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) 1999-2007 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include <numeric>
  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 "integral.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
  43   print_func<print_context>(&pseries::do_print).
  44   print_func<print_latex>(&pseries::do_print_latex).
  45   print_func<print_tree>(&pseries::do_print_tree).
  46   print_func<print_python>(&pseries::do_print_python).
  47   print_func<print_python_repr>(&pseries::do_print_python_repr))
  48
  49
  50 /*
  51  *  Default constructor
  52  */
  53
  54 pseries::pseries() : inherited(&pseries::tinfo_static) { }
  55
  56
  57 /*
  58  *  Other ctors
  59  */
  60
  61 /** Construct pseries from a vector of coefficients and powers.
  62  *  expair.rest holds the coefficient, expair.coeff holds the power.
  63  *  The powers must be integers (positive or negative) and in ascending order;
  64  *  the last coefficient can be Order(_ex1) to represent a truncated,
  65  *  non-terminating series.
  66  *
  67  *  @param rel_  expansion variable and point (must hold a relational)
  68  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  69  *  @return newly constructed pseries */
  70 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(&pseries::tinfo_static), seq(ops_)
  71 {
  72         GINAC_ASSERT(is_a<relational>(rel_));
  73         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  74         point = rel_.rhs();
  75         var = rel_.lhs();
  76 }
  77
  78
  79 /*
  80  *  Archiving
  81  */
  82
  83 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  84 {
  85         archive_node::archive_node_cit first = n.find_first("coeff");
  86         archive_node::archive_node_cit last = n.find_last("power");
  87         ++last;
  88         seq.reserve((last-first)/2);
  89
  90         for (archive_node::archive_node_cit loc = first; loc < last;) {
  91                 ex rest;
  92                 ex coeff;
  93                 n.find_ex_by_loc(loc++, rest, sym_lst);
  94                 n.find_ex_by_loc(loc++, coeff, sym_lst);
  95                 seq.push_back(expair(rest, coeff));
  96         }
  97
  98         n.find_ex("var", var, sym_lst);
  99         n.find_ex("point", point, sym_lst);
 100 }
 101
 102 void pseries::archive(archive_node &n) const
 103 {
 104         inherited::archive(n);
 105         epvector::const_iterator i = seq.begin(), iend = seq.end();
 106         while (i != iend) {
 107                 n.add_ex("coeff", i->rest);
 108                 n.add_ex("power", i->coeff);
 109                 ++i;
 110         }
 111         n.add_ex("var", var);
 112         n.add_ex("point", point);
 113 }
 114
 115 DEFAULT_UNARCHIVE(pseries)
 116
 117 //////////
 118 // functions overriding virtual functions from base classes
 119 //////////
 120
 121 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
 122 {
 123         if (precedence() <= level)
 124                 c.s << '(';
 125
 126         // objects of type pseries must not have any zero entries, so the
 127         // trivial (zero) pseries needs a special treatment here:
 128         if (seq.empty())
 129                 c.s << '0';
 130
 131         epvector::const_iterator i = seq.begin(), end = seq.end();
 132         while (i != end) {
 133
 134                 // print a sign, if needed
 135                 if (i != seq.begin())
 136                         c.s << '+';
 137
 138                 if (!is_order_function(i->rest)) {
 139
 140                         // print 'rest', i.e. the expansion coefficient
 141                         if (i->rest.info(info_flags::numeric) &&
 142                                 i->rest.info(info_flags::positive)) {
 143                                 i->rest.print(c);
 144                         } else {
 145                                 c.s << openbrace << '(';
 146                                 i->rest.print(c);
 147                                 c.s << ')' << closebrace;
 148                         }
 149
 150                         // print 'coeff', something like (x-1)^42
 151                         if (!i->coeff.is_zero()) {
 152                                 c.s << mul_sym;
 153                                 if (!point.is_zero()) {
 154                                         c.s << openbrace << '(';
 155                                         (var-point).print(c);
 156                                         c.s << ')' << closebrace;
 157                                 } else
 158                                         var.print(c);
 159                                 if (i->coeff.compare(_ex1)) {
 160                                         c.s << pow_sym;
 161                                         c.s << openbrace;
 162                                         if (i->coeff.info(info_flags::negative)) {
 163                                                 c.s << '(';
 164                                                 i->coeff.print(c);
 165                                                 c.s << ')';
 166                                         } else
 167                                                 i->coeff.print(c);
 168                                         c.s << closebrace;
 169                                 }
 170                         }
 171                 } else
 172                         Order(power(var-point,i->coeff)).print(c);
 173                 ++i;
 174         }
 175
 176         if (precedence() <= level)
 177                 c.s << ')';
 178 }
 179
 180 void pseries::do_print(const print_context & c, unsigned level) const
 181 {
 182         print_series(c, "", "", "*", "^", level);
 183 }
 184
 185 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 186 {
 187         print_series(c, "{", "}", " ", "^", level);
 188 }
 189
 190 void pseries::do_print_python(const print_python & c, unsigned level) const
 191 {
 192         print_series(c, "", "", "*", "**", level);
 193 }
 194
 195 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 196 {
 197         c.s << std::string(level, ' ') << class_name() << " @" << this
 198             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 199             << std::endl;
 200         size_t num = seq.size();
 201         for (size_t i=0; i<num; ++i) {
 202                 seq[i].rest.print(c, level + c.delta_indent);
 203                 seq[i].coeff.print(c, level + c.delta_indent);
 204                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 205         }
 206         var.print(c, level + c.delta_indent);
 207         point.print(c, level + c.delta_indent);
 208 }
 209
 210 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 211 {
 212         c.s << class_name() << "(relational(";
 213         var.print(c);
 214         c.s << ',';
 215         point.print(c);
 216         c.s << "),[";
 217         size_t num = seq.size();
 218         for (size_t i=0; i<num; ++i) {
 219                 if (i)
 220                         c.s << ',';
 221                 c.s << '(';
 222                 seq[i].rest.print(c);
 223                 c.s << ',';
 224                 seq[i].coeff.print(c);
 225                 c.s << ')';
 226         }
 227         c.s << "])";
 228 }
 229
 230 int pseries::compare_same_type(const basic & other) const
 231 {
 232         GINAC_ASSERT(is_a<pseries>(other));
 233         const pseries &o = static_cast<const pseries &>(other);
 234
 235         // first compare the lengths of the series...
 236         if (seq.size()>o.seq.size())
 237                 return 1;
 238         if (seq.size()<o.seq.size())
 239                 return -1;
 240
 241         // ...then the expansion point...
 242         int cmpval = var.compare(o.var);
 243         if (cmpval)
 244                 return cmpval;
 245         cmpval = point.compare(o.point);
 246         if (cmpval)
 247                 return cmpval;
 248
 249         // ...and if that failed the individual elements
 250         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 251         while (it!=seq.end() && o_it!=o.seq.end()) {
 252                 cmpval = it->compare(*o_it);
 253                 if (cmpval)
 254                         return cmpval;
 255                 ++it;
 256                 ++o_it;
 257         }
 258
 259         // so they are equal.
 260         return 0;
 261 }
 262
 263 /** Return the number of operands including a possible order term. */
 264 size_t pseries::nops() const
 265 {
 266         return seq.size();
 267 }
 268
 269 /** Return the ith term in the series when represented as a sum. */
 270 ex pseries::op(size_t i) const
 271 {
 272         if (i >= seq.size())
 273                 throw (std::out_of_range("op() out of range"));
 274
 275         if (is_order_function(seq[i].rest))
 276                 return Order(power(var-point, seq[i].coeff));
 277         return seq[i].rest * power(var - point, seq[i].coeff);
 278 }
 279
 280 /** Return degree of highest power of the series.  This is usually the exponent
 281  *  of the Order term.  If s is not the expansion variable of the series, the
 282  *  series is examined termwise. */
 283 int pseries::degree(const ex &s) const
 284 {
 285         if (var.is_equal(s)) {
 286                 // Return last exponent
 287                 if (seq.size())
 288                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 289                 else
 290                         return 0;
 291         } else {
 292                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 293                 if (it == itend)
 294                         return 0;
 295                 int max_pow = INT_MIN;
 296                 while (it != itend) {
 297                         int pow = it->rest.degree(s);
 298                         if (pow > max_pow)
 299                                 max_pow = pow;
 300                         ++it;
 301                 }
 302                 return max_pow;
 303         }
 304 }
 305
 306 /** Return degree of lowest power of the series.  This is usually the exponent
 307  *  of the leading term.  If s is not the expansion variable of the series, the
 308  *  series is examined termwise.  If s is the expansion variable but the
 309  *  expansion point is not zero the series is not expanded to find the degree.
 310  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 311 int pseries::ldegree(const ex &s) const
 312 {
 313         if (var.is_equal(s)) {
 314                 // Return first exponent
 315                 if (seq.size())
 316                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 317                 else
 318                         return 0;
 319         } else {
 320                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 321                 if (it == itend)
 322                         return 0;
 323                 int min_pow = INT_MAX;
 324                 while (it != itend) {
 325                         int pow = it->rest.ldegree(s);
 326                         if (pow < min_pow)
 327                                 min_pow = pow;
 328                         ++it;
 329                 }
 330                 return min_pow;
 331         }
 332 }
 333
 334 /** Return coefficient of degree n in power series if s is the expansion
 335  *  variable.  If the expansion point is nonzero, by definition the n=1
 336  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 337  *  the expansion took place in the s in the first place).
 338  *  If s is not the expansion variable, an attempt is made to convert the
 339  *  series to a polynomial and return the corresponding coefficient from
 340  *  there. */
 341 ex pseries::coeff(const ex &s, int n) const
 342 {
 343         if (var.is_equal(s)) {
 344                 if (seq.empty())
 345                         return _ex0;
 346
 347                 // Binary search in sequence for given power
 348                 numeric looking_for = numeric(n);
 349                 int lo = 0, hi = seq.size() - 1;
 350                 while (lo <= hi) {
 351                         int mid = (lo + hi) / 2;
 352                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 353                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 354                         switch (cmp) {
 355                                 case -1:
 356                                         lo = mid + 1;
 357                                         break;
 358                                 case 0:
 359                                         return seq[mid].rest;
 360                                 case 1:
 361                                         hi = mid - 1;
 362                                         break;
 363                                 default:
 364                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 365                         }
 366                 }
 367                 return _ex0;
 368         } else
 369                 return convert_to_poly().coeff(s, n);
 370 }
 371
 372 /** Does nothing. */
 373 ex pseries::collect(const ex &s, bool distributed) const
 374 {
 375         return *this;
 376 }
 377
 378 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 379 ex pseries::eval(int level) const
 380 {
 381         if (level == 1)
 382                 return this->hold();
 383
 384         if (level == -max_recursion_level)
 385                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 386
 387         // Construct a new series with evaluated coefficients
 388         epvector new_seq;
 389         new_seq.reserve(seq.size());
 390         epvector::const_iterator it = seq.begin(), itend = seq.end();
 391         while (it != itend) {
 392                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 393                 ++it;
 394         }
 395         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 396 }
 397
 398 /** Evaluate coefficients numerically. */
 399 ex pseries::evalf(int level) const
 400 {
 401         if (level == 1)
 402                 return *this;
 403
 404         if (level == -max_recursion_level)
 405                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 406
 407         // Construct a new series with evaluated coefficients
 408         epvector new_seq;
 409         new_seq.reserve(seq.size());
 410         epvector::const_iterator it = seq.begin(), itend = seq.end();
 411         while (it != itend) {
 412                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 413                 ++it;
 414         }
 415         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 416 }
 417
 418 ex pseries::conjugate() const
 419 {
 420         if(!var.info(info_flags::real))
 421                 return conjugate_function(*this).hold();
 422
 423         epvector * newseq = conjugateepvector(seq);
 424         ex newpoint = point.conjugate();
 425
 426         if (!newseq     && are_ex_trivially_equal(point, newpoint)) {
 427                 return *this;
 428         }
 429
 430         ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
 431         if (newseq) {
 432                 delete newseq;
 433         }
 434         return result;
 435 }
 436
 437 ex pseries::real_part() const
 438 {
 439         if(!var.info(info_flags::real))
 440                 return real_part_function(*this).hold();
 441         ex newpoint = point.real_part();
 442         if(newpoint != point)
 443                 return real_part_function(*this).hold();
 444
 445         epvector v;
 446         v.reserve(seq.size());
 447         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 448                 v.push_back(expair((i->rest).real_part(), i->coeff));
 449         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 450 }
 451
 452 ex pseries::imag_part() const
 453 {
 454         if(!var.info(info_flags::real))
 455                 return imag_part_function(*this).hold();
 456         ex newpoint = point.real_part();
 457         if(newpoint != point)
 458                 return imag_part_function(*this).hold();
 459
 460         epvector v;
 461         v.reserve(seq.size());
 462         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 463                 v.push_back(expair((i->rest).imag_part(), i->coeff));
 464         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 465 }
 466
 467 ex pseries::eval_integ() const
 468 {
 469         epvector *newseq = NULL;
 470         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 471                 if (newseq) {
 472                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 473                         continue;
 474                 }
 475                 ex newterm = i->rest.eval_integ();
 476                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 477                         newseq = new epvector;
 478                         newseq->reserve(seq.size());
 479                         for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
 480                                 newseq->push_back(*j);
 481                         newseq->push_back(expair(newterm, i->coeff));
 482                 }
 483         }
 484
 485         ex newpoint = point.eval_integ();
 486         if (newseq || !are_ex_trivially_equal(newpoint, point))
 487                 return (new pseries(var==newpoint, *newseq))
 488                        ->setflag(status_flags::dynallocated);
 489         return *this;
 490 }
 491
 492 ex pseries::evalm() const
 493 {
 494         // evalm each coefficient
 495         epvector newseq;
 496         bool something_changed = false;
 497         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 498                 if (something_changed) {
 499                         ex newcoeff = i->rest.evalm();
 500                         if (!newcoeff.is_zero())
 501                                 newseq.push_back(expair(newcoeff, i->coeff));
 502                 }
 503                 else {
 504                         ex newcoeff = i->rest.evalm();
 505                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 506                                 something_changed = true;
 507                                 newseq.reserve(seq.size());
 508                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 509                                 if (!newcoeff.is_zero())
 510                                         newseq.push_back(expair(newcoeff, i->coeff));
 511                         }
 512                 }
 513         }
 514         if (something_changed)
 515                 return (new pseries(var==point, newseq))->setflag(status_flags::dynallocated);
 516         else
 517                 return *this;
 518 }
 519
 520 ex pseries::subs(const exmap & m, unsigned options) const
 521 {
 522         // If expansion variable is being substituted, convert the series to a
 523         // polynomial and do the substitution there because the result might
 524         // no longer be a power series
 525         if (m.find(var) != m.end())
 526                 return convert_to_poly(true).subs(m, options);
 527
 528         // Otherwise construct a new series with substituted coefficients and
 529         // expansion point
 530         epvector newseq;
 531         newseq.reserve(seq.size());
 532         epvector::const_iterator it = seq.begin(), itend = seq.end();
 533         while (it != itend) {
 534                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
 535                 ++it;
 536         }
 537         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
 538 }
 539
 540 /** Implementation of ex::expand() for a power series.  It expands all the
 541  *  terms individually and returns the resulting series as a new pseries. */
 542 ex pseries::expand(unsigned options) const
 543 {
 544         epvector newseq;
 545         epvector::const_iterator i = seq.begin(), end = seq.end();
 546         while (i != end) {
 547                 ex restexp = i->rest.expand();
 548                 if (!restexp.is_zero())
 549                         newseq.push_back(expair(restexp, i->coeff));
 550                 ++i;
 551         }
 552         return (new pseries(relational(var,point), newseq))
 553                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 554 }
 555
 556 /** Implementation of ex::diff() for a power series.
 557  *  @see ex::diff */
 558 ex pseries::derivative(const symbol & s) const
 559 {
 560         epvector new_seq;
 561         epvector::const_iterator it = seq.begin(), itend = seq.end();
 562
 563         if (s == var) {
 564
 565                 // FIXME: coeff might depend on var
 566                 while (it != itend) {
 567                         if (is_order_function(it->rest)) {
 568                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 569                         } else {
 570                                 ex c = it->rest * it->coeff;
 571                                 if (!c.is_zero())
 572                                         new_seq.push_back(expair(c, it->coeff - 1));
 573                         }
 574                         ++it;
 575                 }
 576
 577         } else {
 578
 579                 while (it != itend) {
 580                         if (is_order_function(it->rest)) {
 581                                 new_seq.push_back(*it);
 582                         } else {
 583                                 ex c = it->rest.diff(s);
 584                                 if (!c.is_zero())
 585                                         new_seq.push_back(expair(c, it->coeff));
 586                         }
 587                         ++it;
 588                 }
 589         }
 590
 591         return pseries(relational(var,point), new_seq);
 592 }
 593
 594 ex pseries::convert_to_poly(bool no_order) const
 595 {
 596         ex e;
 597         epvector::const_iterator it = seq.begin(), itend = seq.end();
 598
 599         while (it != itend) {
 600                 if (is_order_function(it->rest)) {
 601                         if (!no_order)
 602                                 e += Order(power(var - point, it->coeff));
 603                 } else
 604                         e += it->rest * power(var - point, it->coeff);
 605                 ++it;
 606         }
 607         return e;
 608 }
 609
 610 bool pseries::is_terminating() const
 611 {
 612         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 613 }
 614
 615 ex pseries::coeffop(size_t i) const
 616 {
 617         if (i >=nops())
 618                 throw (std::out_of_range("coeffop() out of range"));
 619         return seq[i].rest;
 620 }
 621
 622 ex pseries::exponop(size_t i) const
 623 {
 624         if (i >= nops())
 625                 throw (std::out_of_range("exponop() out of range"));
 626         return seq[i].coeff;
 627 }
 628
 629
 630 /*
 631  *  Implementations of series expansion
 632  */
 633
 634 /** Default implementation of ex::series(). This performs Taylor expansion.
 635  *  @see ex::series */
 636 ex basic::series(const relational & r, int order, unsigned options) const
 637 {
 638         epvector seq;
 639         const symbol &s = ex_to<symbol>(r.lhs());
 640
 641         // default for order-values that make no sense for Taylor expansion
 642         if ((order <= 0) && this->has(s)) {
 643                 seq.push_back(expair(Order(_ex1), order));
 644                 return pseries(r, seq);
 645         }
 646
 647         // do Taylor expansion
 648         numeric fac = 1;
 649         ex deriv = *this;
 650         ex coeff = deriv.subs(r, subs_options::no_pattern);
 651
 652         if (!coeff.is_zero()) {
 653                 seq.push_back(expair(coeff, _ex0));
 654         }
 655
 656         int n;
 657         for (n=1; n<order; ++n) {
 658                 fac = fac.mul(n);
 659                 // We need to test for zero in order to see if the series terminates.
 660                 // The problem is that there is no such thing as a perfect test for
 661                 // zero.  Expanding the term occasionally helps a little...
 662                 deriv = deriv.diff(s).expand();
 663                 if (deriv.is_zero())  // Series terminates
 664                         return pseries(r, seq);
 665
 666                 coeff = deriv.subs(r, subs_options::no_pattern);
 667                 if (!coeff.is_zero())
 668                         seq.push_back(expair(fac.inverse() * coeff, n));
 669         }
 670
 671         // Higher-order terms, if present
 672         deriv = deriv.diff(s);
 673         if (!deriv.expand().is_zero())
 674                 seq.push_back(expair(Order(_ex1), n));
 675         return pseries(r, seq);
 676 }
 677
 678
 679 /** Implementation of ex::series() for symbols.
 680  *  @see ex::series */
 681 ex symbol::series(const relational & r, int order, unsigned options) const
 682 {
 683         epvector seq;
 684         const ex point = r.rhs();
 685         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 686
 687         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 688                 if (order > 0 && !point.is_zero())
 689                         seq.push_back(expair(point, _ex0));
 690                 if (order > 1)
 691                         seq.push_back(expair(_ex1, _ex1));
 692                 else
 693                         seq.push_back(expair(Order(_ex1), numeric(order)));
 694         } else
 695                 seq.push_back(expair(*this, _ex0));
 696         return pseries(r, seq);
 697 }
 698
 699
 700 /** Add one series object to another, producing a pseries object that
 701  *  represents the sum.
 702  *
 703  *  @param other  pseries object to add with
 704  *  @return the sum as a pseries */
 705 ex pseries::add_series(const pseries &other) const
 706 {
 707         // Adding two series with different variables or expansion points
 708         // results in an empty (constant) series
 709         if (!is_compatible_to(other)) {
 710                 epvector nul;
 711                 nul.push_back(expair(Order(_ex1), _ex0));
 712                 return pseries(relational(var,point), nul);
 713         }
 714
 715         // Series addition
 716         epvector new_seq;
 717         epvector::const_iterator a = seq.begin();
 718         epvector::const_iterator b = other.seq.begin();
 719         epvector::const_iterator a_end = seq.end();
 720         epvector::const_iterator b_end = other.seq.end();
 721         int pow_a = INT_MAX, pow_b = INT_MAX;
 722         for (;;) {
 723                 // If a is empty, fill up with elements from b and stop
 724                 if (a == a_end) {
 725                         while (b != b_end) {
 726                                 new_seq.push_back(*b);
 727                                 ++b;
 728                         }
 729                         break;
 730                 } else
 731                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 732
 733                 // If b is empty, fill up with elements from a and stop
 734                 if (b == b_end) {
 735                         while (a != a_end) {
 736                                 new_seq.push_back(*a);
 737                                 ++a;
 738                         }
 739                         break;
 740                 } else
 741                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 742
 743                 // a and b are non-empty, compare powers
 744                 if (pow_a < pow_b) {
 745                         // a has lesser power, get coefficient from a
 746                         new_seq.push_back(*a);
 747                         if (is_order_function((*a).rest))
 748                                 break;
 749                         ++a;
 750                 } else if (pow_b < pow_a) {
 751                         // b has lesser power, get coefficient from b
 752                         new_seq.push_back(*b);
 753                         if (is_order_function((*b).rest))
 754                                 break;
 755                         ++b;
 756                 } else {
 757                         // Add coefficient of a and b
 758                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 759                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 760                                 break;  // Order term ends the sequence
 761                         } else {
 762                                 ex sum = (*a).rest + (*b).rest;
 763                                 if (!(sum.is_zero()))
 764                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 765                                 ++a;
 766                                 ++b;
 767                         }
 768                 }
 769         }
 770         return pseries(relational(var,point), new_seq);
 771 }
 772
 773
 774 /** Implementation of ex::series() for sums. This performs series addition when
 775  *  adding pseries objects.
 776  *  @see ex::series */
 777 ex add::series(const relational & r, int order, unsigned options) const
 778 {
 779         ex acc; // Series accumulator
 780
 781         // Get first term from overall_coeff
 782         acc = overall_coeff.series(r, order, options);
 783
 784         // Add remaining terms
 785         epvector::const_iterator it = seq.begin();
 786         epvector::const_iterator itend = seq.end();
 787         for (; it!=itend; ++it) {
 788                 ex op;
 789                 if (is_exactly_a<pseries>(it->rest))
 790                         op = it->rest;
 791                 else
 792                         op = it->rest.series(r, order, options);
 793                 if (!it->coeff.is_equal(_ex1))
 794                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 795
 796                 // Series addition
 797                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 798         }
 799         return acc;
 800 }
 801
 802
 803 /** Multiply a pseries object with a numeric constant, producing a pseries
 804  *  object that represents the product.
 805  *
 806  *  @param other  constant to multiply with
 807  *  @return the product as a pseries */
 808 ex pseries::mul_const(const numeric &other) const
 809 {
 810         epvector new_seq;
 811         new_seq.reserve(seq.size());
 812
 813         epvector::const_iterator it = seq.begin(), itend = seq.end();
 814         while (it != itend) {
 815                 if (!is_order_function(it->rest))
 816                         new_seq.push_back(expair(it->rest * other, it->coeff));
 817                 else
 818                         new_seq.push_back(*it);
 819                 ++it;
 820         }
 821         return pseries(relational(var,point), new_seq);
 822 }
 823
 824
 825 /** Multiply one pseries object to another, producing a pseries object that
 826  *  represents the product.
 827  *
 828  *  @param other  pseries object to multiply with
 829  *  @return the product as a pseries */
 830 ex pseries::mul_series(const pseries &other) const
 831 {
 832         // Multiplying two series with different variables or expansion points
 833         // results in an empty (constant) series
 834         if (!is_compatible_to(other)) {
 835                 epvector nul;
 836                 nul.push_back(expair(Order(_ex1), _ex0));
 837                 return pseries(relational(var,point), nul);
 838         }
 839
 840         if (seq.empty() || other.seq.empty()) {
 841                 return (new pseries(var==point, epvector()))
 842                        ->setflag(status_flags::dynallocated);
 843         }
 844
 845         // Series multiplication
 846         epvector new_seq;
 847         int a_max = degree(var);
 848         int b_max = other.degree(var);
 849         int a_min = ldegree(var);
 850         int b_min = other.ldegree(var);
 851         int cdeg_min = a_min + b_min;
 852         int cdeg_max = a_max + b_max;
 853
 854         int higher_order_a = INT_MAX;
 855         int higher_order_b = INT_MAX;
 856         if (is_order_function(coeff(var, a_max)))
 857                 higher_order_a = a_max + b_min;
 858         if (is_order_function(other.coeff(var, b_max)))
 859                 higher_order_b = b_max + a_min;
 860         int higher_order_c = std::min(higher_order_a, higher_order_b);
 861         if (cdeg_max >= higher_order_c)
 862                 cdeg_max = higher_order_c - 1;
 863
 864         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 865                 ex co = _ex0;
 866                 // c(i)=a(0)b(i)+...+a(i)b(0)
 867                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 868                         ex a_coeff = coeff(var, i);
 869                         ex b_coeff = other.coeff(var, cdeg-i);
 870                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 871                                 co += a_coeff * b_coeff;
 872                 }
 873                 if (!co.is_zero())
 874                         new_seq.push_back(expair(co, numeric(cdeg)));
 875         }
 876         if (higher_order_c < INT_MAX)
 877                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 878         return pseries(relational(var, point), new_seq);
 879 }
 880
 881
 882 /** Implementation of ex::series() for product. This performs series
 883  *  multiplication when multiplying series.
 884  *  @see ex::series */
 885 ex mul::series(const relational & r, int order, unsigned options) const
 886 {
 887         pseries acc; // Series accumulator
 888
 889         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 890         const ex& sym = r.lhs();
 891
 892         // holds ldegrees of the series of individual factors
 893         std::vector<int> ldegrees;
 894
 895         // find minimal degrees
 896         const epvector::const_iterator itbeg = seq.begin();
 897         const epvector::const_iterator itend = seq.end();
 898         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 899
 900                 ex expon = it->coeff;
 901                 int factor = 1;
 902                 ex buf;
 903                 if (expon.info(info_flags::integer)) {
 904                         buf = it->rest;
 905                         factor = ex_to<numeric>(expon).to_int();
 906                 } else {
 907                         buf = recombine_pair_to_ex(*it);
 908                 }
 909
 910                 int real_ldegree = 0;
 911                 try {
 912                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 913                 } catch (std::runtime_error) {}
 914
 915                 if (real_ldegree == 0) {
 916                         int orderloop = 0;
 917                         do {
 918                                 orderloop++;
 919                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 920                         } while (real_ldegree == orderloop);
 921                 }
 922
 923                 ldegrees.push_back(factor * real_ldegree);
 924         }
 925
 926         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 927
 928         if (degsum >= order) {
 929                 epvector epv;
 930                 epv.push_back(expair(Order(_ex1), order));
 931                 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
 932         }
 933
 934         // Multiply with remaining terms
 935         std::vector<int>::const_iterator itd = ldegrees.begin();
 936         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
 937
 938                 // do series expansion with adjusted order
 939                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 940
 941                 // Series multiplication
 942                 if (it == itbeg)
 943                         acc = ex_to<pseries>(op);
 944                 else
 945                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 946         }
 947
 948         return acc.mul_const(ex_to<numeric>(overall_coeff));
 949 }
 950
 951
 952 /** Compute the p-th power of a series.
 953  *
 954  *  @param p  power to compute
 955  *  @param deg  truncation order of series calculation */
 956 ex pseries::power_const(const numeric &p, int deg) const
 957 {
 958         // method:
 959         // (due to Leonhard Euler)
 960         // let A(x) be this series and for the time being let it start with a
 961         // constant (later we'll generalize):
 962         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 963         // We want to compute
 964         //     C(x) = A(x)^p
 965         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 966         // Taking the derivative on both sides and multiplying with A(x) one
 967         // immediately arrives at
 968         //     C'(x)*A(x) = p*C(x)*A'(x)
 969         // Multiplying this out and comparing coefficients we get the recurrence
 970         // formula
 971         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 972         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 973         // which can easily be solved given the starting value c_0 = (a_0)^p.
 974         // For the more general case where the leading coefficient of A(x) is not
 975         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 976         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 977         // then of course x^(p*m) but the recurrence formula still holds.
 978
 979         if (seq.empty()) {
 980                 // as a special case, handle the empty (zero) series honoring the
 981                 // usual power laws such as implemented in power::eval()
 982                 if (p.real().is_zero())
 983                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 984                 else if (p.real().is_negative())
 985                         throw pole_error("pseries::power_const(): division by zero",1);
 986                 else
 987                         return *this;
 988         }
 989
 990         const int ldeg = ldegree(var);
 991         if (!(p*ldeg).is_integer())
 992                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 993
 994         // adjust number of coefficients
 995         int numcoeff = deg - (p*ldeg).to_int();
 996         if (numcoeff <= 0) {
 997                 epvector epv;
 998                 epv.reserve(1);
 999                 epv.push_back(expair(Order(_ex1), deg));
1000                 return (new pseries(relational(var,point), epv))
1001                        ->setflag(status_flags::dynallocated);
1002         }
1003
1004         // O(x^n)^(-m) is undefined
1005         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1006                 throw pole_error("pseries::power_const(): division by zero",1);
1007
1008         // Compute coefficients of the powered series
1009         exvector co;
1010         co.reserve(numcoeff);
1011         co.push_back(power(coeff(var, ldeg), p));
1012         for (int i=1; i<numcoeff; ++i) {
1013                 ex sum = _ex0;
1014                 for (int j=1; j<=i; ++j) {
1015                         ex c = coeff(var, j + ldeg);
1016                         if (is_order_function(c)) {
1017                                 co.push_back(Order(_ex1));
1018                                 break;
1019                         } else
1020                                 sum += (p * j - (i - j)) * co[i - j] * c;
1021                 }
1022                 co.push_back(sum / coeff(var, ldeg) / i);
1023         }
1024
1025         // Construct new series (of non-zero coefficients)
1026         epvector new_seq;
1027         bool higher_order = false;
1028         for (int i=0; i<numcoeff; ++i) {
1029                 if (!co[i].is_zero())
1030                         new_seq.push_back(expair(co[i], p * ldeg + i));
1031                 if (is_order_function(co[i])) {
1032                         higher_order = true;
1033                         break;
1034                 }
1035         }
1036         if (!higher_order)
1037                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1038
1039         return pseries(relational(var,point), new_seq);
1040 }
1041
1042
1043 /** Return a new pseries object with the powers shifted by deg. */
1044 pseries pseries::shift_exponents(int deg) const
1045 {
1046         epvector newseq = seq;
1047         epvector::iterator i = newseq.begin(), end  = newseq.end();
1048         while (i != end) {
1049                 i->coeff += deg;
1050                 ++i;
1051         }
1052         return pseries(relational(var, point), newseq);
1053 }
1054
1055
1056 /** Implementation of ex::series() for powers. This performs Laurent expansion
1057  *  of reciprocals of series at singularities.
1058  *  @see ex::series */
1059 ex power::series(const relational & r, int order, unsigned options) const
1060 {
1061         // If basis is already a series, just power it
1062         if (is_exactly_a<pseries>(basis))
1063                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1064
1065         // Basis is not a series, may there be a singularity?
1066         bool must_expand_basis = false;
1067         try {
1068                 basis.subs(r, subs_options::no_pattern);
1069         } catch (pole_error) {
1070                 must_expand_basis = true;
1071         }
1072
1073         // Is the expression of type something^(-int)?
1074         if (!must_expand_basis && !exponent.info(info_flags::negint)
1075          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1076                 return basic::series(r, order, options);
1077
1078         // Is the expression of type 0^something?
1079         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1080          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1081                 return basic::series(r, order, options);
1082
1083         // Singularity encountered, is the basis equal to (var - point)?
1084         if (basis.is_equal(r.lhs() - r.rhs())) {
1085                 epvector new_seq;
1086                 if (ex_to<numeric>(exponent).to_int() < order)
1087                         new_seq.push_back(expair(_ex1, exponent));
1088                 else
1089                         new_seq.push_back(expair(Order(_ex1), exponent));
1090                 return pseries(r, new_seq);
1091         }
1092
1093         // No, expand basis into series
1094
1095         numeric numexp;
1096         if (is_a<numeric>(exponent)) {
1097                 numexp = ex_to<numeric>(exponent);
1098         } else {
1099                 numexp = 0;
1100         }
1101         const ex& sym = r.lhs();
1102         // find existing minimal degree
1103         int real_ldegree = basis.expand().ldegree(sym-r.rhs());
1104         if (real_ldegree == 0) {
1105                 int orderloop = 0;
1106                 do {
1107                         orderloop++;
1108                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1109                 } while (real_ldegree == orderloop);
1110         }
1111
1112         if (!(real_ldegree*numexp).is_integer())
1113                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1114         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1115
1116         ex result;
1117         try {
1118                 result = ex_to<pseries>(e).power_const(numexp, order);
1119         } catch (pole_error) {
1120                 epvector ser;
1121                 ser.push_back(expair(Order(_ex1), order));
1122                 result = pseries(r, ser);
1123         }
1124
1125         return result;
1126 }
1127
1128
1129 /** Re-expansion of a pseries object. */
1130 ex pseries::series(const relational & r, int order, unsigned options) const
1131 {
1132         const ex p = r.rhs();
1133         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1134         const symbol &s = ex_to<symbol>(r.lhs());
1135
1136         if (var.is_equal(s) && point.is_equal(p)) {
1137                 if (order > degree(s))
1138                         return *this;
1139                 else {
1140                         epvector new_seq;
1141                         epvector::const_iterator it = seq.begin(), itend = seq.end();
1142                         while (it != itend) {
1143                                 int o = ex_to<numeric>(it->coeff).to_int();
1144                                 if (o >= order) {
1145                                         new_seq.push_back(expair(Order(_ex1), o));
1146                                         break;
1147                                 }
1148                                 new_seq.push_back(*it);
1149                                 ++it;
1150                         }
1151                         return pseries(r, new_seq);
1152                 }
1153         } else
1154                 return convert_to_poly().series(r, order, options);
1155 }
1156
1157 ex integral::series(const relational & r, int order, unsigned options) const
1158 {
1159         if (x.subs(r) != x)
1160                 throw std::logic_error("Cannot series expand wrt dummy variable");
1161
1162         // Expanding integrant with r substituted taken in boundaries.
1163         ex fseries = f.series(r, order, options);
1164         epvector fexpansion;
1165         fexpansion.reserve(fseries.nops());
1166         for (size_t i=0; i<fseries.nops(); ++i) {
1167                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1168                 currcoeff = (currcoeff == Order(_ex1))
1169                         ? currcoeff
1170                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1171                 if (currcoeff != 0)
1172                         fexpansion.push_back(
1173                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1174         }
1175
1176         // Expanding lower boundary
1177         ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1178         ex aseries = (a-a.subs(r)).series(r, order, options);
1179         fseries = f.series(x == (a.subs(r)), order, options);
1180         for (size_t i=0; i<fseries.nops(); ++i) {
1181                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1182                 if (is_order_function(currcoeff))
1183                         break;
1184                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1185                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1186                 currcoeff = currcoeff.series(r, orderforf);
1187                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1188                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1189                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1190                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1191         }
1192
1193         // Expanding upper boundary
1194         ex bseries = (b-b.subs(r)).series(r, order, options);
1195         fseries = f.series(x == (b.subs(r)), order, options);
1196         for (size_t i=0; i<fseries.nops(); ++i) {
1197                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1198                 if (is_order_function(currcoeff))
1199                         break;
1200                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1201                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1202                 currcoeff = currcoeff.series(r, orderforf);
1203                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1204                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1205                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1206                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1207         }
1208
1209         return result;
1210 }
1211
1212
1213 /** Compute the truncated series expansion of an expression.
1214  *  This function returns an expression containing an object of class pseries
1215  *  to represent the series. If the series does not terminate within the given
1216  *  truncation order, the last term of the series will be an order term.
1217  *
1218  *  @param r  expansion relation, lhs holds variable and rhs holds point
1219  *  @param order  truncation order of series calculations
1220  *  @param options  of class series_options
1221  *  @return an expression holding a pseries object */
1222 ex ex::series(const ex & r, int order, unsigned options) const
1223 {
1224         ex e;
1225         relational rel_;
1226
1227         if (is_a<relational>(r))
1228                 rel_ = ex_to<relational>(r);
1229         else if (is_a<symbol>(r))
1230                 rel_ = relational(r,_ex0);
1231         else
1232                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1233
1234         try {
1235                 e = bp->series(rel_, order, options);
1236         } catch (std::exception &x) {
1237                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1238         }
1239         return e;
1240 }
1241
1242 } // namespace GiNaC