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" // for Order function
  29 #include "lst.h"
  30 #include "mul.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "symbol.h"
  34 #include "print.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  41
  42
  43 /*
  44  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  45  */
  46
  47 pseries::pseries() : inherited(TINFO_pseries) { }
  48
  49 void pseries::copy(const pseries &other)
  50 {
  51         inherited::copy(other);
  52         seq = other.seq;
  53         var = other.var;
  54         point = other.point;
  55 }
  56
  57 DEFAULT_DESTROY(pseries)
  58
  59
  60 /*
  61  *  Other ctors
  62  */
  63
  64 /** Construct pseries from a vector of coefficients and powers.
  65  *  expair.rest holds the coefficient, expair.coeff holds the power.
  66  *  The powers must be integers (positive or negative) and in ascending order;
  67  *  the last coefficient can be Order(_ex1) to represent a truncated,
  68  *  non-terminating series.
  69  *
  70  *  @param rel_  expansion variable and point (must hold a relational)
  71  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  72  *  @return newly constructed pseries */
  73 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  74 {
  75         GINAC_ASSERT(is_exactly_a<relational>(rel_));
  76         GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
  77         point = rel_.rhs();
  78         var = rel_.lhs();
  79 }
  80
  81
  82 /*
  83  *  Archiving
  84  */
  85
  86 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  87 {
  88         for (unsigned int i=0; true; ++i) {
  89                 ex rest;
  90                 ex coeff;
  91                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  92                         seq.push_back(expair(rest, coeff));
  93                 else
  94                         break;
  95         }
  96         n.find_ex("var", var, sym_lst);
  97         n.find_ex("point", point, sym_lst);
  98 }
  99
 100 void pseries::archive(archive_node &n) const
 101 {
 102         inherited::archive(n);
 103         epvector::const_iterator i = seq.begin(), iend = seq.end();
 104         while (i != iend) {
 105                 n.add_ex("coeff", i->rest);
 106                 n.add_ex("power", i->coeff);
 107                 ++i;
 108         }
 109         n.add_ex("var", var);
 110         n.add_ex("point", point);
 111 }
 112
 113 DEFAULT_UNARCHIVE(pseries)
 114
 115 //////////
 116 // functions overriding virtual functions from base classes
 117 //////////
 118
 119 void pseries::print(const print_context & c, unsigned level) const
 120 {
 121         if (is_a<print_tree>(c)) {
 122
 123                 c.s << std::string(level, ' ') << class_name()
 124                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 125                     << std::endl;
 126                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 127                 unsigned num = seq.size();
 128                 for (unsigned i=0; i<num; ++i) {
 129                         seq[i].rest.print(c, level + delta_indent);
 130                         seq[i].coeff.print(c, level + delta_indent);
 131                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 132                 }
 133                 var.print(c, level + delta_indent);
 134                 point.print(c, level + delta_indent);
 135
 136         } else {
 137
 138                 if (precedence() <= level)
 139                         c.s << "(";
 140
 141                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
 142                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
 143
 144                 // objects of type pseries must not have any zero entries, so the
 145                 // trivial (zero) pseries needs a special treatment here:
 146                 if (seq.empty())
 147                         c.s << '0';
 148                 epvector::const_iterator i = seq.begin(), end = seq.end();
 149                 while (i != end) {
 150                         // print a sign, if needed
 151                         if (i != seq.begin())
 152                                 c.s << '+';
 153                         if (!is_order_function(i->rest)) {
 154                                 // print 'rest', i.e. the expansion coefficient
 155                                 if (i->rest.info(info_flags::numeric) &&
 156                                         i->rest.info(info_flags::positive)) {
 157                                         i->rest.print(c);
 158                                 } else {
 159                                         c.s << par_open;
 160                                         i->rest.print(c);
 161                                         c.s << par_close;
 162                                 }
 163                                 // print 'coeff', something like (x-1)^42
 164                                 if (!i->coeff.is_zero()) {
 165                                         if (is_a<print_latex>(c))
 166                                                 c.s << ' ';
 167                                         else
 168                                                 c.s << '*';
 169                                         if (!point.is_zero()) {
 170                                                 c.s << par_open;
 171                                                 (var-point).print(c);
 172                                                 c.s << par_close;
 173                                         } else
 174                                                 var.print(c);
 175                                         if (i->coeff.compare(_ex1)) {
 176                                                 c.s << '^';
 177                                                 if (i->coeff.info(info_flags::negative)) {
 178                                                         c.s << par_open;
 179                                                         i->coeff.print(c);
 180                                                         c.s << par_close;
 181                                                 } else {
 182                                                         if (is_a<print_latex>(c)) {
 183                                                                 c.s << '{';
 184                                                                 i->coeff.print(c);
 185                                                                 c.s << '}';
 186                                                         } else
 187                                                                 i->coeff.print(c);
 188                                                 }
 189                                         }
 190                                 }
 191                         } else
 192                                 Order(power(var-point,i->coeff)).print(c);
 193                         ++i;
 194                 }
 195
 196                 if (precedence() <= level)
 197                         c.s << ")";
 198         }
 199 }
 200
 201 int pseries::compare_same_type(const basic & other) const
 202 {
 203         GINAC_ASSERT(is_a<pseries>(other));
 204         const pseries &o = static_cast<const pseries &>(other);
 205
 206         // first compare the lengths of the series...
 207         if (seq.size()>o.seq.size())
 208                 return 1;
 209         if (seq.size()<o.seq.size())
 210                 return -1;
 211
 212         // ...then the expansion point...
 213         int cmpval = var.compare(o.var);
 214         if (cmpval)
 215                 return cmpval;
 216         cmpval = point.compare(o.point);
 217         if (cmpval)
 218                 return cmpval;
 219
 220         // ...and if that failed the individual elements
 221         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 222         while (it!=seq.end() && o_it!=o.seq.end()) {
 223                 cmpval = it->compare(*o_it);
 224                 if (cmpval)
 225                         return cmpval;
 226                 ++it;
 227                 ++o_it;
 228         }
 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 /** Return the ith term in the series when represented as a sum. */
 241 ex pseries::op(int i) const
 242 {
 243         if (i < 0 || unsigned(i) >= seq.size())
 244                 throw (std::out_of_range("op() out of range"));
 245         return seq[i].rest * power(var - point, seq[i].coeff);
 246 }
 247
 248 ex &pseries::let_op(int i)
 249 {
 250         throw (std::logic_error("let_op not defined for pseries"));
 251 }
 252
 253 /** Return degree of highest power of the series.  This is usually the exponent
 254  *  of the Order term.  If s is not the expansion variable of the series, the
 255  *  series is examined termwise. */
 256 int pseries::degree(const ex &s) const
 257 {
 258         if (var.is_equal(s)) {
 259                 // Return last exponent
 260                 if (seq.size())
 261                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 262                 else
 263                         return 0;
 264         } else {
 265                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 266                 if (it == itend)
 267                         return 0;
 268                 int max_pow = INT_MIN;
 269                 while (it != itend) {
 270                         int pow = it->rest.degree(s);
 271                         if (pow > max_pow)
 272                                 max_pow = pow;
 273                         ++it;
 274                 }
 275                 return max_pow;
 276         }
 277 }
 278
 279 /** Return degree of lowest power of the series.  This is usually the exponent
 280  *  of the leading term.  If s is not the expansion variable of the series, the
 281  *  series is examined termwise.  If s is the expansion variable but the
 282  *  expansion point is not zero the series is not expanded to find the degree.
 283  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 284 int pseries::ldegree(const ex &s) const
 285 {
 286         if (var.is_equal(s)) {
 287                 // Return first exponent
 288                 if (seq.size())
 289                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 290                 else
 291                         return 0;
 292         } else {
 293                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 294                 if (it == itend)
 295                         return 0;
 296                 int min_pow = INT_MAX;
 297                 while (it != itend) {
 298                         int pow = it->rest.ldegree(s);
 299                         if (pow < min_pow)
 300                                 min_pow = pow;
 301                         ++it;
 302                 }
 303                 return min_pow;
 304         }
 305 }
 306
 307 /** Return coefficient of degree n in power series if s is the expansion
 308  *  variable.  If the expansion point is nonzero, by definition the n=1
 309  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 310  *  the expansion took place in the s in the first place).
 311  *  If s is not the expansion variable, an attempt is made to convert the
 312  *  series to a polynomial and return the corresponding coefficient from
 313  *  there. */
 314 ex pseries::coeff(const ex &s, int n) const
 315 {
 316         if (var.is_equal(s)) {
 317                 if (seq.empty())
 318                         return _ex0;
 319
 320                 // Binary search in sequence for given power
 321                 numeric looking_for = numeric(n);
 322                 int lo = 0, hi = seq.size() - 1;
 323                 while (lo <= hi) {
 324                         int mid = (lo + hi) / 2;
 325                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 326                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 327                         switch (cmp) {
 328                                 case -1:
 329                                         lo = mid + 1;
 330                                         break;
 331                                 case 0:
 332                                         return seq[mid].rest;
 333                                 case 1:
 334                                         hi = mid - 1;
 335                                         break;
 336                                 default:
 337                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 338                         }
 339                 }
 340                 return _ex0;
 341         } else
 342                 return convert_to_poly().coeff(s, n);
 343 }
 344
 345 /** Does nothing. */
 346 ex pseries::collect(const ex &s, bool distributed) const
 347 {
 348         return *this;
 349 }
 350
 351 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 352 ex pseries::eval(int level) const
 353 {
 354         if (level == 1)
 355                 return this->hold();
 356
 357         if (level == -max_recursion_level)
 358                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 359
 360         // Construct a new series with evaluated coefficients
 361         epvector new_seq;
 362         new_seq.reserve(seq.size());
 363         epvector::const_iterator it = seq.begin(), itend = seq.end();
 364         while (it != itend) {
 365                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 366                 ++it;
 367         }
 368         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 369 }
 370
 371 /** Evaluate coefficients numerically. */
 372 ex pseries::evalf(int level) const
 373 {
 374         if (level == 1)
 375                 return *this;
 376
 377         if (level == -max_recursion_level)
 378                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 379
 380         // Construct a new series with evaluated coefficients
 381         epvector new_seq;
 382         new_seq.reserve(seq.size());
 383         epvector::const_iterator it = seq.begin(), itend = seq.end();
 384         while (it != itend) {
 385                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 386                 ++it;
 387         }
 388         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 389 }
 390
 391 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
 392 {
 393         // If expansion variable is being substituted, convert the series to a
 394         // polynomial and do the substitution there because the result might
 395         // no longer be a power series
 396         if (ls.has(var))
 397                 return convert_to_poly(true).subs(ls, lr, no_pattern);
 398
 399         // Otherwise construct a new series with substituted coefficients and
 400         // expansion point
 401         epvector newseq;
 402         newseq.reserve(seq.size());
 403         epvector::const_iterator it = seq.begin(), itend = seq.end();
 404         while (it != itend) {
 405                 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
 406                 ++it;
 407         }
 408         return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
 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         epvector::const_iterator i = seq.begin(), end = seq.end();
 417         while (i != end) {
 418                 ex restexp = i->rest.expand();
 419                 if (!restexp.is_zero())
 420                         newseq.push_back(expair(restexp, i->coeff));
 421                 ++i;
 422         }
 423         return (new pseries(relational(var,point), newseq))
 424                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 425 }
 426
 427 /** Implementation of ex::diff() for a power series.  It treats the series as a
 428  *  polynomial.
 429  *  @see ex::diff */
 430 ex pseries::derivative(const symbol & s) const
 431 {
 432         if (s == var) {
 433                 epvector new_seq;
 434                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 435
 436                 // FIXME: coeff might depend on var
 437                 while (it != itend) {
 438                         if (is_order_function(it->rest)) {
 439                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 440                         } else {
 441                                 ex c = it->rest * it->coeff;
 442                                 if (!c.is_zero())
 443                                         new_seq.push_back(expair(c, it->coeff - 1));
 444                         }
 445                         ++it;
 446                 }
 447                 return pseries(relational(var,point), new_seq);
 448         } else {
 449                 return *this;
 450         }
 451 }
 452
 453 ex pseries::convert_to_poly(bool no_order) const
 454 {
 455         ex e;
 456         epvector::const_iterator it = seq.begin(), itend = seq.end();
 457
 458         while (it != itend) {
 459                 if (is_order_function(it->rest)) {
 460                         if (!no_order)
 461                                 e += Order(power(var - point, it->coeff));
 462                 } else
 463                         e += it->rest * power(var - point, it->coeff);
 464                 ++it;
 465         }
 466         return e;
 467 }
 468
 469 bool pseries::is_terminating(void) const
 470 {
 471         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 472 }
 473
 474
 475 /*
 476  *  Implementations of series expansion
 477  */
 478
 479 /** Default implementation of ex::series(). This performs Taylor expansion.
 480  *  @see ex::series */
 481 ex basic::series(const relational & r, int order, unsigned options) const
 482 {
 483         epvector seq;
 484         numeric fac = 1;
 485         ex deriv = *this;
 486         ex coeff = deriv.subs(r);
 487         const symbol &s = ex_to<symbol>(r.lhs());
 488
 489         if (!coeff.is_zero())
 490                 seq.push_back(expair(coeff, _ex0));
 491
 492         int n;
 493         for (n=1; n<order; ++n) {
 494                 fac = fac.mul(n);
 495                 // We need to test for zero in order to see if the series terminates.
 496                 // The problem is that there is no such thing as a perfect test for
 497                 // zero.  Expanding the term occasionally helps a little...
 498                 deriv = deriv.diff(s).expand();
 499                 if (deriv.is_zero())  // Series terminates
 500                         return pseries(r, seq);
 501
 502                 coeff = deriv.subs(r);
 503                 if (!coeff.is_zero())
 504                         seq.push_back(expair(fac.inverse() * coeff, n));
 505         }
 506
 507         // Higher-order terms, if present
 508         deriv = deriv.diff(s);
 509         if (!deriv.expand().is_zero())
 510                 seq.push_back(expair(Order(_ex1), n));
 511         return pseries(r, seq);
 512 }
 513
 514
 515 /** Implementation of ex::series() for symbols.
 516  *  @see ex::series */
 517 ex symbol::series(const relational & r, int order, unsigned options) const
 518 {
 519         epvector seq;
 520         const ex point = r.rhs();
 521         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 522
 523         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 524                 if (order > 0 && !point.is_zero())
 525                         seq.push_back(expair(point, _ex0));
 526                 if (order > 1)
 527                         seq.push_back(expair(_ex1, _ex1));
 528                 else
 529                         seq.push_back(expair(Order(_ex1), numeric(order)));
 530         } else
 531                 seq.push_back(expair(*this, _ex0));
 532         return pseries(r, seq);
 533 }
 534
 535
 536 /** Add one series object to another, producing a pseries object that
 537  *  represents the sum.
 538  *
 539  *  @param other  pseries object to add with
 540  *  @return the sum as a pseries */
 541 ex pseries::add_series(const pseries &other) const
 542 {
 543         // Adding two series with different variables or expansion points
 544         // results in an empty (constant) series
 545         if (!is_compatible_to(other)) {
 546                 epvector nul;
 547                 nul.push_back(expair(Order(_ex1), _ex0));
 548                 return pseries(relational(var,point), nul);
 549         }
 550
 551         // Series addition
 552         epvector new_seq;
 553         epvector::const_iterator a = seq.begin();
 554         epvector::const_iterator b = other.seq.begin();
 555         epvector::const_iterator a_end = seq.end();
 556         epvector::const_iterator b_end = other.seq.end();
 557         int pow_a = INT_MAX, pow_b = INT_MAX;
 558         for (;;) {
 559                 // If a is empty, fill up with elements from b and stop
 560                 if (a == a_end) {
 561                         while (b != b_end) {
 562                                 new_seq.push_back(*b);
 563                                 ++b;
 564                         }
 565                         break;
 566                 } else
 567                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 568
 569                 // If b is empty, fill up with elements from a and stop
 570                 if (b == b_end) {
 571                         while (a != a_end) {
 572                                 new_seq.push_back(*a);
 573                                 ++a;
 574                         }
 575                         break;
 576                 } else
 577                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 578
 579                 // a and b are non-empty, compare powers
 580                 if (pow_a < pow_b) {
 581                         // a has lesser power, get coefficient from a
 582                         new_seq.push_back(*a);
 583                         if (is_order_function((*a).rest))
 584                                 break;
 585                         ++a;
 586                 } else if (pow_b < pow_a) {
 587                         // b has lesser power, get coefficient from b
 588                         new_seq.push_back(*b);
 589                         if (is_order_function((*b).rest))
 590                                 break;
 591                         ++b;
 592                 } else {
 593                         // Add coefficient of a and b
 594                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 595                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 596                                 break;  // Order term ends the sequence
 597                         } else {
 598                                 ex sum = (*a).rest + (*b).rest;
 599                                 if (!(sum.is_zero()))
 600                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 601                                 ++a;
 602                                 ++b;
 603                         }
 604                 }
 605         }
 606         return pseries(relational(var,point), new_seq);
 607 }
 608
 609
 610 /** Implementation of ex::series() for sums. This performs series addition when
 611  *  adding pseries objects.
 612  *  @see ex::series */
 613 ex add::series(const relational & r, int order, unsigned options) const
 614 {
 615         ex acc; // Series accumulator
 616
 617         // Get first term from overall_coeff
 618         acc = overall_coeff.series(r, order, options);
 619
 620         // Add remaining terms
 621         epvector::const_iterator it = seq.begin();
 622         epvector::const_iterator itend = seq.end();
 623         for (; it!=itend; ++it) {
 624                 ex op;
 625                 if (is_ex_exactly_of_type(it->rest, pseries))
 626                         op = it->rest;
 627                 else
 628                         op = it->rest.series(r, order, options);
 629                 if (!it->coeff.is_equal(_ex1))
 630                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 631
 632                 // Series addition
 633                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 634         }
 635         return acc;
 636 }
 637
 638
 639 /** Multiply a pseries object with a numeric constant, producing a pseries
 640  *  object that represents the product.
 641  *
 642  *  @param other  constant to multiply with
 643  *  @return the product as a pseries */
 644 ex pseries::mul_const(const numeric &other) const
 645 {
 646         epvector new_seq;
 647         new_seq.reserve(seq.size());
 648
 649         epvector::const_iterator it = seq.begin(), itend = seq.end();
 650         while (it != itend) {
 651                 if (!is_order_function(it->rest))
 652                         new_seq.push_back(expair(it->rest * other, it->coeff));
 653                 else
 654                         new_seq.push_back(*it);
 655                 ++it;
 656         }
 657         return pseries(relational(var,point), new_seq);
 658 }
 659
 660
 661 /** Multiply one pseries object to another, producing a pseries object that
 662  *  represents the product.
 663  *
 664  *  @param other  pseries object to multiply with
 665  *  @return the product as a pseries */
 666 ex pseries::mul_series(const pseries &other) const
 667 {
 668         // Multiplying two series with different variables or expansion points
 669         // results in an empty (constant) series
 670         if (!is_compatible_to(other)) {
 671                 epvector nul;
 672                 nul.push_back(expair(Order(_ex1), _ex0));
 673                 return pseries(relational(var,point), nul);
 674         }
 675
 676         // Series multiplication
 677         epvector new_seq;
 678         int a_max = degree(var);
 679         int b_max = other.degree(var);
 680         int a_min = ldegree(var);
 681         int b_min = other.ldegree(var);
 682         int cdeg_min = a_min + b_min;
 683         int cdeg_max = a_max + b_max;
 684
 685         int higher_order_a = INT_MAX;
 686         int higher_order_b = INT_MAX;
 687         if (is_order_function(coeff(var, a_max)))
 688                 higher_order_a = a_max + b_min;
 689         if (is_order_function(other.coeff(var, b_max)))
 690                 higher_order_b = b_max + a_min;
 691         int higher_order_c = std::min(higher_order_a, higher_order_b);
 692         if (cdeg_max >= higher_order_c)
 693                 cdeg_max = higher_order_c - 1;
 694
 695         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 696                 ex co = _ex0;
 697                 // c(i)=a(0)b(i)+...+a(i)b(0)
 698                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 699                         ex a_coeff = coeff(var, i);
 700                         ex b_coeff = other.coeff(var, cdeg-i);
 701                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 702                                 co += a_coeff * b_coeff;
 703                 }
 704                 if (!co.is_zero())
 705                         new_seq.push_back(expair(co, numeric(cdeg)));
 706         }
 707         if (higher_order_c < INT_MAX)
 708                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 709         return pseries(relational(var, point), new_seq);
 710 }
 711
 712
 713 /** Implementation of ex::series() for product. This performs series
 714  *  multiplication when multiplying series.
 715  *  @see ex::series */
 716 ex mul::series(const relational & r, int order, unsigned options) const
 717 {
 718         pseries acc; // Series accumulator
 719
 720         // Multiply with remaining terms
 721         const epvector::const_iterator itbeg = seq.begin();
 722         const epvector::const_iterator itend = seq.end();
 723         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 724                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 725
 726                 // Series multiplication
 727                 if (it==itbeg)
 728                         acc = ex_to<pseries>(op);
 729                 else
 730                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 731         }
 732         return acc.mul_const(ex_to<numeric>(overall_coeff));
 733 }
 734
 735
 736 /** Compute the p-th power of a series.
 737  *
 738  *  @param p  power to compute
 739  *  @param deg  truncation order of series calculation */
 740 ex pseries::power_const(const numeric &p, int deg) const
 741 {
 742         // method:
 743         // (due to Leonhard Euler)
 744         // let A(x) be this series and for the time being let it start with a
 745         // constant (later we'll generalize):
 746         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 747         // We want to compute
 748         //     C(x) = A(x)^p
 749         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 750         // Taking the derivative on both sides and multiplying with A(x) one
 751         // immediately arrives at
 752         //     C'(x)*A(x) = p*C(x)*A'(x)
 753         // Multiplying this out and comparing coefficients we get the recurrence
 754         // formula
 755         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 756         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 757         // which can easily be solved given the starting value c_0 = (a_0)^p.
 758         // For the more general case where the leading coefficient of A(x) is not
 759         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 760         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 761         // then of course x^(p*m) but the recurrence formula still holds.
 762
 763         if (seq.empty()) {
 764                 // as a special case, handle the empty (zero) series honoring the
 765                 // usual power laws such as implemented in power::eval()
 766                 if (p.real().is_zero())
 767                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 768                 else if (p.real().is_negative())
 769                         throw pole_error("pseries::power_const(): division by zero",1);
 770                 else
 771                         return *this;
 772         }
 773
 774         const int ldeg = ldegree(var);
 775         if (!(p*ldeg).is_integer())
 776                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 777
 778         // O(x^n)^(-m) is undefined
 779         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 780                 throw pole_error("pseries::power_const(): division by zero",1);
 781
 782         // Compute coefficients of the powered series
 783         exvector co;
 784         co.reserve(deg);
 785         co.push_back(power(coeff(var, ldeg), p));
 786         bool all_sums_zero = true;
 787         for (int i=1; i<deg; ++i) {
 788                 ex sum = _ex0;
 789                 for (int j=1; j<=i; ++j) {
 790                         ex c = coeff(var, j + ldeg);
 791                         if (is_order_function(c)) {
 792                                 co.push_back(Order(_ex1));
 793                                 break;
 794                         } else
 795                                 sum += (p * j - (i - j)) * co[i - j] * c;
 796                 }
 797                 if (!sum.is_zero())
 798                         all_sums_zero = false;
 799                 co.push_back(sum / coeff(var, ldeg) / i);
 800         }
 801
 802         // Construct new series (of non-zero coefficients)
 803         epvector new_seq;
 804         bool higher_order = false;
 805         for (int i=0; i<deg; ++i) {
 806                 if (!co[i].is_zero())
 807                         new_seq.push_back(expair(co[i], p * ldeg + i));
 808                 if (is_order_function(co[i])) {
 809                         higher_order = true;
 810                         break;
 811                 }
 812         }
 813         if (!higher_order && !all_sums_zero)
 814                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 815         return pseries(relational(var,point), new_seq);
 816 }
 817
 818
 819 /** Return a new pseries object with the powers shifted by deg. */
 820 pseries pseries::shift_exponents(int deg) const
 821 {
 822         epvector newseq = seq;
 823         epvector::iterator i = newseq.begin(), end  = newseq.end();
 824         while (i != end) {
 825                 i->coeff += deg;
 826                 ++i;
 827         }
 828         return pseries(relational(var, point), newseq);
 829 }
 830
 831
 832 /** Implementation of ex::series() for powers. This performs Laurent expansion
 833  *  of reciprocals of series at singularities.
 834  *  @see ex::series */
 835 ex power::series(const relational & r, int order, unsigned options) const
 836 {
 837         // If basis is already a series, just power it
 838         if (is_ex_exactly_of_type(basis, pseries))
 839                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 840
 841         // Basis is not a series, may there be a singularity?
 842         bool must_expand_basis = false;
 843         try {
 844                 basis.subs(r);
 845         } catch (pole_error) {
 846                 must_expand_basis = true;
 847         }
 848
 849         // Is the expression of type something^(-int)?
 850         if (!must_expand_basis && !exponent.info(info_flags::negint))
 851                 return basic::series(r, order, options);
 852
 853         // Is the expression of type 0^something?
 854         if (!must_expand_basis && !basis.subs(r).is_zero())
 855                 return basic::series(r, order, options);
 856
 857         // Singularity encountered, is the basis equal to (var - point)?
 858         if (basis.is_equal(r.lhs() - r.rhs())) {
 859                 epvector new_seq;
 860                 if (ex_to<numeric>(exponent).to_int() < order)
 861                         new_seq.push_back(expair(_ex1, exponent));
 862                 else
 863                         new_seq.push_back(expair(Order(_ex1), exponent));
 864                 return pseries(r, new_seq);
 865         }
 866
 867         // No, expand basis into series
 868         ex e = basis.series(r, order, options);
 869         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 870 }
 871
 872
 873 /** Re-expansion of a pseries object. */
 874 ex pseries::series(const relational & r, int order, unsigned options) const
 875 {
 876         const ex p = r.rhs();
 877         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 878         const symbol &s = ex_to<symbol>(r.lhs());
 879
 880         if (var.is_equal(s) && point.is_equal(p)) {
 881                 if (order > degree(s))
 882                         return *this;
 883                 else {
 884                         epvector new_seq;
 885                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 886                         while (it != itend) {
 887                                 int o = ex_to<numeric>(it->coeff).to_int();
 888                                 if (o >= order) {
 889                                         new_seq.push_back(expair(Order(_ex1), o));
 890                                         break;
 891                                 }
 892                                 new_seq.push_back(*it);
 893                                 ++it;
 894                         }
 895                         return pseries(r, new_seq);
 896                 }
 897         } else
 898                 return convert_to_poly().series(r, order, options);
 899 }
 900
 901
 902 /** Compute the truncated series expansion of an expression.
 903  *  This function returns an expression containing an object of class pseries
 904  *  to represent the series. If the series does not terminate within the given
 905  *  truncation order, the last term of the series will be an order term.
 906  *
 907  *  @param r  expansion relation, lhs holds variable and rhs holds point
 908  *  @param order  truncation order of series calculations
 909  *  @param options  of class series_options
 910  *  @return an expression holding a pseries object */
 911 ex ex::series(const ex & r, int order, unsigned options) const
 912 {
 913         GINAC_ASSERT(bp!=0);
 914         ex e;
 915         relational rel_;
 916
 917         if (is_ex_exactly_of_type(r,relational))
 918                 rel_ = ex_to<relational>(r);
 919         else if (is_ex_exactly_of_type(r,symbol))
 920                 rel_ = relational(r,_ex0);
 921         else
 922                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 923
 924         try {
 925                 e = bp->series(rel_, order, options);
 926         } catch (std::exception &x) {
 927                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 928         }
 929         return e;
 930 }
 931
 932 } // namespace GiNaC