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) 1999-2006 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         for (unsigned int i=0; true; ++i) {
  86                 ex rest;
  87                 ex coeff;
  88                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  89                         seq.push_back(expair(rest, coeff));
  90                 else
  91                         break;
  92         }
  93         n.find_ex("var", var, sym_lst);
  94         n.find_ex("point", point, sym_lst);
  95 }
  96
  97 void pseries::archive(archive_node &n) const
  98 {
  99         inherited::archive(n);
 100         epvector::const_iterator i = seq.begin(), iend = seq.end();
 101         while (i != iend) {
 102                 n.add_ex("coeff", i->rest);
 103                 n.add_ex("power", i->coeff);
 104                 ++i;
 105         }
 106         n.add_ex("var", var);
 107         n.add_ex("point", point);
 108 }
 109
 110 DEFAULT_UNARCHIVE(pseries)
 111
 112 //////////
 113 // functions overriding virtual functions from base classes
 114 //////////
 115
 116 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
 117 {
 118         if (precedence() <= level)
 119                 c.s << '(';
 120
 121         // objects of type pseries must not have any zero entries, so the
 122         // trivial (zero) pseries needs a special treatment here:
 123         if (seq.empty())
 124                 c.s << '0';
 125
 126         epvector::const_iterator i = seq.begin(), end = seq.end();
 127         while (i != end) {
 128
 129                 // print a sign, if needed
 130                 if (i != seq.begin())
 131                         c.s << '+';
 132
 133                 if (!is_order_function(i->rest)) {
 134
 135                         // print 'rest', i.e. the expansion coefficient
 136                         if (i->rest.info(info_flags::numeric) &&
 137                                 i->rest.info(info_flags::positive)) {
 138                                 i->rest.print(c);
 139                         } else {
 140                                 c.s << openbrace << '(';
 141                                 i->rest.print(c);
 142                                 c.s << ')' << closebrace;
 143                         }
 144
 145                         // print 'coeff', something like (x-1)^42
 146                         if (!i->coeff.is_zero()) {
 147                                 c.s << mul_sym;
 148                                 if (!point.is_zero()) {
 149                                         c.s << openbrace << '(';
 150                                         (var-point).print(c);
 151                                         c.s << ')' << closebrace;
 152                                 } else
 153                                         var.print(c);
 154                                 if (i->coeff.compare(_ex1)) {
 155                                         c.s << pow_sym;
 156                                         c.s << openbrace;
 157                                         if (i->coeff.info(info_flags::negative)) {
 158                                                 c.s << '(';
 159                                                 i->coeff.print(c);
 160                                                 c.s << ')';
 161                                         } else
 162                                                 i->coeff.print(c);
 163                                         c.s << closebrace;
 164                                 }
 165                         }
 166                 } else
 167                         Order(power(var-point,i->coeff)).print(c);
 168                 ++i;
 169         }
 170
 171         if (precedence() <= level)
 172                 c.s << ')';
 173 }
 174
 175 void pseries::do_print(const print_context & c, unsigned level) const
 176 {
 177         print_series(c, "", "", "*", "^", level);
 178 }
 179
 180 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 181 {
 182         print_series(c, "{", "}", " ", "^", level);
 183 }
 184
 185 void pseries::do_print_python(const print_python & c, unsigned level) const
 186 {
 187         print_series(c, "", "", "*", "**", level);
 188 }
 189
 190 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 191 {
 192         c.s << std::string(level, ' ') << class_name() << " @" << this
 193             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 194             << std::endl;
 195         size_t num = seq.size();
 196         for (size_t i=0; i<num; ++i) {
 197                 seq[i].rest.print(c, level + c.delta_indent);
 198                 seq[i].coeff.print(c, level + c.delta_indent);
 199                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 200         }
 201         var.print(c, level + c.delta_indent);
 202         point.print(c, level + c.delta_indent);
 203 }
 204
 205 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 206 {
 207         c.s << class_name() << "(relational(";
 208         var.print(c);
 209         c.s << ',';
 210         point.print(c);
 211         c.s << "),[";
 212         size_t num = seq.size();
 213         for (size_t i=0; i<num; ++i) {
 214                 if (i)
 215                         c.s << ',';
 216                 c.s << '(';
 217                 seq[i].rest.print(c);
 218                 c.s << ',';
 219                 seq[i].coeff.print(c);
 220                 c.s << ')';
 221         }
 222         c.s << "])";
 223 }
 224
 225 int pseries::compare_same_type(const basic & other) const
 226 {
 227         GINAC_ASSERT(is_a<pseries>(other));
 228         const pseries &o = static_cast<const pseries &>(other);
 229
 230         // first compare the lengths of the series...
 231         if (seq.size()>o.seq.size())
 232                 return 1;
 233         if (seq.size()<o.seq.size())
 234                 return -1;
 235
 236         // ...then the expansion point...
 237         int cmpval = var.compare(o.var);
 238         if (cmpval)
 239                 return cmpval;
 240         cmpval = point.compare(o.point);
 241         if (cmpval)
 242                 return cmpval;
 243
 244         // ...and if that failed the individual elements
 245         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 246         while (it!=seq.end() && o_it!=o.seq.end()) {
 247                 cmpval = it->compare(*o_it);
 248                 if (cmpval)
 249                         return cmpval;
 250                 ++it;
 251                 ++o_it;
 252         }
 253
 254         // so they are equal.
 255         return 0;
 256 }
 257
 258 /** Return the number of operands including a possible order term. */
 259 size_t pseries::nops() const
 260 {
 261         return seq.size();
 262 }
 263
 264 /** Return the ith term in the series when represented as a sum. */
 265 ex pseries::op(size_t i) const
 266 {
 267         if (i >= seq.size())
 268                 throw (std::out_of_range("op() out of range"));
 269
 270         if (is_order_function(seq[i].rest))
 271                 return Order(power(var-point, seq[i].coeff));
 272         return seq[i].rest * power(var - point, seq[i].coeff);
 273 }
 274
 275 /** Return degree of highest power of the series.  This is usually the exponent
 276  *  of the Order term.  If s is not the expansion variable of the series, the
 277  *  series is examined termwise. */
 278 int pseries::degree(const ex &s) const
 279 {
 280         if (var.is_equal(s)) {
 281                 // Return last exponent
 282                 if (seq.size())
 283                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 284                 else
 285                         return 0;
 286         } else {
 287                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 288                 if (it == itend)
 289                         return 0;
 290                 int max_pow = INT_MIN;
 291                 while (it != itend) {
 292                         int pow = it->rest.degree(s);
 293                         if (pow > max_pow)
 294                                 max_pow = pow;
 295                         ++it;
 296                 }
 297                 return max_pow;
 298         }
 299 }
 300
 301 /** Return degree of lowest power of the series.  This is usually the exponent
 302  *  of the leading term.  If s is not the expansion variable of the series, the
 303  *  series is examined termwise.  If s is the expansion variable but the
 304  *  expansion point is not zero the series is not expanded to find the degree.
 305  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 306 int pseries::ldegree(const ex &s) const
 307 {
 308         if (var.is_equal(s)) {
 309                 // Return first exponent
 310                 if (seq.size())
 311                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 312                 else
 313                         return 0;
 314         } else {
 315                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 316                 if (it == itend)
 317                         return 0;
 318                 int min_pow = INT_MAX;
 319                 while (it != itend) {
 320                         int pow = it->rest.ldegree(s);
 321                         if (pow < min_pow)
 322                                 min_pow = pow;
 323                         ++it;
 324                 }
 325                 return min_pow;
 326         }
 327 }
 328
 329 /** Return coefficient of degree n in power series if s is the expansion
 330  *  variable.  If the expansion point is nonzero, by definition the n=1
 331  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 332  *  the expansion took place in the s in the first place).
 333  *  If s is not the expansion variable, an attempt is made to convert the
 334  *  series to a polynomial and return the corresponding coefficient from
 335  *  there. */
 336 ex pseries::coeff(const ex &s, int n) const
 337 {
 338         if (var.is_equal(s)) {
 339                 if (seq.empty())
 340                         return _ex0;
 341
 342                 // Binary search in sequence for given power
 343                 numeric looking_for = numeric(n);
 344                 int lo = 0, hi = seq.size() - 1;
 345                 while (lo <= hi) {
 346                         int mid = (lo + hi) / 2;
 347                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 348                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 349                         switch (cmp) {
 350                                 case -1:
 351                                         lo = mid + 1;
 352                                         break;
 353                                 case 0:
 354                                         return seq[mid].rest;
 355                                 case 1:
 356                                         hi = mid - 1;
 357                                         break;
 358                                 default:
 359                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 360                         }
 361                 }
 362                 return _ex0;
 363         } else
 364                 return convert_to_poly().coeff(s, n);
 365 }
 366
 367 /** Does nothing. */
 368 ex pseries::collect(const ex &s, bool distributed) const
 369 {
 370         return *this;
 371 }
 372
 373 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 374 ex pseries::eval(int level) const
 375 {
 376         if (level == 1)
 377                 return this->hold();
 378
 379         if (level == -max_recursion_level)
 380                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 381
 382         // Construct a new series with evaluated coefficients
 383         epvector new_seq;
 384         new_seq.reserve(seq.size());
 385         epvector::const_iterator it = seq.begin(), itend = seq.end();
 386         while (it != itend) {
 387                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 388                 ++it;
 389         }
 390         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 391 }
 392
 393 /** Evaluate coefficients numerically. */
 394 ex pseries::evalf(int level) const
 395 {
 396         if (level == 1)
 397                 return *this;
 398
 399         if (level == -max_recursion_level)
 400                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 401
 402         // Construct a new series with evaluated coefficients
 403         epvector new_seq;
 404         new_seq.reserve(seq.size());
 405         epvector::const_iterator it = seq.begin(), itend = seq.end();
 406         while (it != itend) {
 407                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 408                 ++it;
 409         }
 410         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 411 }
 412
 413 ex pseries::conjugate() const
 414 {
 415         if(!var.info(info_flags::real))
 416                 return conjugate_function(*this).hold();
 417
 418         epvector * newseq = conjugateepvector(seq);
 419         ex newpoint = point.conjugate();
 420
 421         if (!newseq     && are_ex_trivially_equal(point, newpoint)) {
 422                 return *this;
 423         }
 424
 425         ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
 426         if (newseq) {
 427                 delete newseq;
 428         }
 429         return result;
 430 }
 431
 432 ex pseries::real_part() const
 433 {
 434         if(!var.info(info_flags::real))
 435                 return real_part_function(*this).hold();
 436         ex newpoint = point.real_part();
 437         if(newpoint != point)
 438                 return real_part_function(*this).hold();
 439
 440         epvector v;
 441         v.reserve(seq.size());
 442         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 443                 v.push_back(expair((i->rest).real_part(), i->coeff));
 444         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 445 }
 446
 447 ex pseries::imag_part() const
 448 {
 449         if(!var.info(info_flags::real))
 450                 return imag_part_function(*this).hold();
 451         ex newpoint = point.real_part();
 452         if(newpoint != point)
 453                 return imag_part_function(*this).hold();
 454
 455         epvector v;
 456         v.reserve(seq.size());
 457         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 458                 v.push_back(expair((i->rest).imag_part(), i->coeff));
 459         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
 460 }
 461
 462 ex pseries::eval_integ() const
 463 {
 464         epvector *newseq = NULL;
 465         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 466                 if (newseq) {
 467                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 468                         continue;
 469                 }
 470                 ex newterm = i->rest.eval_integ();
 471                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 472                         newseq = new epvector;
 473                         newseq->reserve(seq.size());
 474                         for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
 475                                 newseq->push_back(*j);
 476                         newseq->push_back(expair(newterm, i->coeff));
 477                 }
 478         }
 479
 480         ex newpoint = point.eval_integ();
 481         if (newseq || !are_ex_trivially_equal(newpoint, point))
 482                 return (new pseries(var==newpoint, *newseq))
 483                        ->setflag(status_flags::dynallocated);
 484         return *this;
 485 }
 486
 487 ex pseries::subs(const exmap & m, unsigned options) const
 488 {
 489         // If expansion variable is being substituted, convert the series to a
 490         // polynomial and do the substitution there because the result might
 491         // no longer be a power series
 492         if (m.find(var) != m.end())
 493                 return convert_to_poly(true).subs(m, options);
 494
 495         // Otherwise construct a new series with substituted coefficients and
 496         // expansion point
 497         epvector newseq;
 498         newseq.reserve(seq.size());
 499         epvector::const_iterator it = seq.begin(), itend = seq.end();
 500         while (it != itend) {
 501                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
 502                 ++it;
 503         }
 504         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
 505 }
 506
 507 /** Implementation of ex::expand() for a power series.  It expands all the
 508  *  terms individually and returns the resulting series as a new pseries. */
 509 ex pseries::expand(unsigned options) const
 510 {
 511         epvector newseq;
 512         epvector::const_iterator i = seq.begin(), end = seq.end();
 513         while (i != end) {
 514                 ex restexp = i->rest.expand();
 515                 if (!restexp.is_zero())
 516                         newseq.push_back(expair(restexp, i->coeff));
 517                 ++i;
 518         }
 519         return (new pseries(relational(var,point), newseq))
 520                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 521 }
 522
 523 /** Implementation of ex::diff() for a power series.
 524  *  @see ex::diff */
 525 ex pseries::derivative(const symbol & s) const
 526 {
 527         epvector new_seq;
 528         epvector::const_iterator it = seq.begin(), itend = seq.end();
 529
 530         if (s == var) {
 531
 532                 // FIXME: coeff might depend on var
 533                 while (it != itend) {
 534                         if (is_order_function(it->rest)) {
 535                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 536                         } else {
 537                                 ex c = it->rest * it->coeff;
 538                                 if (!c.is_zero())
 539                                         new_seq.push_back(expair(c, it->coeff - 1));
 540                         }
 541                         ++it;
 542                 }
 543
 544         } else {
 545
 546                 while (it != itend) {
 547                         if (is_order_function(it->rest)) {
 548                                 new_seq.push_back(*it);
 549                         } else {
 550                                 ex c = it->rest.diff(s);
 551                                 if (!c.is_zero())
 552                                         new_seq.push_back(expair(c, it->coeff));
 553                         }
 554                         ++it;
 555                 }
 556         }
 557
 558         return pseries(relational(var,point), new_seq);
 559 }
 560
 561 ex pseries::convert_to_poly(bool no_order) const
 562 {
 563         ex e;
 564         epvector::const_iterator it = seq.begin(), itend = seq.end();
 565
 566         while (it != itend) {
 567                 if (is_order_function(it->rest)) {
 568                         if (!no_order)
 569                                 e += Order(power(var - point, it->coeff));
 570                 } else
 571                         e += it->rest * power(var - point, it->coeff);
 572                 ++it;
 573         }
 574         return e;
 575 }
 576
 577 bool pseries::is_terminating() const
 578 {
 579         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 580 }
 581
 582 ex pseries::coeffop(size_t i) const
 583 {
 584         if (i >=nops())
 585                 throw (std::out_of_range("coeffop() out of range"));
 586         return seq[i].rest;
 587 }
 588
 589 ex pseries::exponop(size_t i) const
 590 {
 591         if (i >= nops())
 592                 throw (std::out_of_range("exponop() out of range"));
 593         return seq[i].coeff;
 594 }
 595
 596
 597 /*
 598  *  Implementations of series expansion
 599  */
 600
 601 /** Default implementation of ex::series(). This performs Taylor expansion.
 602  *  @see ex::series */
 603 ex basic::series(const relational & r, int order, unsigned options) const
 604 {
 605         epvector seq;
 606         const symbol &s = ex_to<symbol>(r.lhs());
 607
 608         // default for order-values that make no sense for Taylor expansion
 609         if ((order <= 0) && this->has(s)) {
 610                 seq.push_back(expair(Order(_ex1), order));
 611                 return pseries(r, seq);
 612         }
 613
 614         // do Taylor expansion
 615         numeric fac = 1;
 616         ex deriv = *this;
 617         ex coeff = deriv.subs(r, subs_options::no_pattern);
 618
 619         if (!coeff.is_zero()) {
 620                 seq.push_back(expair(coeff, _ex0));
 621         }
 622
 623         int n;
 624         for (n=1; n<order; ++n) {
 625                 fac = fac.mul(n);
 626                 // We need to test for zero in order to see if the series terminates.
 627                 // The problem is that there is no such thing as a perfect test for
 628                 // zero.  Expanding the term occasionally helps a little...
 629                 deriv = deriv.diff(s).expand();
 630                 if (deriv.is_zero())  // Series terminates
 631                         return pseries(r, seq);
 632
 633                 coeff = deriv.subs(r, subs_options::no_pattern);
 634                 if (!coeff.is_zero())
 635                         seq.push_back(expair(fac.inverse() * coeff, n));
 636         }
 637
 638         // Higher-order terms, if present
 639         deriv = deriv.diff(s);
 640         if (!deriv.expand().is_zero())
 641                 seq.push_back(expair(Order(_ex1), n));
 642         return pseries(r, seq);
 643 }
 644
 645
 646 /** Implementation of ex::series() for symbols.
 647  *  @see ex::series */
 648 ex symbol::series(const relational & r, int order, unsigned options) const
 649 {
 650         epvector seq;
 651         const ex point = r.rhs();
 652         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 653
 654         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 655                 if (order > 0 && !point.is_zero())
 656                         seq.push_back(expair(point, _ex0));
 657                 if (order > 1)
 658                         seq.push_back(expair(_ex1, _ex1));
 659                 else
 660                         seq.push_back(expair(Order(_ex1), numeric(order)));
 661         } else
 662                 seq.push_back(expair(*this, _ex0));
 663         return pseries(r, seq);
 664 }
 665
 666
 667 /** Add one series object to another, producing a pseries object that
 668  *  represents the sum.
 669  *
 670  *  @param other  pseries object to add with
 671  *  @return the sum as a pseries */
 672 ex pseries::add_series(const pseries &other) const
 673 {
 674         // Adding two series with different variables or expansion points
 675         // results in an empty (constant) series
 676         if (!is_compatible_to(other)) {
 677                 epvector nul;
 678                 nul.push_back(expair(Order(_ex1), _ex0));
 679                 return pseries(relational(var,point), nul);
 680         }
 681
 682         // Series addition
 683         epvector new_seq;
 684         epvector::const_iterator a = seq.begin();
 685         epvector::const_iterator b = other.seq.begin();
 686         epvector::const_iterator a_end = seq.end();
 687         epvector::const_iterator b_end = other.seq.end();
 688         int pow_a = INT_MAX, pow_b = INT_MAX;
 689         for (;;) {
 690                 // If a is empty, fill up with elements from b and stop
 691                 if (a == a_end) {
 692                         while (b != b_end) {
 693                                 new_seq.push_back(*b);
 694                                 ++b;
 695                         }
 696                         break;
 697                 } else
 698                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 699
 700                 // If b is empty, fill up with elements from a and stop
 701                 if (b == b_end) {
 702                         while (a != a_end) {
 703                                 new_seq.push_back(*a);
 704                                 ++a;
 705                         }
 706                         break;
 707                 } else
 708                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 709
 710                 // a and b are non-empty, compare powers
 711                 if (pow_a < pow_b) {
 712                         // a has lesser power, get coefficient from a
 713                         new_seq.push_back(*a);
 714                         if (is_order_function((*a).rest))
 715                                 break;
 716                         ++a;
 717                 } else if (pow_b < pow_a) {
 718                         // b has lesser power, get coefficient from b
 719                         new_seq.push_back(*b);
 720                         if (is_order_function((*b).rest))
 721                                 break;
 722                         ++b;
 723                 } else {
 724                         // Add coefficient of a and b
 725                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 726                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 727                                 break;  // Order term ends the sequence
 728                         } else {
 729                                 ex sum = (*a).rest + (*b).rest;
 730                                 if (!(sum.is_zero()))
 731                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 732                                 ++a;
 733                                 ++b;
 734                         }
 735                 }
 736         }
 737         return pseries(relational(var,point), new_seq);
 738 }
 739
 740
 741 /** Implementation of ex::series() for sums. This performs series addition when
 742  *  adding pseries objects.
 743  *  @see ex::series */
 744 ex add::series(const relational & r, int order, unsigned options) const
 745 {
 746         ex acc; // Series accumulator
 747
 748         // Get first term from overall_coeff
 749         acc = overall_coeff.series(r, order, options);
 750
 751         // Add remaining terms
 752         epvector::const_iterator it = seq.begin();
 753         epvector::const_iterator itend = seq.end();
 754         for (; it!=itend; ++it) {
 755                 ex op;
 756                 if (is_exactly_a<pseries>(it->rest))
 757                         op = it->rest;
 758                 else
 759                         op = it->rest.series(r, order, options);
 760                 if (!it->coeff.is_equal(_ex1))
 761                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 762
 763                 // Series addition
 764                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 765         }
 766         return acc;
 767 }
 768
 769
 770 /** Multiply a pseries object with a numeric constant, producing a pseries
 771  *  object that represents the product.
 772  *
 773  *  @param other  constant to multiply with
 774  *  @return the product as a pseries */
 775 ex pseries::mul_const(const numeric &other) const
 776 {
 777         epvector new_seq;
 778         new_seq.reserve(seq.size());
 779
 780         epvector::const_iterator it = seq.begin(), itend = seq.end();
 781         while (it != itend) {
 782                 if (!is_order_function(it->rest))
 783                         new_seq.push_back(expair(it->rest * other, it->coeff));
 784                 else
 785                         new_seq.push_back(*it);
 786                 ++it;
 787         }
 788         return pseries(relational(var,point), new_seq);
 789 }
 790
 791
 792 /** Multiply one pseries object to another, producing a pseries object that
 793  *  represents the product.
 794  *
 795  *  @param other  pseries object to multiply with
 796  *  @return the product as a pseries */
 797 ex pseries::mul_series(const pseries &other) const
 798 {
 799         // Multiplying two series with different variables or expansion points
 800         // results in an empty (constant) series
 801         if (!is_compatible_to(other)) {
 802                 epvector nul;
 803                 nul.push_back(expair(Order(_ex1), _ex0));
 804                 return pseries(relational(var,point), nul);
 805         }
 806
 807         if (seq.empty() || other.seq.empty()) {
 808                 return (new pseries(var==point, epvector()))
 809                        ->setflag(status_flags::dynallocated);
 810         }
 811
 812         // Series multiplication
 813         epvector new_seq;
 814         int a_max = degree(var);
 815         int b_max = other.degree(var);
 816         int a_min = ldegree(var);
 817         int b_min = other.ldegree(var);
 818         int cdeg_min = a_min + b_min;
 819         int cdeg_max = a_max + b_max;
 820
 821         int higher_order_a = INT_MAX;
 822         int higher_order_b = INT_MAX;
 823         if (is_order_function(coeff(var, a_max)))
 824                 higher_order_a = a_max + b_min;
 825         if (is_order_function(other.coeff(var, b_max)))
 826                 higher_order_b = b_max + a_min;
 827         int higher_order_c = std::min(higher_order_a, higher_order_b);
 828         if (cdeg_max >= higher_order_c)
 829                 cdeg_max = higher_order_c - 1;
 830
 831         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 832                 ex co = _ex0;
 833                 // c(i)=a(0)b(i)+...+a(i)b(0)
 834                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 835                         ex a_coeff = coeff(var, i);
 836                         ex b_coeff = other.coeff(var, cdeg-i);
 837                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 838                                 co += a_coeff * b_coeff;
 839                 }
 840                 if (!co.is_zero())
 841                         new_seq.push_back(expair(co, numeric(cdeg)));
 842         }
 843         if (higher_order_c < INT_MAX)
 844                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 845         return pseries(relational(var, point), new_seq);
 846 }
 847
 848
 849 /** Implementation of ex::series() for product. This performs series
 850  *  multiplication when multiplying series.
 851  *  @see ex::series */
 852 ex mul::series(const relational & r, int order, unsigned options) const
 853 {
 854         pseries acc; // Series accumulator
 855
 856         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 857         const ex& sym = r.lhs();
 858
 859         // holds ldegrees of the series of individual factors
 860         std::vector<int> ldegrees;
 861
 862         // find minimal degrees
 863         const epvector::const_iterator itbeg = seq.begin();
 864         const epvector::const_iterator itend = seq.end();
 865         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 866
 867                 ex expon = it->coeff;
 868                 int factor = 1;
 869                 ex buf;
 870                 if (expon.info(info_flags::integer)) {
 871                         buf = it->rest;
 872                         factor = ex_to<numeric>(expon).to_int();
 873                 } else {
 874                         buf = recombine_pair_to_ex(*it);
 875                 }
 876
 877                 int real_ldegree = 0;
 878                 try {
 879                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 880                 } catch (std::runtime_error) {}
 881
 882                 if (real_ldegree == 0) {
 883                         int orderloop = 0;
 884                         do {
 885                                 orderloop++;
 886                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 887                         } while (real_ldegree == orderloop);
 888                 }
 889
 890                 ldegrees.push_back(factor * real_ldegree);
 891         }
 892
 893         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 894
 895         if (degsum >= order) {
 896                 epvector epv;
 897                 epv.push_back(expair(Order(_ex1), order));
 898                 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
 899         }
 900
 901         // Multiply with remaining terms
 902         std::vector<int>::const_iterator itd = ldegrees.begin();
 903         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
 904
 905                 // do series expansion with adjusted order
 906                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 907
 908                 // Series multiplication
 909                 if (it == itbeg)
 910                         acc = ex_to<pseries>(op);
 911                 else
 912                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 913         }
 914
 915         return acc.mul_const(ex_to<numeric>(overall_coeff));
 916 }
 917
 918
 919 /** Compute the p-th power of a series.
 920  *
 921  *  @param p  power to compute
 922  *  @param deg  truncation order of series calculation */
 923 ex pseries::power_const(const numeric &p, int deg) const
 924 {
 925         // method:
 926         // (due to Leonhard Euler)
 927         // let A(x) be this series and for the time being let it start with a
 928         // constant (later we'll generalize):
 929         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 930         // We want to compute
 931         //     C(x) = A(x)^p
 932         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 933         // Taking the derivative on both sides and multiplying with A(x) one
 934         // immediately arrives at
 935         //     C'(x)*A(x) = p*C(x)*A'(x)
 936         // Multiplying this out and comparing coefficients we get the recurrence
 937         // formula
 938         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 939         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 940         // which can easily be solved given the starting value c_0 = (a_0)^p.
 941         // For the more general case where the leading coefficient of A(x) is not
 942         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 943         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 944         // then of course x^(p*m) but the recurrence formula still holds.
 945
 946         if (seq.empty()) {
 947                 // as a special case, handle the empty (zero) series honoring the
 948                 // usual power laws such as implemented in power::eval()
 949                 if (p.real().is_zero())
 950                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 951                 else if (p.real().is_negative())
 952                         throw pole_error("pseries::power_const(): division by zero",1);
 953                 else
 954                         return *this;
 955         }
 956
 957         const int ldeg = ldegree(var);
 958         if (!(p*ldeg).is_integer())
 959                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 960
 961         // adjust number of coefficients
 962         int numcoeff = deg - (p*ldeg).to_int();
 963         if (numcoeff <= 0) {
 964                 epvector epv;
 965                 epv.reserve(1);
 966                 epv.push_back(expair(Order(_ex1), deg));
 967                 return (new pseries(relational(var,point), epv))
 968                        ->setflag(status_flags::dynallocated);
 969         }
 970
 971         // O(x^n)^(-m) is undefined
 972         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 973                 throw pole_error("pseries::power_const(): division by zero",1);
 974
 975         // Compute coefficients of the powered series
 976         exvector co;
 977         co.reserve(numcoeff);
 978         co.push_back(power(coeff(var, ldeg), p));
 979         for (int i=1; i<numcoeff; ++i) {
 980                 ex sum = _ex0;
 981                 for (int j=1; j<=i; ++j) {
 982                         ex c = coeff(var, j + ldeg);
 983                         if (is_order_function(c)) {
 984                                 co.push_back(Order(_ex1));
 985                                 break;
 986                         } else
 987                                 sum += (p * j - (i - j)) * co[i - j] * c;
 988                 }
 989                 co.push_back(sum / coeff(var, ldeg) / i);
 990         }
 991
 992         // Construct new series (of non-zero coefficients)
 993         epvector new_seq;
 994         bool higher_order = false;
 995         for (int i=0; i<numcoeff; ++i) {
 996                 if (!co[i].is_zero())
 997                         new_seq.push_back(expair(co[i], p * ldeg + i));
 998                 if (is_order_function(co[i])) {
 999                         higher_order = true;
1000                         break;
1001                 }
1002         }
1003         if (!higher_order)
1004                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1005
1006         return pseries(relational(var,point), new_seq);
1007 }
1008
1009
1010 /** Return a new pseries object with the powers shifted by deg. */
1011 pseries pseries::shift_exponents(int deg) const
1012 {
1013         epvector newseq = seq;
1014         epvector::iterator i = newseq.begin(), end  = newseq.end();
1015         while (i != end) {
1016                 i->coeff += deg;
1017                 ++i;
1018         }
1019         return pseries(relational(var, point), newseq);
1020 }
1021
1022
1023 /** Implementation of ex::series() for powers. This performs Laurent expansion
1024  *  of reciprocals of series at singularities.
1025  *  @see ex::series */
1026 ex power::series(const relational & r, int order, unsigned options) const
1027 {
1028         // If basis is already a series, just power it
1029         if (is_exactly_a<pseries>(basis))
1030                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1031
1032         // Basis is not a series, may there be a singularity?
1033         bool must_expand_basis = false;
1034         try {
1035                 basis.subs(r, subs_options::no_pattern);
1036         } catch (pole_error) {
1037                 must_expand_basis = true;
1038         }
1039
1040         // Is the expression of type something^(-int)?
1041         if (!must_expand_basis && !exponent.info(info_flags::negint)
1042          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1043                 return basic::series(r, order, options);
1044
1045         // Is the expression of type 0^something?
1046         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1047          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1048                 return basic::series(r, order, options);
1049
1050         // Singularity encountered, is the basis equal to (var - point)?
1051         if (basis.is_equal(r.lhs() - r.rhs())) {
1052                 epvector new_seq;
1053                 if (ex_to<numeric>(exponent).to_int() < order)
1054                         new_seq.push_back(expair(_ex1, exponent));
1055                 else
1056                         new_seq.push_back(expair(Order(_ex1), exponent));
1057                 return pseries(r, new_seq);
1058         }
1059
1060         // No, expand basis into series
1061
1062         numeric numexp;
1063         if (is_a<numeric>(exponent)) {
1064                 numexp = ex_to<numeric>(exponent);
1065         } else {
1066                 numexp = 0;
1067         }
1068         const ex& sym = r.lhs();
1069         // find existing minimal degree
1070         int real_ldegree = basis.expand().ldegree(sym-r.rhs());
1071         if (real_ldegree == 0) {
1072                 int orderloop = 0;
1073                 do {
1074                         orderloop++;
1075                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1076                 } while (real_ldegree == orderloop);
1077         }
1078
1079         if (!(real_ldegree*numexp).is_integer())
1080                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1081         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1082
1083         ex result;
1084         try {
1085                 result = ex_to<pseries>(e).power_const(numexp, order);
1086         } catch (pole_error) {
1087                 epvector ser;
1088                 ser.push_back(expair(Order(_ex1), order));
1089                 result = pseries(r, ser);
1090         }
1091
1092         return result;
1093 }
1094
1095
1096 /** Re-expansion of a pseries object. */
1097 ex pseries::series(const relational & r, int order, unsigned options) const
1098 {
1099         const ex p = r.rhs();
1100         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1101         const symbol &s = ex_to<symbol>(r.lhs());
1102
1103         if (var.is_equal(s) && point.is_equal(p)) {
1104                 if (order > degree(s))
1105                         return *this;
1106                 else {
1107                         epvector new_seq;
1108                         epvector::const_iterator it = seq.begin(), itend = seq.end();
1109                         while (it != itend) {
1110                                 int o = ex_to<numeric>(it->coeff).to_int();
1111                                 if (o >= order) {
1112                                         new_seq.push_back(expair(Order(_ex1), o));
1113                                         break;
1114                                 }
1115                                 new_seq.push_back(*it);
1116                                 ++it;
1117                         }
1118                         return pseries(r, new_seq);
1119                 }
1120         } else
1121                 return convert_to_poly().series(r, order, options);
1122 }
1123
1124 ex integral::series(const relational & r, int order, unsigned options) const
1125 {
1126         if (x.subs(r) != x)
1127                 throw std::logic_error("Cannot series expand wrt dummy variable");
1128
1129         // Expanding integrant with r substituted taken in boundaries.
1130         ex fseries = f.series(r, order, options);
1131         epvector fexpansion;
1132         fexpansion.reserve(fseries.nops());
1133         for (size_t i=0; i<fseries.nops(); ++i) {
1134                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1135                 currcoeff = (currcoeff == Order(_ex1))
1136                         ? currcoeff
1137                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1138                 if (currcoeff != 0)
1139                         fexpansion.push_back(
1140                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1141         }
1142
1143         // Expanding lower boundary
1144         ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1145         ex aseries = (a-a.subs(r)).series(r, order, options);
1146         fseries = f.series(x == (a.subs(r)), order, options);
1147         for (size_t i=0; i<fseries.nops(); ++i) {
1148                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1149                 if (is_order_function(currcoeff))
1150                         break;
1151                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1152                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1153                 currcoeff = currcoeff.series(r, orderforf);
1154                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1155                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1156                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1157                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1158         }
1159
1160         // Expanding upper boundary
1161         ex bseries = (b-b.subs(r)).series(r, order, options);
1162         fseries = f.series(x == (b.subs(r)), order, options);
1163         for (size_t i=0; i<fseries.nops(); ++i) {
1164                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1165                 if (is_order_function(currcoeff))
1166                         break;
1167                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1168                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1169                 currcoeff = currcoeff.series(r, orderforf);
1170                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1171                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1172                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1173                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1174         }
1175
1176         return result;
1177 }
1178
1179
1180 /** Compute the truncated series expansion of an expression.
1181  *  This function returns an expression containing an object of class pseries
1182  *  to represent the series. If the series does not terminate within the given
1183  *  truncation order, the last term of the series will be an order term.
1184  *
1185  *  @param r  expansion relation, lhs holds variable and rhs holds point
1186  *  @param order  truncation order of series calculations
1187  *  @param options  of class series_options
1188  *  @return an expression holding a pseries object */
1189 ex ex::series(const ex & r, int order, unsigned options) const
1190 {
1191         ex e;
1192         relational rel_;
1193
1194         if (is_a<relational>(r))
1195                 rel_ = ex_to<relational>(r);
1196         else if (is_a<symbol>(r))
1197                 rel_ = relational(r,_ex0);
1198         else
1199                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1200
1201         try {
1202                 e = bp->series(rel_, order, options);
1203         } catch (std::exception &x) {
1204                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1205         }
1206         return e;
1207 }
1208
1209 } // namespace GiNaC