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