add with
 530  *  @return the sum as a pseries */
 531 ex pseries::add_series(const pseries &other) const
 532 {
 533         // Adding two series with different variables or expansion points
 534         // results in an empty (constant) series
 535         if (!is_compatible_to(other)) {
 536                 epvector nul;
 537                 nul.push_back(expair(Order(_ex1()), _ex0()));
 538                 return pseries(relational(var,point), nul);
 539         }
 540
 541         // Series addition
 542         epvector new_seq;
 543         epvector::const_iterator a = seq.begin();
 544         epvector::const_iterator b = other.seq.begin();
 545         epvector::const_iterator a_end = seq.end();
 546         epvector::const_iterator b_end = other.seq.end();
 547         int pow_a = INT_MAX, pow_b = INT_MAX;
 548         for (;;) {
 549                 // If a is empty, fill up with elements from b and stop
 550                 if (a == a_end) {
 551                         while (b != b_end) {
 552                                 new_seq.push_back(*b);
 553                                 ++b;
 554                         }
 555                         break;
 556                 } else
 557                         pow_a = ex_to_numeric((*a).coeff).to_int();
 558
 559                 // If b is empty, fill up with elements from a and stop
 560                 if (b == b_end) {
 561                         while (a != a_end) {
 562                                 new_seq.push_back(*a);
 563                                 ++a;
 564                         }
 565                         break;
 566                 } else
 567                         pow_b = ex_to_numeric((*b).coeff).to_int();
 568
 569                 // a and b are non-empty, compare powers
 570                 if (pow_a < pow_b) {
 571                         // a has lesser power, get coefficient from a
 572                         new_seq.push_back(*a);
 573                         if (is_order_function((*a).rest))
 574                                 break;
 575                         ++a;
 576                 } else if (pow_b < pow_a) {
 577                         // b has lesser power, get coefficient from b
 578                         new_seq.push_back(*b);
 579                         if (is_order_function((*b).rest))
 580                                 break;
 581                         ++b;
 582                 } else {
 583                         // Add coefficient of a and b
 584                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 585                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 586                                 break;  // Order term ends the sequence
 587                         } else {
 588                                 ex sum = (*a).rest + (*b).rest;
 589                                 if (!(sum.is_zero()))
 590                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 591                                 ++a;
 592                                 ++b;
 593                         }
 594                 }
 595         }
 596         return pseries(relational(var,point), new_seq);
 597 }
 598
 599
 600 /** Implementation of ex::series() for sums. This performs series addition when
 601  *  adding pseries objects.
 602  *  @see ex::series */
 603 ex add::series(const relational & r, int order, unsigned options) const
 604 {
 605         ex acc; // Series accumulator
 606
 607         // Get first term from overall_coeff
 608         acc = overall_coeff.series(r, order, options);
 609
 610         // Add remaining terms
 611         epvector::const_iterator it = seq.begin();
 612         epvector::const_iterator itend = seq.end();
 613         for (; it!=itend; ++it) {
 614                 ex op;
 615                 if (is_ex_exactly_of_type(it->rest, pseries))
 616                         op = it->rest;
 617                 else
 618                         op = it->rest.series(r, order, options);
 619                 if (!it->coeff.is_equal(_ex1()))
 620                         op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 621
 622                 // Series addition
 623                 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 624         }
 625         return acc;
 626 }
 627
 628
 629 /** Multiply a pseries object with a numeric constant, producing a pseries
 630  *  object that represents the product.
 631  *
 632  *  @param other  constant to multiply with
 633  *  @return the product as a pseries */
 634 ex pseries::mul_const(const numeric &other) const
 635 {
 636         epvector new_seq;
 637         new_seq.reserve(seq.size());
 638
 639         epvector::const_iterator it = seq.begin(), itend = seq.end();
 640         while (it != itend) {
 641                 if (!is_order_function(it->rest))
 642                         new_seq.push_back(expair(it->rest * other, it->coeff));
 643                 else
 644                         new_seq.push_back(*it);
 645                 ++it;
 646         }
 647         return pseries(relational(var,point), new_seq);
 648 }
 649
 650
 651 /** Multiply one pseries object to another, producing a pseries object that
 652  *  represents the product.
 653  *
 654  *  @param other  pseries object to multiply with
 655  *  @return the product as a pseries */
 656 ex pseries::mul_series(const pseries &other) const
 657 {
 658         // Multiplying two series with different variables or expansion points
 659         // results in an empty (constant) series
 660         if (!is_compatible_to(other)) {
 661                 epvector nul;
 662                 nul.push_back(expair(Order(_ex1()), _ex0()));
 663                 return pseries(relational(var,point), nul);
 664         }
 665
 666         // Series multiplication
 667         epvector new_seq;
 668
 669         int a_max = degree(var);
 670         int b_max = other.degree(var);
 671         int a_min = ldegree(var);
 672         int b_min = other.ldegree(var);
 673         int cdeg_min = a_min + b_min;
 674         int cdeg_max = a_max + b_max;
 675
 676         int higher_order_a = INT_MAX;
 677         int higher_order_b = INT_MAX;
 678         if (is_order_function(coeff(var, a_max)))
 679                 higher_order_a = a_max + b_min;
 680         if (is_order_function(other.coeff(var, b_max)))
 681                 higher_order_b = b_max + a_min;
 682         int higher_order_c = std::min(higher_order_a, higher_order_b);
 683         if (cdeg_max >= higher_order_c)
 684                 cdeg_max = higher_order_c - 1;
 685
 686         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 687                 ex co = _ex0();
 688                 // c(i)=a(0)b(i)+...+a(i)b(0)
 689                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 690                         ex a_coeff = coeff(var, i);
 691                         ex b_coeff = other.coeff(var, cdeg-i);
 692                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 693                                 co += a_coeff * b_coeff;
 694                 }
 695                 if (!co.is_zero())
 696                         new_seq.push_back(expair(co, numeric(cdeg)));
 697         }
 698         if (higher_order_c < INT_MAX)
 699                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 700         return pseries(relational(var, point), new_seq);
 701 }
 702
 703
 704 /** Implementation of ex::series() for product. This performs series
 705  *  multiplication when multiplying series.
 706  *  @see ex::series */
 707 ex mul::series(const relational & r, int order, unsigned options) const
 708 {
 709         ex acc; // Series accumulator
 710
 711         // Get first term from overall_coeff
 712         acc = overall_coeff.series(r, order, options);
 713
 714         // Multiply with remaining terms
 715         epvector::const_iterator it = seq.begin();
 716         epvector::const_iterator itend = seq.end();
 717         for (; it!=itend; ++it) {
 718                 ex op = it->rest;
 719                 if (op.info(info_flags::numeric)) {
 720                         // series * const (special case, faster)
 721                         ex f = power(op, it->coeff);
 722                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 723                         continue;
 724                 } else if (!is_ex_exactly_of_type(op, pseries))
 725                         op = op.series(r, order, options);
 726                 if (!it->coeff.is_equal(_ex1()))
 727                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 728
 729                 // Series multiplication
 730                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 731         }
 732         return acc;
 733 }
 734
 735
 736 /** Compute the p-th power of a series.
 737  *
 738  *  @param p  power to compute
 739  *  @param deg  truncation order of series calculation */
 740 ex pseries::power_const(const numeric &p, int deg) const
 741 {
 742         // method:
 743         // let A(x) be this series and for the time being let it start with a
 744         // constant (later we'll generalize):
 745         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 746         // We want to compute
 747         //     C(x) = A(x)^p
 748         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 749         // Taking the derivative on both sides and multiplying with A(x) one
 750         // immediately arrives at
 751         //     C'(x)*A(x) = p*C(x)*A'(x)
 752         // Multiplying this out and comparing coefficients we get the recurrence
 753         // formula
 754         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 755         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 756         // which can easily be solved given the starting value c_0 = (a_0)^p.
 757         // For the more general case where the leading coefficient of A(x) is not
 758         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 759         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 760         // then of course x^(p*m) but the recurrence formula still holds.
 761
 762         if (seq.size()==0) {
 763                 // as a spacial case, handle the empty (zero) series honoring the
 764                 // usual power laws such as implemented in power::eval()
 765                 if (p.real().is_zero())
 766                         throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
 767                 else if (p.real().is_negative())
 768                         throw (pole_error("pseries::power_const(): division by zero",1));
 769                 else
 770                         return *this;
 771         }
 772
 773         int ldeg = ldegree(var);
 774
 775         // Compute coefficients of the powered series
 776         exvector co;
 777         co.reserve(deg);
 778         co.push_back(power(coeff(var, ldeg), p));
 779         bool all_sums_zero = true;
 780         for (int i=1; i<deg; ++i) {
 781                 ex sum = _ex0();
 782                 for (int j=1; j<=i; ++j) {
 783                         ex c = coeff(var, j + ldeg);
 784                         if (is_order_function(c)) {
 785                                 co.push_back(Order(_ex1()));
 786                                 break;
 787                         } else
 788                                 sum += (p * j - (i - j)) * co[i - j] * c;
 789                 }
 790                 if (!sum.is_zero())
 791                         all_sums_zero = false;
 792                 co.push_back(sum / coeff(var, ldeg) / numeric(i));
 793         }
 794
 795         // Construct new series (of non-zero coefficients)
 796         epvector new_seq;
 797         bool higher_order = false;
 798         for (int i=0; i<deg; ++i) {
 799                 if (!co[i].is_zero())
 800                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 801                 if (is_order_function(co[i])) {
 802                         higher_order = true;
 803                         break;
 804                 }
 805         }
 806         if (!higher_order && !all_sums_zero)
 807                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 808         return pseries(relational(var,point), new_seq);
 809 }
 810
 811
 812 /** Return a new pseries object with the powers shifted by deg. */
 813 pseries pseries::shift_exponents(int deg) const
 814 {
 815         epvector newseq(seq);
 816         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
 817                 i->coeff = i->coeff + deg;
 818         return pseries(relational(var, point), newseq);
 819 }
 820
 821
 822 /** Implementation of ex::series() for powers. This performs Laurent expansion
 823  *  of reciprocals of series at singularities.
 824  *  @see ex::series */
 825 ex power::series(const relational & r, int order, unsigned options) const
 826 {
 827         ex e;
 828         if (!is_ex_exactly_of_type(basis, pseries)) {
 829                 // Basis is not a series, may there be a singularity?
 830                 bool must_expand_basis = false;
 831                 try {
 832                         basis.subs(r);
 833                 } catch (pole_error) {
 834                         must_expand_basis = true;
 835                 }
 836
 837                 // Is the expression of type something^(-int)?
 838                 if (!must_expand_basis && !exponent.info(info_flags::negint))
 839                         return basic::series(r, order, options);
 840
 841                 // Is the expression of type 0^something?
 842                 if (!must_expand_basis && !basis.subs(r).is_zero())
 843                         return basic::series(r, order, options);
 844
 845                 // Singularity encountered, expand basis into series
 846                 e = basis.series(r, order, options);
 847         } else {
 848                 // Basis is a series
 849                 e = basis;
 850         }
 851
 852         // Power e
 853         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 854 }
 855
 856
 857 /** Re-expansion of a pseries object. */
 858 ex pseries::series(const relational & r, int order, unsigned options) const
 859 {
 860         const ex p = r.rhs();
 861         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 862         const symbol &s = static_cast<symbol &>(*r.lhs().bp);
 863
 864         if (var.is_equal(s) && point.is_equal(p)) {
 865                 if (order > degree(s))
 866                         return *this;
 867                 else {
 868                         epvector new_seq;
 869                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 870                         while (it != itend) {
 871                                 int o = ex_to_numeric(it->coeff).to_int();
 872                                 if (o >= order) {
 873                                         new_seq.push_back(expair(Order(_ex1()), o));
 874                                         break;
 875                                 }
 876                                 new_seq.push_back(*it);
 877                                 ++it;
 878                         }
 879                         return pseries(r, new_seq);
 880                 }
 881         } else
 882                 return convert_to_poly().series(r, order, options);
 883 }
 884
 885
 886 /** Compute the truncated series expansion of an expression.
 887  *  This function returns an expression containing an object of class pseries
 888  *  to represent the series. If the series does not terminate within the given
 889  *  truncation order, the last term of the series will be an order term.
 890  *
 891  *  @param r  expansion relation, lhs holds variable and rhs holds point
 892  *  @param order  truncation order of series calculations
 893  *  @param options  of class series_options
 894  *  @return an expression holding a pseries object */
 895 ex ex::series(const ex & r, int order, unsigned options) const
 896 {
 897         GINAC_ASSERT(bp!=0);
 898         ex e;
 899         relational rel_;
 900
 901         if (is_ex_exactly_of_type(r,relational))
 902                 rel_ = ex_to_relational(r);
 903         else if (is_ex_exactly_of_type(r,symbol))
 904                 rel_ = relational(r,_ex0());
 905         else
 906                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 907
 908         try {
 909                 e = bp->series(rel_, order, options);
 910         } catch (std::exception &x) {
 911                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 912         }
 913         return e;
 914 }
 915
 916 //////////
 917 // static member variables
 918 //////////
 919
 920 // protected
 921
 922 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
 923
 924 } // namespace GiNaC