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) 1999-2004 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 "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
  42   print_func<print_context>(&pseries::do_print).
  43   print_func<print_latex>(&pseries::do_print_latex).
  44   print_func<print_tree>(&pseries::do_print_tree).
  45   print_func<print_python>(&pseries::do_print_python).
  46   print_func<print_python_repr>(&pseries::do_print_python_repr))
  47
  48
  49 /*
  50  *  Default constructor
  51  */
  52
  53 pseries::pseries() : inherited(TINFO_pseries) { }
  54
  55
  56 /*
  57  *  Other ctors
  58  */
  59
  60 /** Construct pseries from a vector of coefficients and powers.
  61  *  expair.rest holds the coefficient, expair.coeff holds the power.
  62  *  The powers must be integers (positive or negative) and in ascending order;
  63  *  the last coefficient can be Order(_ex1) to represent a truncated,
  64  *  non-terminating series.
  65  *
  66  *  @param rel_  expansion variable and point (must hold a relational)
  67  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  68  *  @return newly constructed pseries */
  69 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  70 {
  71         GINAC_ASSERT(is_a<relational>(rel_));
  72         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  73         point = rel_.rhs();
  74         var = rel_.lhs();
  75 }
  76
  77
  78 /*
  79  *  Archiving
  80  */
  81
  82 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  83 {
  84         for (unsigned int i=0; true; ++i) {
  85                 ex rest;
  86                 ex coeff;
  87                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  88                         seq.push_back(expair(rest, coeff));
  89                 else
  90                         break;
  91         }
  92         n.find_ex("var", var, sym_lst);
  93         n.find_ex("point", point, sym_lst);
  94 }
  95
  96 void pseries::archive(archive_node &n) const
  97 {
  98         inherited::archive(n);
  99         epvector::const_iterator i = seq.begin(), iend = seq.end();
 100         while (i != iend) {
 101                 n.add_ex("coeff", i->rest);
 102                 n.add_ex("power", i->coeff);
 103                 ++i;
 104         }
 105         n.add_ex("var", var);
 106         n.add_ex("point", point);
 107 }
 108
 109 DEFAULT_UNARCHIVE(pseries)
 110
 111 //////////
 112 // functions overriding virtual functions from base classes
 113 //////////
 114
 115 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
 116 {
 117         if (precedence() <= level)
 118                 c.s << '(';
 119
 120         // objects of type pseries must not have any zero entries, so the
 121         // trivial (zero) pseries needs a special treatment here:
 122         if (seq.empty())
 123                 c.s << '0';
 124
 125         epvector::const_iterator i = seq.begin(), end = seq.end();
 126         while (i != end) {
 127
 128                 // print a sign, if needed
 129                 if (i != seq.begin())
 130                         c.s << '+';
 131
 132                 if (!is_order_function(i->rest)) {
 133
 134                         // print 'rest', i.e. the expansion coefficient
 135                         if (i->rest.info(info_flags::numeric) &&
 136                                 i->rest.info(info_flags::positive)) {
 137                                 i->rest.print(c);
 138                         } else {
 139                                 c.s << openbrace << '(';
 140                                 i->rest.print(c);
 141                                 c.s << ')' << closebrace;
 142                         }
 143
 144                         // print 'coeff', something like (x-1)^42
 145                         if (!i->coeff.is_zero()) {
 146                                 c.s << mul_sym;
 147                                 if (!point.is_zero()) {
 148                                         c.s << openbrace << '(';
 149                                         (var-point).print(c);
 150                                         c.s << ')' << closebrace;
 151                                 } else
 152                                         var.print(c);
 153                                 if (i->coeff.compare(_ex1)) {
 154                                         c.s << pow_sym;
 155                                         c.s << openbrace;
 156                                         if (i->coeff.info(info_flags::negative)) {
 157                                                 c.s << '(';
 158                                                 i->coeff.print(c);
 159                                                 c.s << ')';
 160                                         } else
 161                                                 i->coeff.print(c);
 162                                         c.s << closebrace;
 163                                 }
 164                         }
 165                 } else
 166                         Order(power(var-point,i->coeff)).print(c);
 167                 ++i;
 168         }
 169
 170         if (precedence() <= level)
 171                 c.s << ')';
 172 }
 173
 174 void pseries::do_print(const print_context & c, unsigned level) const
 175 {
 176         print_series(c, "", "", "*", "^", level);
 177 }
 178
 179 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 180 {
 181         print_series(c, "{", "}", " ", "^", level);
 182 }
 183
 184 void pseries::do_print_python(const print_python & c, unsigned level) const
 185 {
 186         print_series(c, "", "", "*", "**", level);
 187 }
 188
 189 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 190 {
 191         c.s << std::string(level, ' ') << class_name() << " @" << this
 192             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 193             << std::endl;
 194         size_t num = seq.size();
 195         for (size_t i=0; i<num; ++i) {
 196                 seq[i].rest.print(c, level + c.delta_indent);
 197                 seq[i].coeff.print(c, level + c.delta_indent);
 198                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 199         }
 200         var.print(c, level + c.delta_indent);
 201         point.print(c, level + c.delta_indent);
 202 }
 203
 204 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 205 {
 206         c.s << class_name() << "(relational(";
 207         var.print(c);
 208         c.s << ',';
 209         point.print(c);
 210         c.s << "),[";
 211         size_t num = seq.size();
 212         for (size_t i=0; i<num; ++i) {
 213                 if (i)
 214                         c.s << ',';
 215                 c.s << '(';
 216                 seq[i].rest.print(c);
 217                 c.s << ',';
 218                 seq[i].coeff.print(c);
 219                 c.s << ')';
 220         }
 221         c.s << "])";
 222 }
 223
 224 int pseries::compare_same_type(const basic & other) const
 225 {
 226         GINAC_ASSERT(is_a<pseries>(other));
 227         const pseries &o = static_cast<const pseries &>(other);
 228
 229         // first compare the lengths of the series...
 230         if (seq.size()>o.seq.size())
 231                 return 1;
 232         if (seq.size()<o.seq.size())
 233                 return -1;
 234
 235         // ...then the expansion point...
 236         int cmpval = var.compare(o.var);
 237         if (cmpval)
 238                 return cmpval;
 239         cmpval = point.compare(o.point);
 240         if (cmpval)
 241                 return cmpval;
 242
 243         // ...and if that failed the individual elements
 244         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 245         while (it!=seq.end() && o_it!=o.seq.end()) {
 246                 cmpval = it->compare(*o_it);
 247                 if (cmpval)
 248                         return cmpval;
 249                 ++it;
 250                 ++o_it;
 251         }
 252
 253         // so they are equal.
 254         return 0;
 255 }
 256
 257 /** Return the number of operands including a possible order term. */
 258 size_t pseries::nops() const
 259 {
 260         return seq.size();
 261 }
 262
 263 /** Return the ith term in the series when represented as a sum. */
 264 ex pseries::op(size_t i) const
 265 {
 266         if (i >= seq.size())
 267                 throw (std::out_of_range("op() out of range"));
 268
 269         return seq[i].rest * power(var - point, seq[i].coeff);
 270 }
 271
 272 /** Return degree of highest power of the series.  This is usually the exponent
 273  *  of the Order term.  If s is not the expansion variable of the series, the
 274  *  series is examined termwise. */
 275 int pseries::degree(const ex &s) const
 276 {
 277         if (var.is_equal(s)) {
 278                 // Return last exponent
 279                 if (seq.size())
 280                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 281                 else
 282                         return 0;
 283         } else {
 284                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 285                 if (it == itend)
 286                         return 0;
 287                 int max_pow = INT_MIN;
 288                 while (it != itend) {
 289                         int pow = it->rest.degree(s);
 290                         if (pow > max_pow)
 291                                 max_pow = pow;
 292                         ++it;
 293                 }
 294                 return max_pow;
 295         }
 296 }
 297
 298 /** Return degree of lowest power of the series.  This is usually the exponent
 299  *  of the leading term.  If s is not the expansion variable of the series, the
 300  *  series is examined termwise.  If s is the expansion variable but the
 301  *  expansion point is not zero the series is not expanded to find the degree.
 302  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 303 int pseries::ldegree(const ex &s) const
 304 {
 305         if (var.is_equal(s)) {
 306                 // Return first exponent
 307                 if (seq.size())
 308                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 309                 else
 310                         return 0;
 311         } else {
 312                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 313                 if (it == itend)
 314                         return 0;
 315                 int min_pow = INT_MAX;
 316                 while (it != itend) {
 317                         int pow = it->rest.ldegree(s);
 318                         if (pow < min_pow)
 319                                 min_pow = pow;
 320                         ++it;
 321                 }
 322                 return min_pow;
 323         }
 324 }
 325
 326 /** Return coefficient of degree n in power series if s is the expansion
 327  *  variable.  If the expansion point is nonzero, by definition the n=1
 328  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 329  *  the expansion took place in the s in the first place).
 330  *  If s is not the expansion variable, an attempt is made to convert the
 331  *  series to a polynomial and return the corresponding coefficient from
 332  *  there. */
 333 ex pseries::coeff(const ex &s, int n) const
 334 {
 335         if (var.is_equal(s)) {
 336                 if (seq.empty())
 337                         return _ex0;
 338
 339                 // Binary search in sequence for given power
 340                 numeric looking_for = numeric(n);
 341                 int lo = 0, hi = seq.size() - 1;
 342                 while (lo <= hi) {
 343                         int mid = (lo + hi) / 2;
 344                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 345                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 346                         switch (cmp) {
 347                                 case -1:
 348                                         lo = mid + 1;
 349                                         break;
 350                                 case 0:
 351                                         return seq[mid].rest;
 352                                 case 1:
 353                                         hi = mid - 1;
 354                                         break;
 355                                 default:
 356                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 357                         }
 358                 }
 359                 return _ex0;
 360         } else
 361                 return convert_to_poly().coeff(s, n);
 362 }
 363
 364 /** Does nothing. */
 365 ex pseries::collect(const ex &s, bool distributed) const
 366 {
 367         return *this;
 368 }
 369
 370 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 371 ex pseries::eval(int level) const
 372 {
 373         if (level == 1)
 374                 return this->hold();
 375
 376         if (level == -max_recursion_level)
 377                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 378
 379         // Construct a new series with evaluated coefficients
 380         epvector new_seq;
 381         new_seq.reserve(seq.size());
 382         epvector::const_iterator it = seq.begin(), itend = seq.end();
 383         while (it != itend) {
 384                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 385                 ++it;
 386         }
 387         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 388 }
 389
 390 /** Evaluate coefficients numerically. */
 391 ex pseries::evalf(int level) const
 392 {
 393         if (level == 1)
 394                 return *this;
 395
 396         if (level == -max_recursion_level)
 397                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 398
 399         // Construct a new series with evaluated coefficients
 400         epvector new_seq;
 401         new_seq.reserve(seq.size());
 402         epvector::const_iterator it = seq.begin(), itend = seq.end();
 403         while (it != itend) {
 404                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 405                 ++it;
 406         }
 407         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 408 }
 409
 410 ex pseries::conjugate() const
 411 {
 412         epvector * newseq = conjugateepvector(seq);
 413         ex newvar = var.conjugate();
 414         ex newpoint = point.conjugate();
 415
 416         if (!newseq     && are_ex_trivially_equal(newvar, var) && are_ex_trivially_equal(point, newpoint)) {
 417                 return *this;
 418         }
 419
 420         ex result = (new pseries(newvar==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
 421         if (newseq) {
 422                 delete newseq;
 423         }
 424         return result;
 425 }
 426
 427 ex pseries::subs(const exmap & m, unsigned options) const
 428 {
 429         // If expansion variable is being substituted, convert the series to a
 430         // polynomial and do the substitution there because the result might
 431         // no longer be a power series
 432         if (m.find(var) != m.end())
 433                 return convert_to_poly(true).subs(m, options);
 434
 435         // Otherwise construct a new series with substituted coefficients and
 436         // expansion point
 437         epvector newseq;
 438         newseq.reserve(seq.size());
 439         epvector::const_iterator it = seq.begin(), itend = seq.end();
 440         while (it != itend) {
 441                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
 442                 ++it;
 443         }
 444         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
 445 }
 446
 447 /** Implementation of ex::expand() for a power series.  It expands all the
 448  *  terms individually and returns the resulting series as a new pseries. */
 449 ex pseries::expand(unsigned options) const
 450 {
 451         epvector newseq;
 452         epvector::const_iterator i = seq.begin(), end = seq.end();
 453         while (i != end) {
 454                 ex restexp = i->rest.expand();
 455                 if (!restexp.is_zero())
 456                         newseq.push_back(expair(restexp, i->coeff));
 457                 ++i;
 458         }
 459         return (new pseries(relational(var,point), newseq))
 460                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 461 }
 462
 463 /** Implementation of ex::diff() for a power series.
 464  *  @see ex::diff */
 465 ex pseries::derivative(const symbol & s) const
 466 {
 467         epvector new_seq;
 468         epvector::const_iterator it = seq.begin(), itend = seq.end();
 469
 470         if (s == var) {
 471
 472                 // FIXME: coeff might depend on var
 473                 while (it != itend) {
 474                         if (is_order_function(it->rest)) {
 475                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 476                         } else {
 477                                 ex c = it->rest * it->coeff;
 478                                 if (!c.is_zero())
 479                                         new_seq.push_back(expair(c, it->coeff - 1));
 480                         }
 481                         ++it;
 482                 }
 483
 484         } else {
 485
 486                 while (it != itend) {
 487                         if (is_order_function(it->rest)) {
 488                                 new_seq.push_back(*it);
 489                         } else {
 490                                 ex c = it->rest.diff(s);
 491                                 if (!c.is_zero())
 492                                         new_seq.push_back(expair(c, it->coeff));
 493                         }
 494                         ++it;
 495                 }
 496         }
 497
 498         return pseries(relational(var,point), new_seq);
 499 }
 500
 501 ex pseries::convert_to_poly(bool no_order) const
 502 {
 503         ex e;
 504         epvector::const_iterator it = seq.begin(), itend = seq.end();
 505
 506         while (it != itend) {
 507                 if (is_order_function(it->rest)) {
 508                         if (!no_order)
 509                                 e += Order(power(var - point, it->coeff));
 510                 } else
 511                         e += it->rest * power(var - point, it->coeff);
 512                 ++it;
 513         }
 514         return e;
 515 }
 516
 517 bool pseries::is_terminating() const
 518 {
 519         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 520 }
 521
 522
 523 /*
 524  *  Implementations of series expansion
 525  */
 526
 527 /** Default implementation of ex::series(). This performs Taylor expansion.
 528  *  @see ex::series */
 529 ex basic::series(const relational & r, int order, unsigned options) const
 530 {
 531         epvector seq;
 532         numeric fac = 1;
 533         ex deriv = *this;
 534         ex coeff = deriv.subs(r, subs_options::no_pattern);
 535         const symbol &s = ex_to<symbol>(r.lhs());
 536
 537         if (!coeff.is_zero())
 538                 seq.push_back(expair(coeff, _ex0));
 539
 540         int n;
 541         for (n=1; n<order; ++n) {
 542                 fac = fac.mul(n);
 543                 // We need to test for zero in order to see if the series terminates.
 544                 // The problem is that there is no such thing as a perfect test for
 545                 // zero.  Expanding the term occasionally helps a little...
 546                 deriv = deriv.diff(s).expand();
 547                 if (deriv.is_zero())  // Series terminates
 548                         return pseries(r, seq);
 549
 550                 coeff = deriv.subs(r, subs_options::no_pattern);
 551                 if (!coeff.is_zero())
 552                         seq.push_back(expair(fac.inverse() * coeff, n));
 553         }
 554
 555         // Higher-order terms, if present
 556         deriv = deriv.diff(s);
 557         if (!deriv.expand().is_zero())
 558                 seq.push_back(expair(Order(_ex1), n));
 559         return pseries(r, seq);
 560 }
 561
 562
 563 /** Implementation of ex::series() for symbols.
 564  *  @see ex::series */
 565 ex symbol::series(const relational & r, int order, unsigned options) const
 566 {
 567         epvector seq;
 568         const ex point = r.rhs();
 569         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 570
 571         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 572                 if (order > 0 && !point.is_zero())
 573                         seq.push_back(expair(point, _ex0));
 574                 if (order > 1)
 575                         seq.push_back(expair(_ex1, _ex1));
 576                 else
 577                         seq.push_back(expair(Order(_ex1), numeric(order)));
 578         } else
 579                 seq.push_back(expair(*this, _ex0));
 580         return pseries(r, seq);
 581 }
 582
 583
 584 /** Add one series object to another, producing a pseries object that
 585  *  represents the sum.
 586  *
 587  *  @param other  pseries object to add with
 588  *  @return the sum as a pseries */
 589 ex pseries::add_series(const pseries &other) const
 590 {
 591         // Adding two series with different variables or expansion points
 592         // results in an empty (constant) series
 593         if (!is_compatible_to(other)) {
 594                 epvector nul;
 595                 nul.push_back(expair(Order(_ex1), _ex0));
 596                 return pseries(relational(var,point), nul);
 597         }
 598
 599         // Series addition
 600         epvector new_seq;
 601         epvector::const_iterator a = seq.begin();
 602         epvector::const_iterator b = other.seq.begin();
 603         epvector::const_iterator a_end = seq.end();
 604         epvector::const_iterator b_end = other.seq.end();
 605         int pow_a = INT_MAX, pow_b = INT_MAX;
 606         for (;;) {
 607                 // If a is empty, fill up with elements from b and stop
 608                 if (a == a_end) {
 609                         while (b != b_end) {
 610                                 new_seq.push_back(*b);
 611                                 ++b;
 612                         }
 613                         break;
 614                 } else
 615                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 616
 617                 // If b is empty, fill up with elements from a and stop
 618                 if (b == b_end) {
 619                         while (a != a_end) {
 620                                 new_seq.push_back(*a);
 621                                 ++a;
 622                         }
 623                         break;
 624                 } else
 625                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 626
 627                 // a and b are non-empty, compare powers
 628                 if (pow_a < pow_b) {
 629                         // a has lesser power, get coefficient from a
 630                         new_seq.push_back(*a);
 631                         if (is_order_function((*a).rest))
 632                                 break;
 633                         ++a;
 634                 } else if (pow_b < pow_a) {
 635                         // b has lesser power, get coefficient from b
 636                         new_seq.push_back(*b);
 637                         if (is_order_function((*b).rest))
 638                                 break;
 639                         ++b;
 640                 } else {
 641                         // Add coefficient of a and b
 642                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 643                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 644                                 break;  // Order term ends the sequence
 645                         } else {
 646                                 ex sum = (*a).rest + (*b).rest;
 647                                 if (!(sum.is_zero()))
 648                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 649                                 ++a;
 650                                 ++b;
 651                         }
 652                 }
 653         }
 654         return pseries(relational(var,point), new_seq);
 655 }
 656
 657
 658 /** Implementation of ex::series() for sums. This performs series addition when
 659  *  adding pseries objects.
 660  *  @see ex::series */
 661 ex add::series(const relational & r, int order, unsigned options) const
 662 {
 663         ex acc; // Series accumulator
 664
 665         // Get first term from overall_coeff
 666         acc = overall_coeff.series(r, order, options);
 667
 668         // Add remaining terms
 669         epvector::const_iterator it = seq.begin();
 670         epvector::const_iterator itend = seq.end();
 671         for (; it!=itend; ++it) {
 672                 ex op;
 673                 if (is_exactly_a<pseries>(it->rest))
 674                         op = it->rest;
 675                 else
 676                         op = it->rest.series(r, order, options);
 677                 if (!it->coeff.is_equal(_ex1))
 678                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 679
 680                 // Series addition
 681                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 682         }
 683         return acc;
 684 }
 685
 686
 687 /** Multiply a pseries object with a numeric constant, producing a pseries
 688  *  object that represents the product.
 689  *
 690  *  @param other  constant to multiply with
 691  *  @return the product as a pseries */
 692 ex pseries::mul_const(const numeric &other) const
 693 {
 694         epvector new_seq;
 695         new_seq.reserve(seq.size());
 696
 697         epvector::const_iterator it = seq.begin(), itend = seq.end();
 698         while (it != itend) {
 699                 if (!is_order_function(it->rest))
 700                         new_seq.push_back(expair(it->rest * other, it->coeff));
 701                 else
 702                         new_seq.push_back(*it);
 703                 ++it;
 704         }
 705         return pseries(relational(var,point), new_seq);
 706 }
 707
 708
 709 /** Multiply one pseries object to another, producing a pseries object that
 710  *  represents the product.
 711  *
 712  *  @param other  pseries object to multiply with
 713  *  @return the product as a pseries */
 714 ex pseries::mul_series(const pseries &other) const
 715 {
 716         // Multiplying two series with different variables or expansion points
 717         // results in an empty (constant) series
 718         if (!is_compatible_to(other)) {
 719                 epvector nul;
 720                 nul.push_back(expair(Order(_ex1), _ex0));
 721                 return pseries(relational(var,point), nul);
 722         }
 723
 724         // Series multiplication
 725         epvector new_seq;
 726         int a_max = degree(var);
 727         int b_max = other.degree(var);
 728         int a_min = ldegree(var);
 729         int b_min = other.ldegree(var);
 730         int cdeg_min = a_min + b_min;
 731         int cdeg_max = a_max + b_max;
 732
 733         int higher_order_a = INT_MAX;
 734         int higher_order_b = INT_MAX;
 735         if (is_order_function(coeff(var, a_max)))
 736                 higher_order_a = a_max + b_min;
 737         if (is_order_function(other.coeff(var, b_max)))
 738                 higher_order_b = b_max + a_min;
 739         int higher_order_c = std::min(higher_order_a, higher_order_b);
 740         if (cdeg_max >= higher_order_c)
 741                 cdeg_max = higher_order_c - 1;
 742
 743         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 744                 ex co = _ex0;
 745                 // c(i)=a(0)b(i)+...+a(i)b(0)
 746                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 747                         ex a_coeff = coeff(var, i);
 748                         ex b_coeff = other.coeff(var, cdeg-i);
 749                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 750                                 co += a_coeff * b_coeff;
 751                 }
 752                 if (!co.is_zero())
 753                         new_seq.push_back(expair(co, numeric(cdeg)));
 754         }
 755         if (higher_order_c < INT_MAX)
 756                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 757         return pseries(relational(var, point), new_seq);
 758 }
 759
 760
 761 /** Implementation of ex::series() for product. This performs series
 762  *  multiplication when multiplying series.
 763  *  @see ex::series */
 764 ex mul::series(const relational & r, int order, unsigned options) const
 765 {
 766         pseries acc; // Series accumulator
 767
 768         // Multiply with remaining terms
 769         const epvector::const_iterator itbeg = seq.begin();
 770         const epvector::const_iterator itend = seq.end();
 771         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 772                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 773
 774                 // Series multiplication
 775                 if (it==itbeg)
 776                         acc = ex_to<pseries>(op);
 777                 else
 778                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 779         }
 780         return acc.mul_const(ex_to<numeric>(overall_coeff));
 781 }
 782
 783
 784 /** Compute the p-th power of a series.
 785  *
 786  *  @param p  power to compute
 787  *  @param deg  truncation order of series calculation */
 788 ex pseries::power_const(const numeric &p, int deg) const
 789 {
 790         // method:
 791         // (due to Leonhard Euler)
 792         // let A(x) be this series and for the time being let it start with a
 793         // constant (later we'll generalize):
 794         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 795         // We want to compute
 796         //     C(x) = A(x)^p
 797         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 798         // Taking the derivative on both sides and multiplying with A(x) one
 799         // immediately arrives at
 800         //     C'(x)*A(x) = p*C(x)*A'(x)
 801         // Multiplying this out and comparing coefficients we get the recurrence
 802         // formula
 803         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 804         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 805         // which can easily be solved given the starting value c_0 = (a_0)^p.
 806         // For the more general case where the leading coefficient of A(x) is not
 807         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 808         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 809         // then of course x^(p*m) but the recurrence formula still holds.
 810
 811         if (seq.empty()) {
 812                 // as a special case, handle the empty (zero) series honoring the
 813                 // usual power laws such as implemented in power::eval()
 814                 if (p.real().is_zero())
 815                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 816                 else if (p.real().is_negative())
 817                         throw pole_error("pseries::power_const(): division by zero",1);
 818                 else
 819                         return *this;
 820         }
 821
 822         const int ldeg = ldegree(var);
 823         if (!(p*ldeg).is_integer())
 824                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 825
 826         // O(x^n)^(-m) is undefined
 827         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 828                 throw pole_error("pseries::power_const(): division by zero",1);
 829
 830         // Compute coefficients of the powered series
 831         exvector co;
 832         co.reserve(deg);
 833         co.push_back(power(coeff(var, ldeg), p));
 834         bool all_sums_zero = true;
 835         for (int i=1; i<deg; ++i) {
 836                 ex sum = _ex0;
 837                 for (int j=1; j<=i; ++j) {
 838                         ex c = coeff(var, j + ldeg);
 839                         if (is_order_function(c)) {
 840                                 co.push_back(Order(_ex1));
 841                                 break;
 842                         } else
 843                                 sum += (p * j - (i - j)) * co[i - j] * c;
 844                 }
 845                 if (!sum.is_zero())
 846                         all_sums_zero = false;
 847                 co.push_back(sum / coeff(var, ldeg) / i);
 848         }
 849
 850         // Construct new series (of non-zero coefficients)
 851         epvector new_seq;
 852         bool higher_order = false;
 853         for (int i=0; i<deg; ++i) {
 854                 if (!co[i].is_zero())
 855                         new_seq.push_back(expair(co[i], p * ldeg + i));
 856                 if (is_order_function(co[i])) {
 857                         higher_order = true;
 858                         break;
 859                 }
 860         }
 861         if (!higher_order && !all_sums_zero)
 862                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 863         return pseries(relational(var,point), new_seq);
 864 }
 865
 866
 867 /** Return a new pseries object with the powers shifted by deg. */
 868 pseries pseries::shift_exponents(int deg) const
 869 {
 870         epvector newseq = seq;
 871         epvector::iterator i = newseq.begin(), end  = newseq.end();
 872         while (i != end) {
 873                 i->coeff += deg;
 874                 ++i;
 875         }
 876         return pseries(relational(var, point), newseq);
 877 }
 878
 879
 880 /** Implementation of ex::series() for powers. This performs Laurent expansion
 881  *  of reciprocals of series at singularities.
 882  *  @see ex::series */
 883 ex power::series(const relational & r, int order, unsigned options) const
 884 {
 885         // If basis is already a series, just power it
 886         if (is_exactly_a<pseries>(basis))
 887                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 888
 889         // Basis is not a series, may there be a singularity?
 890         bool must_expand_basis = false;
 891         try {
 892                 basis.subs(r, subs_options::no_pattern);
 893         } catch (pole_error) {
 894                 must_expand_basis = true;
 895         }
 896
 897         // Is the expression of type something^(-int)?
 898         if (!must_expand_basis && !exponent.info(info_flags::negint))
 899                 return basic::series(r, order, options);
 900
 901         // Is the expression of type 0^something?
 902         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero())
 903                 return basic::series(r, order, options);
 904
 905         // Singularity encountered, is the basis equal to (var - point)?
 906         if (basis.is_equal(r.lhs() - r.rhs())) {
 907                 epvector new_seq;
 908                 if (ex_to<numeric>(exponent).to_int() < order)
 909                         new_seq.push_back(expair(_ex1, exponent));
 910                 else
 911                         new_seq.push_back(expair(Order(_ex1), exponent));
 912                 return pseries(r, new_seq);
 913         }
 914
 915         // No, expand basis into series
 916         ex e = basis.series(r, order, options);
 917         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 918 }
 919
 920
 921 /** Re-expansion of a pseries object. */
 922 ex pseries::series(const relational & r, int order, unsigned options) const
 923 {
 924         const ex p = r.rhs();
 925         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 926         const symbol &s = ex_to<symbol>(r.lhs());
 927
 928         if (var.is_equal(s) && point.is_equal(p)) {
 929                 if (order > degree(s))
 930                         return *this;
 931                 else {
 932                         epvector new_seq;
 933                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 934                         while (it != itend) {
 935                                 int o = ex_to<numeric>(it->coeff).to_int();
 936                                 if (o >= order) {
 937                                         new_seq.push_back(expair(Order(_ex1), o));
 938                                         break;
 939                                 }
 940                                 new_seq.push_back(*it);
 941                                 ++it;
 942                         }
 943                         return pseries(r, new_seq);
 944                 }
 945         } else
 946                 return convert_to_poly().series(r, order, options);
 947 }
 948
 949
 950 /** Compute the truncated series expansion of an expression.
 951  *  This function returns an expression containing an object of class pseries
 952  *  to represent the series. If the series does not terminate within the given
 953  *  truncation order, the last term of the series will be an order term.
 954  *
 955  *  @param r  expansion relation, lhs holds variable and rhs holds point
 956  *  @param order  truncation order of series calculations
 957  *  @param options  of class series_options
 958  *  @return an expression holding a pseries object */
 959 ex ex::series(const ex & r, int order, unsigned options) const
 960 {
 961         ex e;
 962         relational rel_;
 963
 964         if (is_a<relational>(r))
 965                 rel_ = ex_to<relational>(r);
 966         else if (is_a<symbol>(r))
 967                 rel_ = relational(r,_ex0);
 968         else
 969                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 970
 971         try {
 972                 e = bp->series(rel_, order, options);
 973         } catch (std::exception &x) {
 974                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 975         }
 976         return e;
 977 }
 978
 979 } // namespace GiNaC