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 "archive.h"
  35 #include "utils.h"
  36 #include "debugmsg.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  41
  42 /*
  43  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  44  */
  45
  46 pseries::pseries() : basic(TINFO_pseries)
  47 {
  48         debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
  49 }
  50
  51 void pseries::copy(const pseries &other)
  52 {
  53         inherited::copy(other);
  54         seq = other.seq;
  55         var = other.var;
  56         point = other.point;
  57 }
  58
  59 void pseries::destroy(bool call_parent)
  60 {
  61         if (call_parent)
  62                 inherited::destroy(call_parent);
  63 }
  64
  65
  66 /*
  67  *  Other ctors
  68  */
  69
  70 /** Construct pseries from a vector of coefficients and powers.
  71  *  expair.rest holds the coefficient, expair.coeff holds the power.
  72  *  The powers must be integers (positive or negative) and in ascending order;
  73  *  the last coefficient can be Order(_ex1()) to represent a truncated,
  74  *  non-terminating series.
  75  *
  76  *  @param rel_  expansion variable and point (must hold a relational)
  77  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  78  *  @return newly constructed pseries */
  79 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  80 {
  81         debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
  82         GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
  83         GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
  84         point = rel_.rhs();
  85         var = *static_cast<symbol *>(rel_.lhs().bp);
  86 }
  87
  88
  89 /*
  90  *  Archiving
  91  */
  92
  93 /** Construct object from archive_node. */
  94 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  95 {
  96         debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
  97         for (unsigned int i=0; true; ++i) {
  98                 ex rest;
  99                 ex coeff;
 100                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
 101                         seq.push_back(expair(rest, coeff));
 102                 else
 103                         break;
 104         }
 105         n.find_ex("var", var, sym_lst);
 106         n.find_ex("point", point, sym_lst);
 107 }
 108
 109 /** Unarchive the object. */
 110 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
 111 {
 112         return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
 113 }
 114
 115 /** Archive the object. */
 116 void pseries::archive(archive_node &n) const
 117 {
 118         inherited::archive(n);
 119         epvector::const_iterator i = seq.begin(), iend = seq.end();
 120         while (i != iend) {
 121                 n.add_ex("coeff", i->rest);
 122                 n.add_ex("power", i->coeff);
 123                 ++i;
 124         }
 125         n.add_ex("var", var);
 126         n.add_ex("point", point);
 127 }
 128
 129 //////////
 130 // functions overriding virtual functions from bases classes
 131 //////////
 132
 133 void pseries::print(std::ostream &os, unsigned upper_precedence) const
 134 {
 135         debugmsg("pseries print", LOGLEVEL_PRINT);
 136         if (precedence<=upper_precedence) os << "(";
 137         // objects of type pseries must not have any zero entries, so the
 138         // trivial (zero) pseries needs a special treatment here:
 139         if (seq.size()==0)
 140                 os << '0';
 141         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 142                 // print a sign, if needed
 143                 if (i!=seq.begin())
 144                         os << '+';
 145                 if (!is_order_function(i->rest)) {
 146                         // print 'rest', i.e. the expansion coefficient
 147                         if (i->rest.info(info_flags::numeric) &&
 148                                 i->rest.info(info_flags::positive)) {
 149                                 os << i->rest;
 150                         } else
 151                                 os << "(" << i->rest << ')';
 152                         // print 'coeff', something like (x-1)^42
 153                         if (!i->coeff.is_zero()) {
 154                                 os << '*';
 155                                 if (!point.is_zero())
 156                                         os << '(' << var-point << ')';
 157                                 else
 158                                         os << var;
 159                                 if (i->coeff.compare(_ex1())) {
 160                                         os << '^';
 161                                         if (i->coeff.info(info_flags::negative))
 162                                                 os << '(' << i->coeff << ')';
 163                                         else
 164                                                 os << i->coeff;
 165                                 }
 166                         }
 167                 } else {
 168                         os << Order(power(var-point,i->coeff));
 169                 }
 170         }
 171         if (precedence<=upper_precedence) os << ")";
 172 }
 173
 174
 175 void pseries::printraw(std::ostream &os) const
 176 {
 177         debugmsg("pseries printraw", LOGLEVEL_PRINT);
 178         os << "pseries(" << var << ";" << point << ";";
 179         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 180                 os << "(" << (*i).rest << "," << (*i).coeff << "),";
 181         os << ")";
 182 }
 183
 184
 185 void pseries::printtree(std::ostream & os, unsigned indent) const
 186 {
 187         debugmsg("pseries printtree",LOGLEVEL_PRINT);
 188         os << std::string(indent,' ') << "pseries "
 189            << ", hash=" << hashvalue
 190            << " (0x" << std::hex << hashvalue << std::dec << ")"
 191            << ", flags=" << flags << std::endl;
 192         for (unsigned i=0; i<seq.size(); ++i) {
 193                 seq[i].rest.printtree(os,indent+delta_indent);
 194                 seq[i].coeff.printtree(os,indent+delta_indent);
 195                 if (i!=seq.size()-1)
 196                         os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
 197         }
 198         var.printtree(os, indent+delta_indent);
 199         point.printtree(os, indent+delta_indent);
 200 }
 201
 202 int pseries::compare_same_type(const basic & other) const
 203 {
 204         GINAC_ASSERT(is_of_type(other, pseries));
 205         const pseries &o = static_cast<const pseries &>(other);
 206
 207         int cmpval = var.compare(o.var);
 208         if (cmpval)
 209                 return cmpval;
 210         cmpval = point.compare(o.point);
 211         if (cmpval)
 212                 return cmpval;
 213
 214         epvector::const_iterator it1 = seq.begin(), it2 = o.seq.begin(), it1end = seq.end(), it2end = o.seq.end();
 215         while ((it1 != it1end) && (it2 != it2end)) {
 216                 cmpval = it1->compare(*it2);
 217                 if (cmpval)
 218                         return cmpval;
 219                 it1++; it2++;
 220         }
 221         if (it1 == it1end)
 222                 return it2 == it2end ? 0 : -1;
 223
 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 symbol &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 symbol &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 symbol &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 symbol &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         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 527
 528         if (this->is_equal(*s)) {
 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         const symbol *s = static_cast<symbol *>(var.bp);
 685         int a_max = degree(*s);
 686         int b_max = other.degree(*s);
 687         int a_min = ldegree(*s);
 688         int b_min = other.ldegree(*s);
 689         int cdeg_min = a_min + b_min;
 690         int cdeg_max = a_max + b_max;
 691
 692         int higher_order_a = INT_MAX;
 693         int higher_order_b = INT_MAX;
 694         if (is_order_function(coeff(*s, a_max)))
 695                 higher_order_a = a_max + b_min;
 696         if (is_order_function(other.coeff(*s, b_max)))
 697                 higher_order_b = b_max + a_min;
 698         int higher_order_c = std::min(higher_order_a, higher_order_b);
 699         if (cdeg_max >= higher_order_c)
 700                 cdeg_max = higher_order_c - 1;
 701
 702         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 703                 ex co = _ex0();
 704                 // c(i)=a(0)b(i)+...+a(i)b(0)
 705                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 706                         ex a_coeff = coeff(*s, i);
 707                         ex b_coeff = other.coeff(*s, cdeg-i);
 708                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 709                                 co += a_coeff * b_coeff;
 710                 }
 711                 if (!co.is_zero())
 712                         new_seq.push_back(expair(co, numeric(cdeg)));
 713         }
 714         if (higher_order_c < INT_MAX)
 715                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 716         return pseries(relational(var,point), new_seq);
 717 }
 718
 719
 720 /** Implementation of ex::series() for product. This performs series
 721  *  multiplication when multiplying series.
 722  *  @see ex::series */
 723 ex mul::series(const relational & r, int order, unsigned options) const
 724 {
 725         ex acc; // Series accumulator
 726
 727         // Get first term from overall_coeff
 728         acc = overall_coeff.series(r, order, options);
 729
 730         // Multiply with remaining terms
 731         epvector::const_iterator it = seq.begin();
 732         epvector::const_iterator itend = seq.end();
 733         for (; it!=itend; ++it) {
 734                 ex op = it->rest;
 735                 if (op.info(info_flags::numeric)) {
 736                         // series * const (special case, faster)
 737                         ex f = power(op, it->coeff);
 738                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 739                         continue;
 740                 } else if (!is_ex_exactly_of_type(op, pseries))
 741                         op = op.series(r, order, options);
 742                 if (!it->coeff.is_equal(_ex1()))
 743                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 744
 745                 // Series multiplication
 746                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 747         }
 748         return acc;
 749 }
 750
 751
 752 /** Compute the p-th power of a series.
 753  *
 754  *  @param p  power to compute
 755  *  @param deg  truncation order of series calculation */
 756 ex pseries::power_const(const numeric &p, int deg) const
 757 {
 758         // method:
 759         // let A(x) be this series and for the time being let it start with a
 760         // constant (later we'll generalize):
 761         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 762         // We want to compute
 763         //     C(x) = A(x)^p
 764         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 765         // Taking the derivative on both sides and multiplying with A(x) one
 766         // immediately arrives at
 767         //     C'(x)*A(x) = p*C(x)*A'(x)
 768         // Multiplying this out and comparing coefficients we get the recurrence
 769         // formula
 770         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 771         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 772         // which can easily be solved given the starting value c_0 = (a_0)^p.
 773         // For the more general case where the leading coefficient of A(x) is not
 774         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 775         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 776         // then of course x^(p*m) but the recurrence formula still holds.
 777
 778         if (seq.size()==0) {
 779                 // as a spacial case, handle the empty (zero) series honoring the
 780                 // usual power laws such as implemented in power::eval()
 781                 if (p.real().is_zero())
 782                         throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
 783                 else if (p.real().is_negative())
 784                         throw (pole_error("pseries::power_const(): division by zero",1));
 785                 else
 786                         return *this;
 787         }
 788
 789         const symbol *s = static_cast<symbol *>(var.bp);
 790         int ldeg = ldegree(*s);
 791
 792         // Compute coefficients of the powered series
 793         exvector co;
 794         co.reserve(deg);
 795         co.push_back(power(coeff(*s, ldeg), p));
 796         bool all_sums_zero = true;
 797         for (int i=1; i<deg; ++i) {
 798                 ex sum = _ex0();
 799                 for (int j=1; j<=i; ++j) {
 800                         ex c = coeff(*s, j + ldeg);
 801                         if (is_order_function(c)) {
 802                                 co.push_back(Order(_ex1()));
 803                                 break;
 804                         } else
 805                                 sum += (p * j - (i - j)) * co[i - j] * c;
 806                 }
 807                 if (!sum.is_zero())
 808                         all_sums_zero = false;
 809                 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
 810         }
 811
 812         // Construct new series (of non-zero coefficients)
 813         epvector new_seq;
 814         bool higher_order = false;
 815         for (int i=0; i<deg; ++i) {
 816                 if (!co[i].is_zero())
 817                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 818                 if (is_order_function(co[i])) {
 819                         higher_order = true;
 820                         break;
 821                 }
 822         }
 823         if (!higher_order && !all_sums_zero)
 824                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 825         return pseries(relational(var,point), new_seq);
 826 }
 827
 828
 829 /** Return a new pseries object with the powers shifted by deg. */
 830 pseries pseries::shift_exponents(int deg) const
 831 {
 832         epvector newseq(seq);
 833         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
 834                 i->coeff = i->coeff + deg;
 835         return pseries(relational(var, point), newseq);
 836 }
 837
 838
 839 /** Implementation of ex::series() for powers. This performs Laurent expansion
 840  *  of reciprocals of series at singularities.
 841  *  @see ex::series */
 842 ex power::series(const relational & r, int order, unsigned options) const
 843 {
 844         ex e;
 845         if (!is_ex_exactly_of_type(basis, pseries)) {
 846                 // Basis is not a series, may there be a singularity?
 847                 bool must_expand_basis = false;
 848                 try {
 849                         basis.subs(r);
 850                 } catch (pole_error) {
 851                         must_expand_basis = true;
 852                 }
 853
 854                 // Is the expression of type something^(-int)?
 855                 if (!must_expand_basis && !exponent.info(info_flags::negint))
 856                         return basic::series(r, order, options);
 857
 858                 // Is the expression of type 0^something?
 859                 if (!must_expand_basis && !basis.subs(r).is_zero())
 860                         return basic::series(r, order, options);
 861
 862                 // Singularity encountered, expand basis into series
 863                 e = basis.series(r, order, options);
 864         } else {
 865                 // Basis is a series
 866                 e = basis;
 867         }
 868
 869         // Power e
 870         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 871 }
 872
 873
 874 /** Re-expansion of a pseries object. */
 875 ex pseries::series(const relational & r, int order, unsigned options) const
 876 {
 877         const ex p = r.rhs();
 878         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 879         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 880
 881         if (var.is_equal(*s) && point.is_equal(p)) {
 882                 if (order > degree(*s))
 883                         return *this;
 884                 else {
 885                         epvector new_seq;
 886                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 887                         while (it != itend) {
 888                                 int o = ex_to_numeric(it->coeff).to_int();
 889                                 if (o >= order) {
 890                                         new_seq.push_back(expair(Order(_ex1()), o));
 891                                         break;
 892                                 }
 893                                 new_seq.push_back(*it);
 894                                 ++it;
 895                         }
 896                         return pseries(r, new_seq);
 897                 }
 898         } else
 899                 return convert_to_poly().series(r, order, options);
 900 }
 901
 902
 903 /** Compute the truncated series expansion of an expression.
 904  *  This function returns an expression containing an object of class pseries
 905  *  to represent the series. If the series does not terminate within the given
 906  *  truncation order, the last term of the series will be an order term.
 907  *
 908  *  @param r  expansion relation, lhs holds variable and rhs holds point
 909  *  @param order  truncation order of series calculations
 910  *  @param options  of class series_options
 911  *  @return an expression holding a pseries object */
 912 ex ex::series(const ex & r, int order, unsigned options) const
 913 {
 914         GINAC_ASSERT(bp!=0);
 915         ex e;
 916         relational rel_;
 917
 918         if (is_ex_exactly_of_type(r,relational))
 919                 rel_ = ex_to_relational(r);
 920         else if (is_ex_exactly_of_type(r,symbol))
 921                 rel_ = relational(r,_ex0());
 922         else
 923                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 924
 925         try {
 926                 e = bp->series(rel_, order, options);
 927         } catch (std::exception &x) {
 928                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 929         }
 930         return e;
 931 }
 932
 933 //////////
 934 // static member variables
 935 //////////
 936
 937 // protected
 938
 939 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
 940
 941 } // namespace GiNaC