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) 1999-2005 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(TINFO_pseries) { }
  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(TINFO_pseries), 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         epvector * newseq = conjugateepvector(seq);
 416         ex newvar = var.conjugate();
 417         ex newpoint = point.conjugate();
 418
 419         if (!newseq     && are_ex_trivially_equal(newvar, var) && are_ex_trivially_equal(point, newpoint)) {
 420                 return *this;
 421         }
 422
 423         ex result = (new pseries(newvar==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
 424         if (newseq) {
 425                 delete newseq;
 426         }
 427         return result;
 428 }
 429
 430 ex pseries::eval_integ() const
 431 {
 432         epvector *newseq = NULL;
 433         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 434                 if (newseq) {
 435                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 436                         continue;
 437                 }
 438                 ex newterm = i->rest.eval_integ();
 439                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 440                         newseq = new epvector;
 441                         newseq->reserve(seq.size());
 442                         for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
 443                                 newseq->push_back(*j);
 444                         newseq->push_back(expair(newterm, i->coeff));
 445                 }
 446         }
 447
 448         ex newpoint = point.eval_integ();
 449         if (newseq || !are_ex_trivially_equal(newpoint, point))
 450                 return (new pseries(var==newpoint, *newseq))
 451                        ->setflag(status_flags::dynallocated);
 452         return *this;
 453 }
 454
 455 ex pseries::subs(const exmap & m, unsigned options) const
 456 {
 457         // If expansion variable is being substituted, convert the series to a
 458         // polynomial and do the substitution there because the result might
 459         // no longer be a power series
 460         if (m.find(var) != m.end())
 461                 return convert_to_poly(true).subs(m, options);
 462
 463         // Otherwise construct a new series with substituted coefficients and
 464         // expansion point
 465         epvector newseq;
 466         newseq.reserve(seq.size());
 467         epvector::const_iterator it = seq.begin(), itend = seq.end();
 468         while (it != itend) {
 469                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
 470                 ++it;
 471         }
 472         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
 473 }
 474
 475 /** Implementation of ex::expand() for a power series.  It expands all the
 476  *  terms individually and returns the resulting series as a new pseries. */
 477 ex pseries::expand(unsigned options) const
 478 {
 479         epvector newseq;
 480         epvector::const_iterator i = seq.begin(), end = seq.end();
 481         while (i != end) {
 482                 ex restexp = i->rest.expand();
 483                 if (!restexp.is_zero())
 484                         newseq.push_back(expair(restexp, i->coeff));
 485                 ++i;
 486         }
 487         return (new pseries(relational(var,point), newseq))
 488                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 489 }
 490
 491 /** Implementation of ex::diff() for a power series.
 492  *  @see ex::diff */
 493 ex pseries::derivative(const symbol & s) const
 494 {
 495         epvector new_seq;
 496         epvector::const_iterator it = seq.begin(), itend = seq.end();
 497
 498         if (s == var) {
 499
 500                 // FIXME: coeff might depend on var
 501                 while (it != itend) {
 502                         if (is_order_function(it->rest)) {
 503                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 504                         } else {
 505                                 ex c = it->rest * it->coeff;
 506                                 if (!c.is_zero())
 507                                         new_seq.push_back(expair(c, it->coeff - 1));
 508                         }
 509                         ++it;
 510                 }
 511
 512         } else {
 513
 514                 while (it != itend) {
 515                         if (is_order_function(it->rest)) {
 516                                 new_seq.push_back(*it);
 517                         } else {
 518                                 ex c = it->rest.diff(s);
 519                                 if (!c.is_zero())
 520                                         new_seq.push_back(expair(c, it->coeff));
 521                         }
 522                         ++it;
 523                 }
 524         }
 525
 526         return pseries(relational(var,point), new_seq);
 527 }
 528
 529 ex pseries::convert_to_poly(bool no_order) const
 530 {
 531         ex e;
 532         epvector::const_iterator it = seq.begin(), itend = seq.end();
 533
 534         while (it != itend) {
 535                 if (is_order_function(it->rest)) {
 536                         if (!no_order)
 537                                 e += Order(power(var - point, it->coeff));
 538                 } else
 539                         e += it->rest * power(var - point, it->coeff);
 540                 ++it;
 541         }
 542         return e;
 543 }
 544
 545 bool pseries::is_terminating() const
 546 {
 547         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 548 }
 549
 550 ex pseries::coeffop(size_t i) const
 551 {
 552         if (i >=nops())
 553                 throw (std::out_of_range("coeffop() out of range"));
 554         return seq[i].rest;
 555 }
 556
 557 ex pseries::exponop(size_t i) const
 558 {
 559         if (i >= nops())
 560                 throw (std::out_of_range("exponop() out of range"));
 561         return seq[i].coeff;
 562 }
 563
 564
 565 /*
 566  *  Implementations of series expansion
 567  */
 568
 569 /** Default implementation of ex::series(). This performs Taylor expansion.
 570  *  @see ex::series */
 571 ex basic::series(const relational & r, int order, unsigned options) const
 572 {
 573         epvector seq;
 574         const symbol &s = ex_to<symbol>(r.lhs());
 575
 576         // default for order-values that make no sense for Taylor expansion
 577         if ((order <= 0) && this->has(s)) {
 578                 seq.push_back(expair(Order(_ex1), order));
 579                 return pseries(r, seq);
 580         }
 581
 582         // do Taylor expansion
 583         numeric fac = 1;
 584         ex deriv = *this;
 585         ex coeff = deriv.subs(r, subs_options::no_pattern);
 586
 587         if (!coeff.is_zero()) {
 588                 seq.push_back(expair(coeff, _ex0));
 589         }
 590
 591         int n;
 592         for (n=1; n<order; ++n) {
 593                 fac = fac.mul(n);
 594                 // We need to test for zero in order to see if the series terminates.
 595                 // The problem is that there is no such thing as a perfect test for
 596                 // zero.  Expanding the term occasionally helps a little...
 597                 deriv = deriv.diff(s).expand();
 598                 if (deriv.is_zero())  // Series terminates
 599                         return pseries(r, seq);
 600
 601                 coeff = deriv.subs(r, subs_options::no_pattern);
 602                 if (!coeff.is_zero())
 603                         seq.push_back(expair(fac.inverse() * coeff, n));
 604         }
 605
 606         // Higher-order terms, if present
 607         deriv = deriv.diff(s);
 608         if (!deriv.expand().is_zero())
 609                 seq.push_back(expair(Order(_ex1), n));
 610         return pseries(r, seq);
 611 }
 612
 613
 614 /** Implementation of ex::series() for symbols.
 615  *  @see ex::series */
 616 ex symbol::series(const relational & r, int order, unsigned options) const
 617 {
 618         epvector seq;
 619         const ex point = r.rhs();
 620         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 621
 622         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 623                 if (order > 0 && !point.is_zero())
 624                         seq.push_back(expair(point, _ex0));
 625                 if (order > 1)
 626                         seq.push_back(expair(_ex1, _ex1));
 627                 else
 628                         seq.push_back(expair(Order(_ex1), numeric(order)));
 629         } else
 630                 seq.push_back(expair(*this, _ex0));
 631         return pseries(r, seq);
 632 }
 633
 634
 635 /** Add one series object to another, producing a pseries object that
 636  *  represents the sum.
 637  *
 638  *  @param other  pseries object to add with
 639  *  @return the sum as a pseries */
 640 ex pseries::add_series(const pseries &other) const
 641 {
 642         // Adding two series with different variables or expansion points
 643         // results in an empty (constant) series
 644         if (!is_compatible_to(other)) {
 645                 epvector nul;
 646                 nul.push_back(expair(Order(_ex1), _ex0));
 647                 return pseries(relational(var,point), nul);
 648         }
 649
 650         // Series addition
 651         epvector new_seq;
 652         epvector::const_iterator a = seq.begin();
 653         epvector::const_iterator b = other.seq.begin();
 654         epvector::const_iterator a_end = seq.end();
 655         epvector::const_iterator b_end = other.seq.end();
 656         int pow_a = INT_MAX, pow_b = INT_MAX;
 657         for (;;) {
 658                 // If a is empty, fill up with elements from b and stop
 659                 if (a == a_end) {
 660                         while (b != b_end) {
 661                                 new_seq.push_back(*b);
 662                                 ++b;
 663                         }
 664                         break;
 665                 } else
 666                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 667
 668                 // If b is empty, fill up with elements from a and stop
 669                 if (b == b_end) {
 670                         while (a != a_end) {
 671                                 new_seq.push_back(*a);
 672                                 ++a;
 673                         }
 674                         break;
 675                 } else
 676                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 677
 678                 // a and b are non-empty, compare powers
 679                 if (pow_a < pow_b) {
 680                         // a has lesser power, get coefficient from a
 681                         new_seq.push_back(*a);
 682                         if (is_order_function((*a).rest))
 683                                 break;
 684                         ++a;
 685                 } else if (pow_b < pow_a) {
 686                         // b has lesser power, get coefficient from b
 687                         new_seq.push_back(*b);
 688                         if (is_order_function((*b).rest))
 689                                 break;
 690                         ++b;
 691                 } else {
 692                         // Add coefficient of a and b
 693                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 694                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 695                                 break;  // Order term ends the sequence
 696                         } else {
 697                                 ex sum = (*a).rest + (*b).rest;
 698                                 if (!(sum.is_zero()))
 699                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 700                                 ++a;
 701                                 ++b;
 702                         }
 703                 }
 704         }
 705         return pseries(relational(var,point), new_seq);
 706 }
 707
 708
 709 /** Implementation of ex::series() for sums. This performs series addition when
 710  *  adding pseries objects.
 711  *  @see ex::series */
 712 ex add::series(const relational & r, int order, unsigned options) const
 713 {
 714         ex acc; // Series accumulator
 715
 716         // Get first term from overall_coeff
 717         acc = overall_coeff.series(r, order, options);
 718
 719         // Add remaining terms
 720         epvector::const_iterator it = seq.begin();
 721         epvector::const_iterator itend = seq.end();
 722         for (; it!=itend; ++it) {
 723                 ex op;
 724                 if (is_exactly_a<pseries>(it->rest))
 725                         op = it->rest;
 726                 else
 727                         op = it->rest.series(r, order, options);
 728                 if (!it->coeff.is_equal(_ex1))
 729                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 730
 731                 // Series addition
 732                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 733         }
 734         return acc;
 735 }
 736
 737
 738 /** Multiply a pseries object with a numeric constant, producing a pseries
 739  *  object that represents the product.
 740  *
 741  *  @param other  constant to multiply with
 742  *  @return the product as a pseries */
 743 ex pseries::mul_const(const numeric &other) const
 744 {
 745         epvector new_seq;
 746         new_seq.reserve(seq.size());
 747
 748         epvector::const_iterator it = seq.begin(), itend = seq.end();
 749         while (it != itend) {
 750                 if (!is_order_function(it->rest))
 751                         new_seq.push_back(expair(it->rest * other, it->coeff));
 752                 else
 753                         new_seq.push_back(*it);
 754                 ++it;
 755         }
 756         return pseries(relational(var,point), new_seq);
 757 }
 758
 759
 760 /** Multiply one pseries object to another, producing a pseries object that
 761  *  represents the product.
 762  *
 763  *  @param other  pseries object to multiply with
 764  *  @return the product as a pseries */
 765 ex pseries::mul_series(const pseries &other) const
 766 {
 767         // Multiplying two series with different variables or expansion points
 768         // results in an empty (constant) series
 769         if (!is_compatible_to(other)) {
 770                 epvector nul;
 771                 nul.push_back(expair(Order(_ex1), _ex0));
 772                 return pseries(relational(var,point), nul);
 773         }
 774
 775         if (seq.empty() || other.seq.empty()) {
 776                 return (new pseries(var==point, epvector()))
 777                        ->setflag(status_flags::dynallocated);
 778         }
 779
 780         // Series multiplication
 781         epvector new_seq;
 782         int a_max = degree(var);
 783         int b_max = other.degree(var);
 784         int a_min = ldegree(var);
 785         int b_min = other.ldegree(var);
 786         int cdeg_min = a_min + b_min;
 787         int cdeg_max = a_max + b_max;
 788
 789         int higher_order_a = INT_MAX;
 790         int higher_order_b = INT_MAX;
 791         if (is_order_function(coeff(var, a_max)))
 792                 higher_order_a = a_max + b_min;
 793         if (is_order_function(other.coeff(var, b_max)))
 794                 higher_order_b = b_max + a_min;
 795         int higher_order_c = std::min(higher_order_a, higher_order_b);
 796         if (cdeg_max >= higher_order_c)
 797                 cdeg_max = higher_order_c - 1;
 798
 799         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 800                 ex co = _ex0;
 801                 // c(i)=a(0)b(i)+...+a(i)b(0)
 802                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 803                         ex a_coeff = coeff(var, i);
 804                         ex b_coeff = other.coeff(var, cdeg-i);
 805                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 806                                 co += a_coeff * b_coeff;
 807                 }
 808                 if (!co.is_zero())
 809                         new_seq.push_back(expair(co, numeric(cdeg)));
 810         }
 811         if (higher_order_c < INT_MAX)
 812                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 813         return pseries(relational(var, point), new_seq);
 814 }
 815
 816
 817 /** Implementation of ex::series() for product. This performs series
 818  *  multiplication when multiplying series.
 819  *  @see ex::series */
 820 ex mul::series(const relational & r, int order, unsigned options) const
 821 {
 822         pseries acc; // Series accumulator
 823
 824         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 825         const ex& sym = r.lhs();
 826
 827         // holds ldegrees of the series of individual factors
 828         std::vector<int> ldegrees;
 829
 830         // find minimal degrees
 831         const epvector::const_iterator itbeg = seq.begin();
 832         const epvector::const_iterator itend = seq.end();
 833         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 834
 835                 ex expon = it->coeff;
 836                 int factor = 1;
 837                 ex buf;
 838                 if (expon.info(info_flags::integer)) {
 839                         buf = it->rest;
 840                         factor = ex_to<numeric>(expon).to_int();
 841                 } else {
 842                         buf = recombine_pair_to_ex(*it);
 843                 }
 844
 845                 int real_ldegree = 0;
 846                 try {
 847                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 848                 } catch (std::runtime_error) {}
 849
 850                 if (real_ldegree == 0) {
 851                         int orderloop = 0;
 852                         do {
 853                                 orderloop++;
 854                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 855                         } while (real_ldegree == orderloop);
 856                 }
 857
 858                 ldegrees.push_back(factor * real_ldegree);
 859         }
 860
 861         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 862
 863         if (degsum >= order) {
 864                 epvector epv;
 865                 epv.push_back(expair(Order(_ex1), order));
 866                 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
 867         }
 868
 869         // Multiply with remaining terms
 870         std::vector<int>::const_iterator itd = ldegrees.begin();
 871         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
 872
 873                 // do series expansion with adjusted order
 874                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 875
 876                 // Series multiplication
 877                 if (it == itbeg)
 878                         acc = ex_to<pseries>(op);
 879                 else
 880                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 881         }
 882
 883         return acc.mul_const(ex_to<numeric>(overall_coeff));
 884 }
 885
 886
 887 /** Compute the p-th power of a series.
 888  *
 889  *  @param p  power to compute
 890  *  @param deg  truncation order of series calculation */
 891 ex pseries::power_const(const numeric &p, int deg) const
 892 {
 893         // method:
 894         // (due to Leonhard Euler)
 895         // let A(x) be this series and for the time being let it start with a
 896         // constant (later we'll generalize):
 897         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 898         // We want to compute
 899         //     C(x) = A(x)^p
 900         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 901         // Taking the derivative on both sides and multiplying with A(x) one
 902         // immediately arrives at
 903         //     C'(x)*A(x) = p*C(x)*A'(x)
 904         // Multiplying this out and comparing coefficients we get the recurrence
 905         // formula
 906         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 907         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 908         // which can easily be solved given the starting value c_0 = (a_0)^p.
 909         // For the more general case where the leading coefficient of A(x) is not
 910         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 911         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 912         // then of course x^(p*m) but the recurrence formula still holds.
 913
 914         if (seq.empty()) {
 915                 // as a special case, handle the empty (zero) series honoring the
 916                 // usual power laws such as implemented in power::eval()
 917                 if (p.real().is_zero())
 918                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 919                 else if (p.real().is_negative())
 920                         throw pole_error("pseries::power_const(): division by zero",1);
 921                 else
 922                         return *this;
 923         }
 924
 925         const int ldeg = ldegree(var);
 926         if (!(p*ldeg).is_integer())
 927                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 928
 929         // adjust number of coefficients
 930         int numcoeff = deg - (p*ldeg).to_int();
 931         if (numcoeff <= 0) {
 932                 epvector epv;
 933                 epv.reserve(1);
 934                 epv.push_back(expair(Order(_ex1), deg));
 935                 return (new pseries(relational(var,point), epv))
 936                        ->setflag(status_flags::dynallocated);
 937         }
 938
 939         // O(x^n)^(-m) is undefined
 940         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 941                 throw pole_error("pseries::power_const(): division by zero",1);
 942
 943         // Compute coefficients of the powered series
 944         exvector co;
 945         co.reserve(numcoeff);
 946         co.push_back(power(coeff(var, ldeg), p));
 947         for (int i=1; i<numcoeff; ++i) {
 948                 ex sum = _ex0;
 949                 for (int j=1; j<=i; ++j) {
 950                         ex c = coeff(var, j + ldeg);
 951                         if (is_order_function(c)) {
 952                                 co.push_back(Order(_ex1));
 953                                 break;
 954                         } else
 955                                 sum += (p * j - (i - j)) * co[i - j] * c;
 956                 }
 957                 co.push_back(sum / coeff(var, ldeg) / i);
 958         }
 959
 960         // Construct new series (of non-zero coefficients)
 961         epvector new_seq;
 962         bool higher_order = false;
 963         for (int i=0; i<numcoeff; ++i) {
 964                 if (!co[i].is_zero())
 965                         new_seq.push_back(expair(co[i], p * ldeg + i));
 966                 if (is_order_function(co[i])) {
 967                         higher_order = true;
 968                         break;
 969                 }
 970         }
 971         if (!higher_order)
 972                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
 973
 974         return pseries(relational(var,point), new_seq);
 975 }
 976
 977
 978 /** Return a new pseries object with the powers shifted by deg. */
 979 pseries pseries::shift_exponents(int deg) const
 980 {
 981         epvector newseq = seq;
 982         epvector::iterator i = newseq.begin(), end  = newseq.end();
 983         while (i != end) {
 984                 i->coeff += deg;
 985                 ++i;
 986         }
 987         return pseries(relational(var, point), newseq);
 988 }
 989
 990
 991 /** Implementation of ex::series() for powers. This performs Laurent expansion
 992  *  of reciprocals of series at singularities.
 993  *  @see ex::series */
 994 ex power::series(const relational & r, int order, unsigned options) const
 995 {
 996         // If basis is already a series, just power it
 997         if (is_exactly_a<pseries>(basis))
 998                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 999
1000         // Basis is not a series, may there be a singularity?
1001         bool must_expand_basis = false;
1002         try {
1003                 basis.subs(r, subs_options::no_pattern);
1004         } catch (pole_error) {
1005                 must_expand_basis = true;
1006         }
1007
1008         // Is the expression of type something^(-int)?
1009         if (!must_expand_basis && !exponent.info(info_flags::negint)
1010          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1011                 return basic::series(r, order, options);
1012
1013         // Is the expression of type 0^something?
1014         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1015          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1016                 return basic::series(r, order, options);
1017
1018         // Singularity encountered, is the basis equal to (var - point)?
1019         if (basis.is_equal(r.lhs() - r.rhs())) {
1020                 epvector new_seq;
1021                 if (ex_to<numeric>(exponent).to_int() < order)
1022                         new_seq.push_back(expair(_ex1, exponent));
1023                 else
1024                         new_seq.push_back(expair(Order(_ex1), exponent));
1025                 return pseries(r, new_seq);
1026         }
1027
1028         // No, expand basis into series
1029
1030         numeric numexp;
1031         if (is_a<numeric>(exponent)) {
1032                 numexp = ex_to<numeric>(exponent);
1033         } else {
1034                 numexp = 0;
1035         }
1036         const ex& sym = r.lhs();
1037         // find existing minimal degree
1038         int real_ldegree = basis.expand().ldegree(sym-r.rhs());
1039         if (real_ldegree == 0) {
1040                 int orderloop = 0;
1041                 do {
1042                         orderloop++;
1043                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1044                 } while (real_ldegree == orderloop);
1045         }
1046
1047         if (!(real_ldegree*numexp).is_integer())
1048                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1049         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1050
1051         ex result;
1052         try {
1053                 result = ex_to<pseries>(e).power_const(numexp, order);
1054         } catch (pole_error) {
1055                 epvector ser;
1056                 ser.push_back(expair(Order(_ex1), order));
1057                 result = pseries(r, ser);
1058         }
1059
1060         return result;
1061 }
1062
1063
1064 /** Re-expansion of a pseries object. */
1065 ex pseries::series(const relational & r, int order, unsigned options) const
1066 {
1067         const ex p = r.rhs();
1068         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1069         const symbol &s = ex_to<symbol>(r.lhs());
1070
1071         if (var.is_equal(s) && point.is_equal(p)) {
1072                 if (order > degree(s))
1073                         return *this;
1074                 else {
1075                         epvector new_seq;
1076                         epvector::const_iterator it = seq.begin(), itend = seq.end();
1077                         while (it != itend) {
1078                                 int o = ex_to<numeric>(it->coeff).to_int();
1079                                 if (o >= order) {
1080                                         new_seq.push_back(expair(Order(_ex1), o));
1081                                         break;
1082                                 }
1083                                 new_seq.push_back(*it);
1084                                 ++it;
1085                         }
1086                         return pseries(r, new_seq);
1087                 }
1088         } else
1089                 return convert_to_poly().series(r, order, options);
1090 }
1091
1092 ex integral::series(const relational & r, int order, unsigned options) const
1093 {
1094         if (x.subs(r) != x)
1095                 throw std::logic_error("Cannot series expand wrt dummy variable");
1096
1097         // Expanding integrant with r substituted taken in boundaries.
1098         ex fseries = f.series(r, order, options);
1099         epvector fexpansion;
1100         fexpansion.reserve(fseries.nops());
1101         for (size_t i=0; i<fseries.nops(); ++i) {
1102                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1103                 currcoeff = (currcoeff == Order(_ex1))
1104                         ? currcoeff
1105                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1106                 if (currcoeff != 0)
1107                         fexpansion.push_back(
1108                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1109         }
1110
1111         // Expanding lower boundary
1112         ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1113         ex aseries = (a-a.subs(r)).series(r, order, options);
1114         fseries = f.series(x == (a.subs(r)), order, options);
1115         for (size_t i=0; i<fseries.nops(); ++i) {
1116                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1117                 if (is_order_function(currcoeff))
1118                         break;
1119                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1120                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1121                 currcoeff = currcoeff.series(r, orderforf);
1122                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1123                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1124                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1125                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1126         }
1127
1128         // Expanding upper boundary
1129         ex bseries = (b-b.subs(r)).series(r, order, options);
1130         fseries = f.series(x == (b.subs(r)), order, options);
1131         for (size_t i=0; i<fseries.nops(); ++i) {
1132                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1133                 if (is_order_function(currcoeff))
1134                         break;
1135                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1136                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1137                 currcoeff = currcoeff.series(r, orderforf);
1138                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1139                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1140                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1141                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1142         }
1143
1144         return result;
1145 }
1146
1147
1148 /** Compute the truncated series expansion of an expression.
1149  *  This function returns an expression containing an object of class pseries
1150  *  to represent the series. If the series does not terminate within the given
1151  *  truncation order, the last term of the series will be an order term.
1152  *
1153  *  @param r  expansion relation, lhs holds variable and rhs holds point
1154  *  @param order  truncation order of series calculations
1155  *  @param options  of class series_options
1156  *  @return an expression holding a pseries object */
1157 ex ex::series(const ex & r, int order, unsigned options) const
1158 {
1159         ex e;
1160         relational rel_;
1161
1162         if (is_a<relational>(r))
1163                 rel_ = ex_to<relational>(r);
1164         else if (is_a<symbol>(r))
1165                 rel_ = relational(r,_ex0);
1166         else
1167                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1168
1169         try {
1170                 e = bp->series(rel_, order, options);
1171         } catch (std::exception &x) {
1172                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1173         }
1174         return e;
1175 }
1176
1177 } // namespace GiNaC