ginac/pseries.cpp

   1 /** @file pseries.cpp
   2  *
   3  *  Implementation of class for extended truncated power series and
   4  *  methods for series expansion. */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  22  */
  23
  24 #include <iostream>
  25 #include <stdexcept>
  26
  27 #include "pseries.h"
  28 #include "add.h"
  29 #include "inifcns.h" // for Order function
  30 #include "lst.h"
  31 #include "mul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "operators.h"
  35 #include "symbol.h"
  36 #include "print.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  43
  44
  45 /*
  46  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  47  */
  48
  49 pseries::pseries() : inherited(TINFO_pseries) { }
  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 DEFAULT_DESTROY(pseries)
  60
  61
  62 /*
  63  *  Other ctors
  64  */
  65
  66 /** Construct pseries from a vector of coefficients and powers.
  67  *  expair.rest holds the coefficient, expair.coeff holds the power.
  68  *  The powers must be integers (positive or negative) and in ascending order;
  69  *  the last coefficient can be Order(_ex1) to represent a truncated,
  70  *  non-terminating series.
  71  *
  72  *  @param rel_  expansion variable and point (must hold a relational)
  73  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  74  *  @return newly constructed pseries */
  75 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  76 {
  77         GINAC_ASSERT(is_a<relational>(rel_));
  78         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  79         point = rel_.rhs();
  80         var = rel_.lhs();
  81 }
  82
  83
  84 /*
  85  *  Archiving
  86  */
  87
  88 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  89 {
  90         for (unsigned int i=0; true; ++i) {
  91                 ex rest;
  92                 ex coeff;
  93                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  94                         seq.push_back(expair(rest, coeff));
  95                 else
  96                         break;
  97         }
  98         n.find_ex("var", var, sym_lst);
  99         n.find_ex("point", point, sym_lst);
 100 }
 101
 102 void pseries::archive(archive_node &n) const
 103 {
 104         inherited::archive(n);
 105         epvector::const_iterator i = seq.begin(), iend = seq.end();
 106         while (i != iend) {
 107                 n.add_ex("coeff", i->rest);
 108                 n.add_ex("power", i->coeff);
 109                 ++i;
 110         }
 111         n.add_ex("var", var);
 112         n.add_ex("point", point);
 113 }
 114
 115 DEFAULT_UNARCHIVE(pseries)
 116
 117 //////////
 118 // functions overriding virtual functions from base classes
 119 //////////
 120
 121 void pseries::print(const print_context & c, unsigned level) const
 122 {
 123         if (is_a<print_tree>(c)) {
 124
 125                 c.s << std::string(level, ' ') << class_name()
 126                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 127                     << std::endl;
 128                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 129                 size_t num = seq.size();
 130                 for (size_t i=0; i<num; ++i) {
 131                         seq[i].rest.print(c, level + delta_indent);
 132                         seq[i].coeff.print(c, level + delta_indent);
 133                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 134                 }
 135                 var.print(c, level + delta_indent);
 136                 point.print(c, level + delta_indent);
 137
 138         } else if (is_a<print_python_repr>(c)) {
 139                 c.s << class_name() << "(relational(";
 140                 var.print(c);
 141                 c.s << ',';
 142                 point.print(c);
 143                 c.s << "),[";
 144                 size_t num = seq.size();
 145                 for (size_t i=0; i<num; ++i) {
 146                         if (i)
 147                                 c.s << ',';
 148                         c.s << '(';
 149                         seq[i].rest.print(c);
 150                         c.s << ',';
 151                         seq[i].coeff.print(c);
 152                         c.s << ')';
 153                 }
 154                 c.s << "])";
 155         } else {
 156
 157                 if (precedence() <= level)
 158                         c.s << "(";
 159
 160                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
 161                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
 162
 163                 // objects of type pseries must not have any zero entries, so the
 164                 // trivial (zero) pseries needs a special treatment here:
 165                 if (seq.empty())
 166                         c.s << '0';
 167                 epvector::const_iterator i = seq.begin(), end = seq.end();
 168                 while (i != end) {
 169                         // print a sign, if needed
 170                         if (i != seq.begin())
 171                                 c.s << '+';
 172                         if (!is_order_function(i->rest)) {
 173                                 // print 'rest', i.e. the expansion coefficient
 174                                 if (i->rest.info(info_flags::numeric) &&
 175                                         i->rest.info(info_flags::positive)) {
 176                                         i->rest.print(c);
 177                                 } else {
 178                                         c.s << par_open;
 179                                         i->rest.print(c);
 180                                         c.s << par_close;
 181                                 }
 182                                 // print 'coeff', something like (x-1)^42
 183                                 if (!i->coeff.is_zero()) {
 184                                         if (is_a<print_latex>(c))
 185                                                 c.s << ' ';
 186                                         else
 187                                                 c.s << '*';
 188                                         if (!point.is_zero()) {
 189                                                 c.s << par_open;
 190                                                 (var-point).print(c);
 191                                                 c.s << par_close;
 192                                         } else
 193                                                 var.print(c);
 194                                         if (i->coeff.compare(_ex1)) {
 195                                                 if (is_a<print_python>(c))
 196                                                         c.s << "**";
 197                                                 else
 198                                                         c.s << '^';
 199                                                 if (i->coeff.info(info_flags::negative)) {
 200                                                         c.s << par_open;
 201                                                         i->coeff.print(c);
 202                                                         c.s << par_close;
 203                                                 } else {
 204                                                         if (is_a<print_latex>(c)) {
 205                                                                 c.s << '{';
 206                                                                 i->coeff.print(c);
 207                                                                 c.s << '}';
 208                                                         } else
 209                                                                 i->coeff.print(c);
 210                                                 }
 211                                         }
 212                                 }
 213                         } else
 214                                 Order(power(var-point,i->coeff)).print(c);
 215                         ++i;
 216                 }
 217
 218                 if (precedence() <= level)
 219                         c.s << ")";
 220         }
 221 }
 222
 223 int pseries::compare_same_type(const basic & other) const
 224 {
 225         GINAC_ASSERT(is_a<pseries>(other));
 226         const pseries &o = static_cast<const pseries &>(other);
 227
 228         // first compare the lengths of the series...
 229         if (seq.size()>o.seq.size())
 230                 return 1;
 231         if (seq.size()<o.seq.size())
 232                 return -1;
 233
 234         // ...then the expansion point...
 235         int cmpval = var.compare(o.var);
 236         if (cmpval)
 237                 return cmpval;
 238         cmpval = point.compare(o.point);
 239         if (cmpval)
 240                 return cmpval;
 241
 242         // ...and if that failed the individual elements
 243         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 244         while (it!=seq.end() && o_it!=o.seq.end()) {
 245                 cmpval = it->compare(*o_it);
 246                 if (cmpval)
 247                         return cmpval;
 248                 ++it;
 249                 ++o_it;
 250         }
 251
 252         // so they are equal.
 253         return 0;
 254 }
 255
 256 /** Return the number of operands including a possible order term. */
 257 size_t pseries::nops(void) const
 258 {
 259         return seq.size();
 260 }
 261
 262 /** Return the ith term in the series when represented as a sum. */
 263 ex pseries::op(size_t i) const
 264 {
 265         if (i >= seq.size())
 266                 throw (std::out_of_range("op() out of range"));
 267
 268         return seq[i].rest * power(var - point, seq[i].coeff);
 269 }
 270
 271 /** Return degree of highest power of the series.  This is usually the exponent
 272  *  of the Order term.  If s is not the expansion variable of the series, the
 273  *  series is examined termwise. */
 274 int pseries::degree(const ex &s) const
 275 {
 276         if (var.is_equal(s)) {
 277                 // Return last exponent
 278                 if (seq.size())
 279                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 280                 else
 281                         return 0;
 282         } else {
 283                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 284                 if (it == itend)
 285                         return 0;
 286                 int max_pow = INT_MIN;
 287                 while (it != itend) {
 288                         int pow = it->rest.degree(s);
 289                         if (pow > max_pow)
 290                                 max_pow = pow;
 291                         ++it;
 292                 }
 293                 return max_pow;
 294         }
 295 }
 296
 297 /** Return degree of lowest power of the series.  This is usually the exponent
 298  *  of the leading term.  If s is not the expansion variable of the series, the
 299  *  series is examined termwise.  If s is the expansion variable but the
 300  *  expansion point is not zero the series is not expanded to find the degree.
 301  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 302 int pseries::ldegree(const ex &s) const
 303 {
 304         if (var.is_equal(s)) {
 305                 // Return first exponent
 306                 if (seq.size())
 307                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 308                 else
 309                         return 0;
 310         } else {
 311                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 312                 if (it == itend)
 313                         return 0;
 314                 int min_pow = INT_MAX;
 315                 while (it != itend) {
 316                         int pow = it->rest.ldegree(s);
 317                         if (pow < min_pow)
 318                                 min_pow = pow;
 319                         ++it;
 320                 }
 321                 return min_pow;
 322         }
 323 }
 324
 325 /** Return coefficient of degree n in power series if s is the expansion
 326  *  variable.  If the expansion point is nonzero, by definition the n=1
 327  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 328  *  the expansion took place in the s in the first place).
 329  *  If s is not the expansion variable, an attempt is made to convert the
 330  *  series to a polynomial and return the corresponding coefficient from
 331  *  there. */
 332 ex pseries::coeff(const ex &s, int n) const
 333 {
 334         if (var.is_equal(s)) {
 335                 if (seq.empty())
 336                         return _ex0;
 337
 338                 // Binary search in sequence for given power
 339                 numeric looking_for = numeric(n);
 340                 int lo = 0, hi = seq.size() - 1;
 341                 while (lo <= hi) {
 342                         int mid = (lo + hi) / 2;
 343                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 344                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 345                         switch (cmp) {
 346                                 case -1:
 347                                         lo = mid + 1;
 348                                         break;
 349                                 case 0:
 350                                         return seq[mid].rest;
 351                                 case 1:
 352                                         hi = mid - 1;
 353                                         break;
 354                                 default:
 355                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 356                         }
 357                 }
 358                 return _ex0;
 359         } else
 360                 return convert_to_poly().coeff(s, n);
 361 }
 362
 363 /** Does nothing. */
 364 ex pseries::collect(const ex &s, bool distributed) const
 365 {
 366         return *this;
 367 }
 368
 369 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 370 ex pseries::eval(int level) const
 371 {
 372         if (level == 1)
 373                 return this->hold();
 374
 375         if (level == -max_recursion_level)
 376                 throw (std::runtime_error("pseries::eval(): 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.eval(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 /** Evaluate coefficients numerically. */
 390 ex pseries::evalf(int level) const
 391 {
 392         if (level == 1)
 393                 return *this;
 394
 395         if (level == -max_recursion_level)
 396                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 397
 398         // Construct a new series with evaluated coefficients
 399         epvector new_seq;
 400         new_seq.reserve(seq.size());
 401         epvector::const_iterator it = seq.begin(), itend = seq.end();
 402         while (it != itend) {
 403                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 404                 ++it;
 405         }
 406         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 407 }
 408
 409 ex pseries::subs(const lst & ls, const lst & lr, unsigned options) const
 410 {
 411         // If expansion variable is being substituted, convert the series to a
 412         // polynomial and do the substitution there because the result might
 413         // no longer be a power series
 414         if (ls.has(var))
 415                 return convert_to_poly(true).subs(ls, lr, options);
 416
 417         // Otherwise construct a new series with substituted coefficients and
 418         // expansion point
 419         epvector newseq;
 420         newseq.reserve(seq.size());
 421         epvector::const_iterator it = seq.begin(), itend = seq.end();
 422         while (it != itend) {
 423                 newseq.push_back(expair(it->rest.subs(ls, lr, options), it->coeff));
 424                 ++it;
 425         }
 426         return (new pseries(relational(var,point.subs(ls, lr, options)), newseq))->setflag(status_flags::dynallocated);
 427 }
 428
 429 /** Implementation of ex::expand() for a power series.  It expands all the
 430  *  terms individually and returns the resulting series as a new pseries. */
 431 ex pseries::expand(unsigned options) const
 432 {
 433         epvector newseq;
 434         epvector::const_iterator i = seq.begin(), end = seq.end();
 435         while (i != end) {
 436                 ex restexp = i->rest.expand();
 437                 if (!restexp.is_zero())
 438                         newseq.push_back(expair(restexp, i->coeff));
 439                 ++i;
 440         }
 441         return (new pseries(relational(var,point), newseq))
 442                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 443 }
 444
 445 /** Implementation of ex::diff() for a power series.
 446  *  @see ex::diff */
 447 ex pseries::derivative(const symbol & s) const
 448 {
 449         epvector new_seq;
 450         epvector::const_iterator it = seq.begin(), itend = seq.end();
 451
 452         if (s == var) {
 453
 454                 // FIXME: coeff might depend on var
 455                 while (it != itend) {
 456                         if (is_order_function(it->rest)) {
 457                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 458                         } else {
 459                                 ex c = it->rest * it->coeff;
 460                                 if (!c.is_zero())
 461                                         new_seq.push_back(expair(c, it->coeff - 1));
 462                         }
 463                         ++it;
 464                 }
 465
 466         } else {
 467
 468                 while (it != itend) {
 469                         if (is_order_function(it->rest)) {
 470                                 new_seq.push_back(*it);
 471                         } else {
 472                                 ex c = it->rest.diff(s);
 473                                 if (!c.is_zero())
 474                                         new_seq.push_back(expair(c, it->coeff));
 475                         }
 476                         ++it;
 477                 }
 478         }
 479
 480         return pseries(relational(var,point), new_seq);
 481 }
 482
 483 ex pseries::convert_to_poly(bool no_order) const
 484 {
 485         ex e;
 486         epvector::const_iterator it = seq.begin(), itend = seq.end();
 487
 488         while (it != itend) {
 489                 if (is_order_function(it->rest)) {
 490                         if (!no_order)
 491                                 e += Order(power(var - point, it->coeff));
 492                 } else
 493                         e += it->rest * power(var - point, it->coeff);
 494                 ++it;
 495         }
 496         return e;
 497 }
 498
 499 bool pseries::is_terminating(void) const
 500 {
 501         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 502 }
 503
 504
 505 /*
 506  *  Implementations of series expansion
 507  */
 508
 509 /** Default implementation of ex::series(). This performs Taylor expansion.
 510  *  @see ex::series */
 511 ex basic::series(const relational & r, int order, unsigned options) const
 512 {
 513         epvector seq;
 514         numeric fac = 1;
 515         ex deriv = *this;
 516         ex coeff = deriv.subs(r);
 517         const symbol &s = ex_to<symbol>(r.lhs());
 518
 519         if (!coeff.is_zero())
 520                 seq.push_back(expair(coeff, _ex0));
 521
 522         int n;
 523         for (n=1; n<order; ++n) {
 524                 fac = fac.mul(n);
 525                 // We need to test for zero in order to see if the series terminates.
 526                 // The problem is that there is no such thing as a perfect test for
 527                 // zero.  Expanding the term occasionally helps a little...
 528                 deriv = deriv.diff(s).expand();
 529                 if (deriv.is_zero())  // Series terminates
 530                         return pseries(r, seq);
 531
 532                 coeff = deriv.subs(r);
 533                 if (!coeff.is_zero())
 534                         seq.push_back(expair(fac.inverse() * coeff, n));
 535         }
 536
 537         // Higher-order terms, if present
 538         deriv = deriv.diff(s);
 539         if (!deriv.expand().is_zero())
 540                 seq.push_back(expair(Order(_ex1), n));
 541         return pseries(r, seq);
 542 }
 543
 544
 545 /** Implementation of ex::series() for symbols.
 546  *  @see ex::series */
 547 ex symbol::series(const relational & r, int order, unsigned options) const
 548 {
 549         epvector seq;
 550         const ex point = r.rhs();
 551         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 552
 553         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 554                 if (order > 0 && !point.is_zero())
 555                         seq.push_back(expair(point, _ex0));
 556                 if (order > 1)
 557                         seq.push_back(expair(_ex1, _ex1));
 558                 else
 559                         seq.push_back(expair(Order(_ex1), numeric(order)));
 560         } else
 561                 seq.push_back(expair(*this, _ex0));
 562         return pseries(r, seq);
 563 }
 564
 565
 566 /** Add one series object to another, producing a pseries object that
 567  *  represents the sum.
 568  *
 569  *  @param other  pseries object to add with
 570  *  @return the sum as a pseries */
 571 ex pseries::add_series(const pseries &other) const
 572 {
 573         // Adding two series with different variables or expansion points
 574         // results in an empty (constant) series
 575         if (!is_compatible_to(other)) {
 576                 epvector nul;
 577                 nul.push_back(expair(Order(_ex1), _ex0));
 578                 return pseries(relational(var,point), nul);
 579         }
 580
 581         // Series addition
 582         epvector new_seq;
 583         epvector::const_iterator a = seq.begin();
 584         epvector::const_iterator b = other.seq.begin();
 585         epvector::const_iterator a_end = seq.end();
 586         epvector::const_iterator b_end = other.seq.end();
 587         int pow_a = INT_MAX, pow_b = INT_MAX;
 588         for (;;) {
 589                 // If a is empty, fill up with elements from b and stop
 590                 if (a == a_end) {
 591                         while (b != b_end) {
 592                                 new_seq.push_back(*b);
 593                                 ++b;
 594                         }
 595                         break;
 596                 } else
 597                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 598
 599                 // If b is empty, fill up with elements from a and stop
 600                 if (b == b_end) {
 601                         while (a != a_end) {
 602                                 new_seq.push_back(*a);
 603                                 ++a;
 604                         }
 605                         break;
 606                 } else
 607                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 608
 609                 // a and b are non-empty, compare powers
 610                 if (pow_a < pow_b) {
 611                         // a has lesser power, get coefficient from a
 612                         new_seq.push_back(*a);
 613                         if (is_order_function((*a).rest))
 614                                 break;
 615                         ++a;
 616                 } else if (pow_b < pow_a) {
 617                         // b has lesser power, get coefficient from b
 618                         new_seq.push_back(*b);
 619                         if (is_order_function((*b).rest))
 620                                 break;
 621                         ++b;
 622                 } else {
 623                         // Add coefficient of a and b
 624                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 625                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 626                                 break;  // Order term ends the sequence
 627                         } else {
 628                                 ex sum = (*a).rest + (*b).rest;
 629                                 if (!(sum.is_zero()))
 630                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 631                                 ++a;
 632                                 ++b;
 633                         }
 634                 }
 635         }
 636         return pseries(relational(var,point), new_seq);
 637 }
 638
 639
 640 /** Implementation of ex::series() for sums. This performs series addition when
 641  *  adding pseries objects.
 642  *  @see ex::series */
 643 ex add::series(const relational & r, int order, unsigned options) const
 644 {
 645         ex acc; // Series accumulator
 646
 647         // Get first term from overall_coeff
 648         acc = overall_coeff.series(r, order, options);
 649
 650         // Add remaining terms
 651         epvector::const_iterator it = seq.begin();
 652         epvector::const_iterator itend = seq.end();
 653         for (; it!=itend; ++it) {
 654                 ex op;
 655                 if (is_exactly_a<pseries>(it->rest))
 656                         op = it->rest;
 657                 else
 658                         op = it->rest.series(r, order, options);
 659                 if (!it->coeff.is_equal(_ex1))
 660                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 661
 662                 // Series addition
 663                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 664         }
 665         return acc;
 666 }
 667
 668
 669 /** Multiply a pseries object with a numeric constant, producing a pseries
 670  *  object that represents the product.
 671  *
 672  *  @param other  constant to multiply with
 673  *  @return the product as a pseries */
 674 ex pseries::mul_const(const numeric &other) const
 675 {
 676         epvector new_seq;
 677         new_seq.reserve(seq.size());
 678
 679         epvector::const_iterator it = seq.begin(), itend = seq.end();
 680         while (it != itend) {
 681                 if (!is_order_function(it->rest))
 682                         new_seq.push_back(expair(it->rest * other, it->coeff));
 683                 else
 684                         new_seq.push_back(*it);
 685                 ++it;
 686         }
 687         return pseries(relational(var,point), new_seq);
 688 }
 689
 690
 691 /** Multiply one pseries object to another, producing a pseries object that
 692  *  represents the product.
 693  *
 694  *  @param other  pseries object to multiply with
 695  *  @return the product as a pseries */
 696 ex pseries::mul_series(const pseries &other) const
 697 {
 698         // Multiplying two series with different variables or expansion points
 699         // results in an empty (constant) series
 700         if (!is_compatible_to(other)) {
 701                 epvector nul;
 702                 nul.push_back(expair(Order(_ex1), _ex0));
 703                 return pseries(relational(var,point), nul);
 704         }
 705
 706         // Series multiplication
 707         epvector new_seq;
 708         int a_max = degree(var);
 709         int b_max = other.degree(var);
 710         int a_min = ldegree(var);
 711         int b_min = other.ldegree(var);
 712         int cdeg_min = a_min + b_min;
 713         int cdeg_max = a_max + b_max;
 714
 715         int higher_order_a = INT_MAX;
 716         int higher_order_b = INT_MAX;
 717         if (is_order_function(coeff(var, a_max)))
 718                 higher_order_a = a_max + b_min;
 719         if (is_order_function(other.coeff(var, b_max)))
 720                 higher_order_b = b_max + a_min;
 721         int higher_order_c = std::min(higher_order_a, higher_order_b);
 722         if (cdeg_max >= higher_order_c)
 723                 cdeg_max = higher_order_c - 1;
 724
 725         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 726                 ex co = _ex0;
 727                 // c(i)=a(0)b(i)+...+a(i)b(0)
 728                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 729                         ex a_coeff = coeff(var, i);
 730                         ex b_coeff = other.coeff(var, cdeg-i);
 731                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 732                                 co += a_coeff * b_coeff;
 733                 }
 734                 if (!co.is_zero())
 735                         new_seq.push_back(expair(co, numeric(cdeg)));
 736         }
 737         if (higher_order_c < INT_MAX)
 738                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 739         return pseries(relational(var, point), new_seq);
 740 }
 741
 742
 743 /** Implementation of ex::series() for product. This performs series
 744  *  multiplication when multiplying series.
 745  *  @see ex::series */
 746 ex mul::series(const relational & r, int order, unsigned options) const
 747 {
 748         pseries acc; // Series accumulator
 749
 750         // Multiply with remaining terms
 751         const epvector::const_iterator itbeg = seq.begin();
 752         const epvector::const_iterator itend = seq.end();
 753         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 754                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 755
 756                 // Series multiplication
 757                 if (it==itbeg)
 758                         acc = ex_to<pseries>(op);
 759                 else
 760                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 761         }
 762         return acc.mul_const(ex_to<numeric>(overall_coeff));
 763 }
 764
 765
 766 /** Compute the p-th power of a series.
 767  *
 768  *  @param p  power to compute
 769  *  @param deg  truncation order of series calculation */
 770 ex pseries::power_const(const numeric &p, int deg) const
 771 {
 772         // method:
 773         // (due to Leonhard Euler)
 774         // let A(x) be this series and for the time being let it start with a
 775         // constant (later we'll generalize):
 776         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 777         // We want to compute
 778         //     C(x) = A(x)^p
 779         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 780         // Taking the derivative on both sides and multiplying with A(x) one
 781         // immediately arrives at
 782         //     C'(x)*A(x) = p*C(x)*A'(x)
 783         // Multiplying this out and comparing coefficients we get the recurrence
 784         // formula
 785         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 786         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 787         // which can easily be solved given the starting value c_0 = (a_0)^p.
 788         // For the more general case where the leading coefficient of A(x) is not
 789         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 790         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 791         // then of course x^(p*m) but the recurrence formula still holds.
 792
 793         if (seq.empty()) {
 794                 // as a special case, handle the empty (zero) series honoring the
 795                 // usual power laws such as implemented in power::eval()
 796                 if (p.real().is_zero())
 797                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 798                 else if (p.real().is_negative())
 799                         throw pole_error("pseries::power_const(): division by zero",1);
 800                 else
 801                         return *this;
 802         }
 803
 804         const int ldeg = ldegree(var);
 805         if (!(p*ldeg).is_integer())
 806                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 807
 808         // O(x^n)^(-m) is undefined
 809         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 810                 throw pole_error("pseries::power_const(): division by zero",1);
 811
 812         // Compute coefficients of the powered series
 813         exvector co;
 814         co.reserve(deg);
 815         co.push_back(power(coeff(var, ldeg), p));
 816         bool all_sums_zero = true;
 817         for (int i=1; i<deg; ++i) {
 818                 ex sum = _ex0;
 819                 for (int j=1; j<=i; ++j) {
 820                         ex c = coeff(var, j + ldeg);
 821                         if (is_order_function(c)) {
 822                                 co.push_back(Order(_ex1));
 823                                 break;
 824                         } else
 825                                 sum += (p * j - (i - j)) * co[i - j] * c;
 826                 }
 827                 if (!sum.is_zero())
 828                         all_sums_zero = false;
 829                 co.push_back(sum / coeff(var, ldeg) / i);
 830         }
 831
 832         // Construct new series (of non-zero coefficients)
 833         epvector new_seq;
 834         bool higher_order = false;
 835         for (int i=0; i<deg; ++i) {
 836                 if (!co[i].is_zero())
 837                         new_seq.push_back(expair(co[i], p * ldeg + i));
 838                 if (is_order_function(co[i])) {
 839                         higher_order = true;
 840                         break;
 841                 }
 842         }
 843         if (!higher_order && !all_sums_zero)
 844                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 845         return pseries(relational(var,point), new_seq);
 846 }
 847
 848
 849 /** Return a new pseries object with the powers shifted by deg. */
 850 pseries pseries::shift_exponents(int deg) const
 851 {
 852         epvector newseq = seq;
 853         epvector::iterator i = newseq.begin(), end  = newseq.end();
 854         while (i != end) {
 855                 i->coeff += deg;
 856                 ++i;
 857         }
 858         return pseries(relational(var, point), newseq);
 859 }
 860
 861
 862 /** Implementation of ex::series() for powers. This performs Laurent expansion
 863  *  of reciprocals of series at singularities.
 864  *  @see ex::series */
 865 ex power::series(const relational & r, int order, unsigned options) const
 866 {
 867         // If basis is already a series, just power it
 868         if (is_exactly_a<pseries>(basis))
 869                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 870
 871         // Basis is not a series, may there be a singularity?
 872         bool must_expand_basis = false;
 873         try {
 874                 basis.subs(r);
 875         } catch (pole_error) {
 876                 must_expand_basis = true;
 877         }
 878
 879         // Is the expression of type something^(-int)?
 880         if (!must_expand_basis && !exponent.info(info_flags::negint))
 881                 return basic::series(r, order, options);
 882
 883         // Is the expression of type 0^something?
 884         if (!must_expand_basis && !basis.subs(r).is_zero())
 885                 return basic::series(r, order, options);
 886
 887         // Singularity encountered, is the basis equal to (var - point)?
 888         if (basis.is_equal(r.lhs() - r.rhs())) {
 889                 epvector new_seq;
 890                 if (ex_to<numeric>(exponent).to_int() < order)
 891                         new_seq.push_back(expair(_ex1, exponent));
 892                 else
 893                         new_seq.push_back(expair(Order(_ex1), exponent));
 894                 return pseries(r, new_seq);
 895         }
 896
 897         // No, expand basis into series
 898         ex e = basis.series(r, order, options);
 899         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 900 }
 901
 902
 903 /** Re-expansion of a pseries object. */
 904 ex pseries::series(const relational & r, int order, unsigned options) const
 905 {
 906         const ex p = r.rhs();
 907         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 908         const symbol &s = ex_to<symbol>(r.lhs());
 909
 910         if (var.is_equal(s) && point.is_equal(p)) {
 911                 if (order > degree(s))
 912                         return *this;
 913                 else {
 914                         epvector new_seq;
 915                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 916                         while (it != itend) {
 917                                 int o = ex_to<numeric>(it->coeff).to_int();
 918                                 if (o >= order) {
 919                                         new_seq.push_back(expair(Order(_ex1), o));
 920                                         break;
 921                                 }
 922                                 new_seq.push_back(*it);
 923                                 ++it;
 924                         }
 925                         return pseries(r, new_seq);
 926                 }
 927         } else
 928                 return convert_to_poly().series(r, order, options);
 929 }
 930
 931
 932 /** Compute the truncated series expansion of an expression.
 933  *  This function returns an expression containing an object of class pseries
 934  *  to represent the series. If the series does not terminate within the given
 935  *  truncation order, the last term of the series will be an order term.
 936  *
 937  *  @param r  expansion relation, lhs holds variable and rhs holds point
 938  *  @param order  truncation order of series calculations
 939  *  @param options  of class series_options
 940  *  @return an expression holding a pseries object */
 941 ex ex::series(const ex & r, int order, unsigned options) const
 942 {
 943         GINAC_ASSERT(bp!=0);
 944         ex e;
 945         relational rel_;
 946
 947         if (is_a<relational>(r))
 948                 rel_ = ex_to<relational>(r);
 949         else if (is_a<symbol>(r))
 950                 rel_ = relational(r,_ex0);
 951         else
 952                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 953
 954         try {
 955                 e = bp->series(rel_, order, options);
 956         } catch (std::exception &x) {
 957                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 958         }
 959         return e;
 960 }
 961
 962 } // namespace GiNaC