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 <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 "symbol.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  42
  43
  44 /*
  45  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  46  */
  47
  48 pseries::pseries() : inherited(TINFO_pseries) { }
  49
  50 void pseries::copy(const pseries &other)
  51 {
  52         inherited::copy(other);
  53         seq = other.seq;
  54         var = other.var;
  55         point = other.point;
  56 }
  57
  58 DEFAULT_DESTROY(pseries)
  59
  60
  61 /*
  62  *  Other ctors
  63  */
  64
  65 /** Construct pseries from a vector of coefficients and powers.
  66  *  expair.rest holds the coefficient, expair.coeff holds the power.
  67  *  The powers must be integers (positive or negative) and in ascending order;
  68  *  the last coefficient can be Order(_ex1) to represent a truncated,
  69  *  non-terminating series.
  70  *
  71  *  @param rel_  expansion variable and point (must hold a relational)
  72  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  73  *  @return newly constructed pseries */
  74 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  75 {
  76         GINAC_ASSERT(is_exactly_a<relational>(rel_));
  77         GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
  78         point = rel_.rhs();
  79         var = rel_.lhs();
  80 }
  81
  82
  83 /*
  84  *  Archiving
  85  */
  86
  87 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  88 {
  89         for (unsigned int i=0; true; ++i) {
  90                 ex rest;
  91                 ex coeff;
  92                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  93                         seq.push_back(expair(rest, coeff));
  94                 else
  95                         break;
  96         }
  97         n.find_ex("var", var, sym_lst);
  98         n.find_ex("point", point, sym_lst);
  99 }
 100
 101 void pseries::archive(archive_node &n) const
 102 {
 103         inherited::archive(n);
 104         epvector::const_iterator i = seq.begin(), iend = seq.end();
 105         while (i != iend) {
 106                 n.add_ex("coeff", i->rest);
 107                 n.add_ex("power", i->coeff);
 108                 ++i;
 109         }
 110         n.add_ex("var", var);
 111         n.add_ex("point", point);
 112 }
 113
 114 DEFAULT_UNARCHIVE(pseries)
 115
 116 //////////
 117 // functions overriding virtual functions from base classes
 118 //////////
 119
 120 void pseries::print(const print_context & c, unsigned level) const
 121 {
 122         if (is_a<print_tree>(c)) {
 123
 124                 c.s << std::string(level, ' ') << class_name()
 125                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 126                     << std::endl;
 127                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 128                 unsigned num = seq.size();
 129                 for (unsigned i=0; i<num; ++i) {
 130                         seq[i].rest.print(c, level + delta_indent);
 131                         seq[i].coeff.print(c, level + delta_indent);
 132                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 133                 }
 134                 var.print(c, level + delta_indent);
 135                 point.print(c, level + delta_indent);
 136
 137         } else {
 138
 139                 if (precedence() <= level)
 140                         c.s << "(";
 141
 142                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
 143                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
 144
 145                 // objects of type pseries must not have any zero entries, so the
 146                 // trivial (zero) pseries needs a special treatment here:
 147                 if (seq.empty())
 148                         c.s << '0';
 149                 epvector::const_iterator i = seq.begin(), end = seq.end();
 150                 while (i != end) {
 151                         // print a sign, if needed
 152                         if (i != seq.begin())
 153                                 c.s << '+';
 154                         if (!is_order_function(i->rest)) {
 155                                 // print 'rest', i.e. the expansion coefficient
 156                                 if (i->rest.info(info_flags::numeric) &&
 157                                         i->rest.info(info_flags::positive)) {
 158                                         i->rest.print(c);
 159                                 } else {
 160                                         c.s << par_open;
 161                                         i->rest.print(c);
 162                                         c.s << par_close;
 163                                 }
 164                                 // print 'coeff', something like (x-1)^42
 165                                 if (!i->coeff.is_zero()) {
 166                                         if (is_a<print_latex>(c))
 167                                                 c.s << ' ';
 168                                         else
 169                                                 c.s << '*';
 170                                         if (!point.is_zero()) {
 171                                                 c.s << par_open;
 172                                                 (var-point).print(c);
 173                                                 c.s << par_close;
 174                                         } else
 175                                                 var.print(c);
 176                                         if (i->coeff.compare(_ex1)) {
 177                                                 c.s << '^';
 178                                                 if (i->coeff.info(info_flags::negative)) {
 179                                                         c.s << par_open;
 180                                                         i->coeff.print(c);
 181                                                         c.s << par_close;
 182                                                 } else {
 183                                                         if (is_a<print_latex>(c)) {
 184                                                                 c.s << '{';
 185                                                                 i->coeff.print(c);
 186                                                                 c.s << '}';
 187                                                         } else
 188                                                                 i->coeff.print(c);
 189                                                 }
 190                                         }
 191                                 }
 192                         } else
 193                                 Order(power(var-point,i->coeff)).print(c);
 194                         ++i;
 195                 }
 196
 197                 if (precedence() <= level)
 198                         c.s << ")";
 199         }
 200 }
 201
 202 int pseries::compare_same_type(const basic & other) const
 203 {
 204         GINAC_ASSERT(is_a<pseries>(other));
 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
 231         // so they are equal.
 232         return 0;
 233 }
 234
 235 /** Return the number of operands including a possible order term. */
 236 unsigned pseries::nops(void) const
 237 {
 238         return seq.size();
 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 ex &pseries::let_op(int i)
 250 {
 251         throw (std::logic_error("let_op not defined for pseries"));
 252 }
 253
 254 /** Return degree of highest power of the series.  This is usually the exponent
 255  *  of the Order term.  If s is not the expansion variable of the series, the
 256  *  series is examined termwise. */
 257 int pseries::degree(const ex &s) const
 258 {
 259         if (var.is_equal(s)) {
 260                 // Return last exponent
 261                 if (seq.size())
 262                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 263                 else
 264                         return 0;
 265         } else {
 266                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 267                 if (it == itend)
 268                         return 0;
 269                 int max_pow = INT_MIN;
 270                 while (it != itend) {
 271                         int pow = it->rest.degree(s);
 272                         if (pow > max_pow)
 273                                 max_pow = pow;
 274                         ++it;
 275                 }
 276                 return max_pow;
 277         }
 278 }
 279
 280 /** Return degree of lowest power of the series.  This is usually the exponent
 281  *  of the leading term.  If s is not the expansion variable of the series, the
 282  *  series is examined termwise.  If s is the expansion variable but the
 283  *  expansion point is not zero the series is not expanded to find the degree.
 284  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 285 int pseries::ldegree(const ex &s) const
 286 {
 287         if (var.is_equal(s)) {
 288                 // Return first exponent
 289                 if (seq.size())
 290                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 291                 else
 292                         return 0;
 293         } else {
 294                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 295                 if (it == itend)
 296                         return 0;
 297                 int min_pow = INT_MAX;
 298                 while (it != itend) {
 299                         int pow = it->rest.ldegree(s);
 300                         if (pow < min_pow)
 301                                 min_pow = pow;
 302                         ++it;
 303                 }
 304                 return min_pow;
 305         }
 306 }
 307
 308 /** Return coefficient of degree n in power series if s is the expansion
 309  *  variable.  If the expansion point is nonzero, by definition the n=1
 310  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 311  *  the expansion took place in the s in the first place).
 312  *  If s is not the expansion variable, an attempt is made to convert the
 313  *  series to a polynomial and return the corresponding coefficient from
 314  *  there. */
 315 ex pseries::coeff(const ex &s, int n) const
 316 {
 317         if (var.is_equal(s)) {
 318                 if (seq.empty())
 319                         return _ex0;
 320
 321                 // Binary search in sequence for given power
 322                 numeric looking_for = numeric(n);
 323                 int lo = 0, hi = seq.size() - 1;
 324                 while (lo <= hi) {
 325                         int mid = (lo + hi) / 2;
 326                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 327                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 328                         switch (cmp) {
 329                                 case -1:
 330                                         lo = mid + 1;
 331                                         break;
 332                                 case 0:
 333                                         return seq[mid].rest;
 334                                 case 1:
 335                                         hi = mid - 1;
 336                                         break;
 337                                 default:
 338                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 339                         }
 340                 }
 341                 return _ex0;
 342         } else
 343                 return convert_to_poly().coeff(s, n);
 344 }
 345
 346 /** Does nothing. */
 347 ex pseries::collect(const ex &s, bool distributed) const
 348 {
 349         return *this;
 350 }
 351
 352 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 353 ex pseries::eval(int level) const
 354 {
 355         if (level == 1)
 356                 return this->hold();
 357
 358         if (level == -max_recursion_level)
 359                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 360
 361         // Construct a new series with evaluated coefficients
 362         epvector new_seq;
 363         new_seq.reserve(seq.size());
 364         epvector::const_iterator it = seq.begin(), itend = seq.end();
 365         while (it != itend) {
 366                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 367                 ++it;
 368         }
 369         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 370 }
 371
 372 /** Evaluate coefficients numerically. */
 373 ex pseries::evalf(int level) const
 374 {
 375         if (level == 1)
 376                 return *this;
 377
 378         if (level == -max_recursion_level)
 379                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 380
 381         // Construct a new series with evaluated coefficients
 382         epvector new_seq;
 383         new_seq.reserve(seq.size());
 384         epvector::const_iterator it = seq.begin(), itend = seq.end();
 385         while (it != itend) {
 386                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 387                 ++it;
 388         }
 389         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 390 }
 391
 392 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
 393 {
 394         // If expansion variable is being substituted, convert the series to a
 395         // polynomial and do the substitution there because the result might
 396         // no longer be a power series
 397         if (ls.has(var))
 398                 return convert_to_poly(true).subs(ls, lr, no_pattern);
 399
 400         // Otherwise construct a new series with substituted coefficients and
 401         // expansion point
 402         epvector newseq;
 403         newseq.reserve(seq.size());
 404         epvector::const_iterator it = seq.begin(), itend = seq.end();
 405         while (it != itend) {
 406                 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
 407                 ++it;
 408         }
 409         return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
 410 }
 411
 412 /** Implementation of ex::expand() for a power series.  It expands all the
 413  *  terms individually and returns the resulting series as a new pseries. */
 414 ex pseries::expand(unsigned options) const
 415 {
 416         epvector newseq;
 417         epvector::const_iterator i = seq.begin(), end = seq.end();
 418         while (i != end) {
 419                 ex restexp = i->rest.expand();
 420                 if (!restexp.is_zero())
 421                         newseq.push_back(expair(restexp, i->coeff));
 422                 ++i;
 423         }
 424         return (new pseries(relational(var,point), newseq))
 425                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 426 }
 427
 428 /** Implementation of ex::diff() for a power series.  It treats the series as a
 429  *  polynomial.
 430  *  @see ex::diff */
 431 ex pseries::derivative(const symbol & s) const
 432 {
 433         if (s == var) {
 434                 epvector new_seq;
 435                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 436
 437                 // FIXME: coeff might depend on var
 438                 while (it != itend) {
 439                         if (is_order_function(it->rest)) {
 440                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 441                         } else {
 442                                 ex c = it->rest * it->coeff;
 443                                 if (!c.is_zero())
 444                                         new_seq.push_back(expair(c, it->coeff - 1));
 445                         }
 446                         ++it;
 447                 }
 448                 return pseries(relational(var,point), new_seq);
 449         } else {
 450                 return *this;
 451         }
 452 }
 453
 454 ex pseries::convert_to_poly(bool no_order) const
 455 {
 456         ex e;
 457         epvector::const_iterator it = seq.begin(), itend = seq.end();
 458
 459         while (it != itend) {
 460                 if (is_order_function(it->rest)) {
 461                         if (!no_order)
 462                                 e += Order(power(var - point, it->coeff));
 463                 } else
 464                         e += it->rest * power(var - point, it->coeff);
 465                 ++it;
 466         }
 467         return e;
 468 }
 469
 470 bool pseries::is_terminating(void) const
 471 {
 472         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 473 }
 474
 475
 476 /*
 477  *  Implementations of series expansion
 478  */
 479
 480 /** Default implementation of ex::series(). This performs Taylor expansion.
 481  *  @see ex::series */
 482 ex basic::series(const relational & r, int order, unsigned options) const
 483 {
 484         epvector seq;
 485         numeric fac = 1;
 486         ex deriv = *this;
 487         ex coeff = deriv.subs(r);
 488         const symbol &s = ex_to<symbol>(r.lhs());
 489
 490         if (!coeff.is_zero())
 491                 seq.push_back(expair(coeff, _ex0));
 492
 493         int n;
 494         for (n=1; n<order; ++n) {
 495                 fac = fac.mul(n);
 496                 // We need to test for zero in order to see if the series terminates.
 497                 // The problem is that there is no such thing as a perfect test for
 498                 // zero.  Expanding the term occasionally helps a little...
 499                 deriv = deriv.diff(s).expand();
 500                 if (deriv.is_zero())  // Series terminates
 501                         return pseries(r, seq);
 502
 503                 coeff = deriv.subs(r);
 504                 if (!coeff.is_zero())
 505                         seq.push_back(expair(fac.inverse() * coeff, n));
 506         }
 507
 508         // Higher-order terms, if present
 509         deriv = deriv.diff(s);
 510         if (!deriv.expand().is_zero())
 511                 seq.push_back(expair(Order(_ex1), n));
 512         return pseries(r, seq);
 513 }
 514
 515
 516 /** Implementation of ex::series() for symbols.
 517  *  @see ex::series */
 518 ex symbol::series(const relational & r, int order, unsigned options) const
 519 {
 520         epvector seq;
 521         const ex point = r.rhs();
 522         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 523
 524         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 525                 if (order > 0 && !point.is_zero())
 526                         seq.push_back(expair(point, _ex0));
 527                 if (order > 1)
 528                         seq.push_back(expair(_ex1, _ex1));
 529                 else
 530                         seq.push_back(expair(Order(_ex1), numeric(order)));
 531         } else
 532                 seq.push_back(expair(*this, _ex0));
 533         return pseries(r, seq);
 534 }
 535
 536
 537 /** Add one series object to another, producing a pseries object that
 538  *  represents the sum.
 539  *
 540  *  @param other  pseries object to add with
 541  *  @return the sum as a pseries */
 542 ex pseries::add_series(const pseries &other) const
 543 {
 544         // Adding two series with different variables or expansion points
 545         // results in an empty (constant) series
 546         if (!is_compatible_to(other)) {
 547                 epvector nul;
 548                 nul.push_back(expair(Order(_ex1), _ex0));
 549                 return pseries(relational(var,point), nul);
 550         }
 551
 552         // Series addition
 553         epvector new_seq;
 554         epvector::const_iterator a = seq.begin();
 555         epvector::const_iterator b = other.seq.begin();
 556         epvector::const_iterator a_end = seq.end();
 557         epvector::const_iterator b_end = other.seq.end();
 558         int pow_a = INT_MAX, pow_b = INT_MAX;
 559         for (;;) {
 560                 // If a is empty, fill up with elements from b and stop
 561                 if (a == a_end) {
 562                         while (b != b_end) {
 563                                 new_seq.push_back(*b);
 564                                 ++b;
 565                         }
 566                         break;
 567                 } else
 568                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 569
 570                 // If b is empty, fill up with elements from a and stop
 571                 if (b == b_end) {
 572                         while (a != a_end) {
 573                                 new_seq.push_back(*a);
 574                                 ++a;
 575                         }
 576                         break;
 577                 } else
 578                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 579
 580                 // a and b are non-empty, compare powers
 581                 if (pow_a < pow_b) {
 582                         // a has lesser power, get coefficient from a
 583                         new_seq.push_back(*a);
 584                         if (is_order_function((*a).rest))
 585                                 break;
 586                         ++a;
 587                 } else if (pow_b < pow_a) {
 588                         // b has lesser power, get coefficient from b
 589                         new_seq.push_back(*b);
 590                         if (is_order_function((*b).rest))
 591                                 break;
 592                         ++b;
 593                 } else {
 594                         // Add coefficient of a and b
 595                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 596                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 597                                 break;  // Order term ends the sequence
 598                         } else {
 599                                 ex sum = (*a).rest + (*b).rest;
 600                                 if (!(sum.is_zero()))
 601                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 602                                 ++a;
 603                                 ++b;
 604                         }
 605                 }
 606         }
 607         return pseries(relational(var,point), new_seq);
 608 }
 609
 610
 611 /** Implementation of ex::series() for sums. This performs series addition when
 612  *  adding pseries objects.
 613  *  @see ex::series */
 614 ex add::series(const relational & r, int order, unsigned options) const
 615 {
 616         ex acc; // Series accumulator
 617
 618         // Get first term from overall_coeff
 619         acc = overall_coeff.series(r, order, options);
 620
 621         // Add remaining terms
 622         epvector::const_iterator it = seq.begin();
 623         epvector::const_iterator itend = seq.end();
 624         for (; it!=itend; ++it) {
 625                 ex op;
 626                 if (is_ex_exactly_of_type(it->rest, pseries))
 627                         op = it->rest;
 628                 else
 629                         op = it->rest.series(r, order, options);
 630                 if (!it->coeff.is_equal(_ex1))
 631                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 632
 633                 // Series addition
 634                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 635         }
 636         return acc;
 637 }
 638
 639
 640 /** Multiply a pseries object with a numeric constant, producing a pseries
 641  *  object that represents the product.
 642  *
 643  *  @param other  constant to multiply with
 644  *  @return the product as a pseries */
 645 ex pseries::mul_const(const numeric &other) const
 646 {
 647         epvector new_seq;
 648         new_seq.reserve(seq.size());
 649
 650         epvector::const_iterator it = seq.begin(), itend = seq.end();
 651         while (it != itend) {
 652                 if (!is_order_function(it->rest))
 653                         new_seq.push_back(expair(it->rest * other, it->coeff));
 654                 else
 655                         new_seq.push_back(*it);
 656                 ++it;
 657         }
 658         return pseries(relational(var,point), new_seq);
 659 }
 660
 661
 662 /** Multiply one pseries object to another, producing a pseries object that
 663  *  represents the product.
 664  *
 665  *  @param other  pseries object to multiply with
 666  *  @return the product as a pseries */
 667 ex pseries::mul_series(const pseries &other) const
 668 {
 669         // Multiplying two series with different variables or expansion points
 670         // results in an empty (constant) series
 671         if (!is_compatible_to(other)) {
 672                 epvector nul;
 673                 nul.push_back(expair(Order(_ex1), _ex0));
 674                 return pseries(relational(var,point), nul);
 675         }
 676
 677         // Series multiplication
 678         epvector new_seq;
 679         int a_max = degree(var);
 680         int b_max = other.degree(var);
 681         int a_min = ldegree(var);
 682         int b_min = other.ldegree(var);
 683         int cdeg_min = a_min + b_min;
 684         int cdeg_max = a_max + b_max;
 685
 686         int higher_order_a = INT_MAX;
 687         int higher_order_b = INT_MAX;
 688         if (is_order_function(coeff(var, a_max)))
 689                 higher_order_a = a_max + b_min;
 690         if (is_order_function(other.coeff(var, b_max)))
 691                 higher_order_b = b_max + a_min;
 692         int higher_order_c = std::min(higher_order_a, higher_order_b);
 693         if (cdeg_max >= higher_order_c)
 694                 cdeg_max = higher_order_c - 1;
 695
 696         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 697                 ex co = _ex0;
 698                 // c(i)=a(0)b(i)+...+a(i)b(0)
 699                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 700                         ex a_coeff = coeff(var, i);
 701                         ex b_coeff = other.coeff(var, cdeg-i);
 702                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 703                                 co += a_coeff * b_coeff;
 704                 }
 705                 if (!co.is_zero())
 706                         new_seq.push_back(expair(co, numeric(cdeg)));
 707         }
 708         if (higher_order_c < INT_MAX)
 709                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 710         return pseries(relational(var, point), new_seq);
 711 }
 712
 713
 714 /** Implementation of ex::series() for product. This performs series
 715  *  multiplication when multiplying series.
 716  *  @see ex::series */
 717 ex mul::series(const relational & r, int order, unsigned options) const
 718 {
 719         pseries acc; // Series accumulator
 720
 721         // Multiply with remaining terms
 722         const epvector::const_iterator itbeg = seq.begin();
 723         const epvector::const_iterator itend = seq.end();
 724         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 725                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 726
 727                 // Series multiplication
 728                 if (it==itbeg)
 729                         acc = ex_to<pseries>(op);
 730                 else
 731                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 732         }
 733         return acc.mul_const(ex_to<numeric>(overall_coeff));
 734 }
 735
 736
 737 /** Compute the p-th power of a series.
 738  *
 739  *  @param p  power to compute
 740  *  @param deg  truncation order of series calculation */
 741 ex pseries::power_const(const numeric &p, int deg) const
 742 {
 743         // method:
 744         // (due to Leonhard Euler)
 745         // let A(x) be this series and for the time being let it start with a
 746         // constant (later we'll generalize):
 747         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 748         // We want to compute
 749         //     C(x) = A(x)^p
 750         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 751         // Taking the derivative on both sides and multiplying with A(x) one
 752         // immediately arrives at
 753         //     C'(x)*A(x) = p*C(x)*A'(x)
 754         // Multiplying this out and comparing coefficients we get the recurrence
 755         // formula
 756         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 757         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 758         // which can easily be solved given the starting value c_0 = (a_0)^p.
 759         // For the more general case where the leading coefficient of A(x) is not
 760         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 761         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 762         // then of course x^(p*m) but the recurrence formula still holds.
 763
 764         if (seq.empty()) {
 765                 // as a special case, handle the empty (zero) series honoring the
 766                 // usual power laws such as implemented in power::eval()
 767                 if (p.real().is_zero())
 768                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 769                 else if (p.real().is_negative())
 770                         throw pole_error("pseries::power_const(): division by zero",1);
 771                 else
 772                         return *this;
 773         }
 774
 775         const int ldeg = ldegree(var);
 776         if (!(p*ldeg).is_integer())
 777                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 778
 779         // O(x^n)^(-m) is undefined
 780         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 781                 throw pole_error("pseries::power_const(): division by zero",1);
 782
 783         // Compute coefficients of the powered series
 784         exvector co;
 785         co.reserve(deg);
 786         co.push_back(power(coeff(var, ldeg), p));
 787         bool all_sums_zero = true;
 788         for (int i=1; i<deg; ++i) {
 789                 ex sum = _ex0;
 790                 for (int j=1; j<=i; ++j) {
 791                         ex c = coeff(var, j + ldeg);
 792                         if (is_order_function(c)) {
 793                                 co.push_back(Order(_ex1));
 794                                 break;
 795                         } else
 796                                 sum += (p * j - (i - j)) * co[i - j] * c;
 797                 }
 798                 if (!sum.is_zero())
 799                         all_sums_zero = false;
 800                 co.push_back(sum / coeff(var, ldeg) / i);
 801         }
 802
 803         // Construct new series (of non-zero coefficients)
 804         epvector new_seq;
 805         bool higher_order = false;
 806         for (int i=0; i<deg; ++i) {
 807                 if (!co[i].is_zero())
 808                         new_seq.push_back(expair(co[i], p * ldeg + i));
 809                 if (is_order_function(co[i])) {
 810                         higher_order = true;
 811                         break;
 812                 }
 813         }
 814         if (!higher_order && !all_sums_zero)
 815                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 816         return pseries(relational(var,point), new_seq);
 817 }
 818
 819
 820 /** Return a new pseries object with the powers shifted by deg. */
 821 pseries pseries::shift_exponents(int deg) const
 822 {
 823         epvector newseq = seq;
 824         epvector::iterator i = newseq.begin(), end  = newseq.end();
 825         while (i != end) {
 826                 i->coeff += deg;
 827                 ++i;
 828         }
 829         return pseries(relational(var, point), newseq);
 830 }
 831
 832
 833 /** Implementation of ex::series() for powers. This performs Laurent expansion
 834  *  of reciprocals of series at singularities.
 835  *  @see ex::series */
 836 ex power::series(const relational & r, int order, unsigned options) const
 837 {
 838         // If basis is already a series, just power it
 839         if (is_ex_exactly_of_type(basis, pseries))
 840                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 841
 842         // Basis is not a series, may there be a singularity?
 843         bool must_expand_basis = false;
 844         try {
 845                 basis.subs(r);
 846         } catch (pole_error) {
 847                 must_expand_basis = true;
 848         }
 849
 850         // Is the expression of type something^(-int)?
 851         if (!must_expand_basis && !exponent.info(info_flags::negint))
 852                 return basic::series(r, order, options);
 853
 854         // Is the expression of type 0^something?
 855         if (!must_expand_basis && !basis.subs(r).is_zero())
 856                 return basic::series(r, order, options);
 857
 858         // Singularity encountered, is the basis equal to (var - point)?
 859         if (basis.is_equal(r.lhs() - r.rhs())) {
 860                 epvector new_seq;
 861                 if (ex_to<numeric>(exponent).to_int() < order)
 862                         new_seq.push_back(expair(_ex1, exponent));
 863                 else
 864                         new_seq.push_back(expair(Order(_ex1), exponent));
 865                 return pseries(r, new_seq);
 866         }
 867
 868         // No, expand basis into series
 869         ex e = basis.series(r, order, options);
 870         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 871 }
 872
 873
 874 /** Re-expansion of a pseries object. */
 875 ex pseries::series(const relational & r, int order, unsigned options) const
 876 {
 877         const ex p = r.rhs();
 878         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 879         const symbol &s = ex_to<symbol>(r.lhs());
 880
 881         if (var.is_equal(s) && point.is_equal(p)) {
 882                 if (order > degree(s))
 883                         return *this;
 884                 else {
 885                         epvector new_seq;
 886                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 887                         while (it != itend) {
 888                                 int o = ex_to<numeric>(it->coeff).to_int();
 889                                 if (o >= order) {
 890                                         new_seq.push_back(expair(Order(_ex1), o));
 891                                         break;
 892                                 }
 893                                 new_seq.push_back(*it);
 894                                 ++it;
 895                         }
 896                         return pseries(r, new_seq);
 897                 }
 898         } else
 899                 return convert_to_poly().series(r, order, options);
 900 }
 901
 902
 903 /** Compute the truncated series expansion of an expression.
 904  *  This function returns an expression containing an object of class pseries
 905  *  to represent the series. If the series does not terminate within the given
 906  *  truncation order, the last term of the series will be an order term.
 907  *
 908  *  @param r  expansion relation, lhs holds variable and rhs holds point
 909  *  @param order  truncation order of series calculations
 910  *  @param options  of class series_options
 911  *  @return an expression holding a pseries object */
 912 ex ex::series(const ex & r, int order, unsigned options) const
 913 {
 914         GINAC_ASSERT(bp!=0);
 915         ex e;
 916         relational rel_;
 917
 918         if (is_ex_exactly_of_type(r,relational))
 919                 rel_ = ex_to<relational>(r);
 920         else if (is_ex_exactly_of_type(r,symbol))
 921                 rel_ = relational(r,_ex0);
 922         else
 923                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 924
 925         try {
 926                 e = bp->series(rel_, order, options);
 927         } catch (std::exception &x) {
 928                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 929         }
 930         return e;
 931 }
 932
 933 } // namespace GiNaC