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) 1999-2015 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 "pseries.h"
  25 #include "add.h"
  26 #include "inifcns.h" // for Order function
  27 #include "lst.h"
  28 #include "mul.h"
  29 #include "power.h"
  30 #include "relational.h"
  31 #include "operators.h"
  32 #include "symbol.h"
  33 #include "integral.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36
  37 #include <limits>
  38 #include <numeric>
  39 #include <stdexcept>
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
  44   print_func<print_context>(&pseries::do_print).
  45   print_func<print_latex>(&pseries::do_print_latex).
  46   print_func<print_tree>(&pseries::do_print_tree).
  47   print_func<print_python>(&pseries::do_print_python).
  48   print_func<print_python_repr>(&pseries::do_print_python_repr))
  49
  50
  51 /*
  52  *  Default constructor
  53  */
  54
  55 pseries::pseries() { }
  56
  57
  58 /*
  59  *  Other ctors
  60  */
  61
  62 /** Construct pseries from a vector of coefficients and powers.
  63  *  expair.rest holds the coefficient, expair.coeff holds the power.
  64  *  The powers must be integers (positive or negative) and in ascending order;
  65  *  the last coefficient can be Order(_ex1) to represent a truncated,
  66  *  non-terminating series.
  67  *
  68  *  @param rel_  expansion variable and point (must hold a relational)
  69  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  70  *  @return newly constructed pseries */
  71 pseries::pseries(const ex &rel_, const epvector &ops_)
  72   : seq(ops_)
  73 {
  74         GINAC_ASSERT(is_a<relational>(rel_));
  75         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  76         point = rel_.rhs();
  77         var = rel_.lhs();
  78 }
  79 pseries::pseries(const ex &rel_, epvector &&ops_)
  80   : seq(std::move(ops_))
  81 {
  82         GINAC_ASSERT(is_a<relational>(rel_));
  83         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  84         point = rel_.rhs();
  85         var = rel_.lhs();
  86 }
  87
  88
  89 /*
  90  *  Archiving
  91  */
  92
  93 void pseries::read_archive(const archive_node &n, lst &sym_lst)
  94 {
  95         inherited::read_archive(n, sym_lst);
  96         auto first = n.find_first("coeff");
  97         auto last = n.find_last("power");
  98         ++last;
  99         seq.reserve((last-first)/2);
 100
 101         for (auto loc = first; loc < last;) {
 102                 ex rest;
 103                 ex coeff;
 104                 n.find_ex_by_loc(loc++, rest, sym_lst);
 105                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 106                 seq.push_back(expair(rest, coeff));
 107         }
 108
 109         n.find_ex("var", var, sym_lst);
 110         n.find_ex("point", point, sym_lst);
 111 }
 112
 113 void pseries::archive(archive_node &n) const
 114 {
 115         inherited::archive(n);
 116         for (auto & it : seq) {
 117                 n.add_ex("coeff", it.rest);
 118                 n.add_ex("power", it.coeff);
 119         }
 120         n.add_ex("var", var);
 121         n.add_ex("point", point);
 122 }
 123
 124
 125 //////////
 126 // functions overriding virtual functions from base classes
 127 //////////
 128
 129 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
 130 {
 131         if (precedence() <= level)
 132                 c.s << '(';
 133
 134         // objects of type pseries must not have any zero entries, so the
 135         // trivial (zero) pseries needs a special treatment here:
 136         if (seq.empty())
 137                 c.s << '0';
 138
 139         auto i = seq.begin(), end = seq.end();
 140         while (i != end) {
 141
 142                 // print a sign, if needed
 143                 if (i != seq.begin())
 144                         c.s << '+';
 145
 146                 if (!is_order_function(i->rest)) {
 147
 148                         // print 'rest', i.e. the expansion coefficient
 149                         if (i->rest.info(info_flags::numeric) &&
 150                                 i->rest.info(info_flags::positive)) {
 151                                 i->rest.print(c);
 152                         } else {
 153                                 c.s << openbrace << '(';
 154                                 i->rest.print(c);
 155                                 c.s << ')' << closebrace;
 156                         }
 157
 158                         // print 'coeff', something like (x-1)^42
 159                         if (!i->coeff.is_zero()) {
 160                                 c.s << mul_sym;
 161                                 if (!point.is_zero()) {
 162                                         c.s << openbrace << '(';
 163                                         (var-point).print(c);
 164                                         c.s << ')' << closebrace;
 165                                 } else
 166                                         var.print(c);
 167                                 if (i->coeff.compare(_ex1)) {
 168                                         c.s << pow_sym;
 169                                         c.s << openbrace;
 170                                         if (i->coeff.info(info_flags::negative)) {
 171                                                 c.s << '(';
 172                                                 i->coeff.print(c);
 173                                                 c.s << ')';
 174                                         } else
 175                                                 i->coeff.print(c);
 176                                         c.s << closebrace;
 177                                 }
 178                         }
 179                 } else
 180                         Order(pow(var - point, i->coeff)).print(c);
 181                 ++i;
 182         }
 183
 184         if (precedence() <= level)
 185                 c.s << ')';
 186 }
 187
 188 void pseries::do_print(const print_context & c, unsigned level) const
 189 {
 190         print_series(c, "", "", "*", "^", level);
 191 }
 192
 193 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 194 {
 195         print_series(c, "{", "}", " ", "^", level);
 196 }
 197
 198 void pseries::do_print_python(const print_python & c, unsigned level) const
 199 {
 200         print_series(c, "", "", "*", "**", level);
 201 }
 202
 203 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 204 {
 205         c.s << std::string(level, ' ') << class_name() << " @" << this
 206             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 207             << std::endl;
 208         size_t num = seq.size();
 209         for (size_t i=0; i<num; ++i) {
 210                 seq[i].rest.print(c, level + c.delta_indent);
 211                 seq[i].coeff.print(c, level + c.delta_indent);
 212                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 213         }
 214         var.print(c, level + c.delta_indent);
 215         point.print(c, level + c.delta_indent);
 216 }
 217
 218 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 219 {
 220         c.s << class_name() << "(relational(";
 221         var.print(c);
 222         c.s << ',';
 223         point.print(c);
 224         c.s << "),[";
 225         size_t num = seq.size();
 226         for (size_t i=0; i<num; ++i) {
 227                 if (i)
 228                         c.s << ',';
 229                 c.s << '(';
 230                 seq[i].rest.print(c);
 231                 c.s << ',';
 232                 seq[i].coeff.print(c);
 233                 c.s << ')';
 234         }
 235         c.s << "])";
 236 }
 237
 238 int pseries::compare_same_type(const basic & other) const
 239 {
 240         GINAC_ASSERT(is_a<pseries>(other));
 241         const pseries &o = static_cast<const pseries &>(other);
 242
 243         // first compare the lengths of the series...
 244         if (seq.size()>o.seq.size())
 245                 return 1;
 246         if (seq.size()<o.seq.size())
 247                 return -1;
 248
 249         // ...then the expansion point...
 250         int cmpval = var.compare(o.var);
 251         if (cmpval)
 252                 return cmpval;
 253         cmpval = point.compare(o.point);
 254         if (cmpval)
 255                 return cmpval;
 256
 257         // ...and if that failed the individual elements
 258         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 259         while (it!=seq.end() && o_it!=o.seq.end()) {
 260                 cmpval = it->compare(*o_it);
 261                 if (cmpval)
 262                         return cmpval;
 263                 ++it;
 264                 ++o_it;
 265         }
 266
 267         // so they are equal.
 268         return 0;
 269 }
 270
 271 /** Return the number of operands including a possible order term. */
 272 size_t pseries::nops() const
 273 {
 274         return seq.size();
 275 }
 276
 277 /** Return the ith term in the series when represented as a sum. */
 278 ex pseries::op(size_t i) const
 279 {
 280         if (i >= seq.size())
 281                 throw (std::out_of_range("op() out of range"));
 282
 283         if (is_order_function(seq[i].rest))
 284                 return Order(pow(var-point, seq[i].coeff));
 285         return seq[i].rest * pow(var - point, seq[i].coeff);
 286 }
 287
 288 /** Return degree of highest power of the series.  This is usually the exponent
 289  *  of the Order term.  If s is not the expansion variable of the series, the
 290  *  series is examined termwise. */
 291 int pseries::degree(const ex &s) const
 292 {
 293         if (var.is_equal(s)) {
 294                 // Return last exponent
 295                 if (seq.size())
 296                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 297                 else
 298                         return 0;
 299         } else {
 300                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 301                 if (it == itend)
 302                         return 0;
 303                 int max_pow = std::numeric_limits<int>::min();
 304                 while (it != itend) {
 305                         int pow = it->rest.degree(s);
 306                         if (pow > max_pow)
 307                                 max_pow = pow;
 308                         ++it;
 309                 }
 310                 return max_pow;
 311         }
 312 }
 313
 314 /** Return degree of lowest power of the series.  This is usually the exponent
 315  *  of the leading term.  If s is not the expansion variable of the series, the
 316  *  series is examined termwise.  If s is the expansion variable but the
 317  *  expansion point is not zero the series is not expanded to find the degree.
 318  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 319 int pseries::ldegree(const ex &s) const
 320 {
 321         if (var.is_equal(s)) {
 322                 // Return first exponent
 323                 if (seq.size())
 324                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 325                 else
 326                         return 0;
 327         } else {
 328                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 329                 if (it == itend)
 330                         return 0;
 331                 int min_pow = std::numeric_limits<int>::max();
 332                 while (it != itend) {
 333                         int pow = it->rest.ldegree(s);
 334                         if (pow < min_pow)
 335                                 min_pow = pow;
 336                         ++it;
 337                 }
 338                 return min_pow;
 339         }
 340 }
 341
 342 /** Return coefficient of degree n in power series if s is the expansion
 343  *  variable.  If the expansion point is nonzero, by definition the n=1
 344  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 345  *  the expansion took place in the s in the first place).
 346  *  If s is not the expansion variable, an attempt is made to convert the
 347  *  series to a polynomial and return the corresponding coefficient from
 348  *  there. */
 349 ex pseries::coeff(const ex &s, int n) const
 350 {
 351         if (var.is_equal(s)) {
 352                 if (seq.empty())
 353                         return _ex0;
 354
 355                 // Binary search in sequence for given power
 356                 numeric looking_for = numeric(n);
 357                 int lo = 0, hi = seq.size() - 1;
 358                 while (lo <= hi) {
 359                         int mid = (lo + hi) / 2;
 360                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 361                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 362                         switch (cmp) {
 363                                 case -1:
 364                                         lo = mid + 1;
 365                                         break;
 366                                 case 0:
 367                                         return seq[mid].rest;
 368                                 case 1:
 369                                         hi = mid - 1;
 370                                         break;
 371                                 default:
 372                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 373                         }
 374                 }
 375                 return _ex0;
 376         } else
 377                 return convert_to_poly().coeff(s, n);
 378 }
 379
 380 /** Does nothing. */
 381 ex pseries::collect(const ex &s, bool distributed) const
 382 {
 383         return *this;
 384 }
 385
 386 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 387 ex pseries::eval() const
 388 {
 389         if (flags & status_flags::evaluated) {
 390                 return *this;
 391         }
 392
 393         // Construct a new series with evaluated coefficients
 394         epvector new_seq;
 395         new_seq.reserve(seq.size());
 396         epvector::const_iterator it = seq.begin(), itend = seq.end();
 397         while (it != itend) {
 398                 new_seq.push_back(expair(it->rest, it->coeff));
 399                 ++it;
 400         }
 401         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 402 }
 403
 404 /** Evaluate coefficients numerically. */
 405 ex pseries::evalf(int level) const
 406 {
 407         if (level == 1)
 408                 return *this;
 409
 410         if (level == -max_recursion_level)
 411                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 412
 413         // Construct a new series with evaluated coefficients
 414         epvector new_seq;
 415         new_seq.reserve(seq.size());
 416         epvector::const_iterator it = seq.begin(), itend = seq.end();
 417         while (it != itend) {
 418                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 419                 ++it;
 420         }
 421         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 422 }
 423
 424 ex pseries::conjugate() const
 425 {
 426         if(!var.info(info_flags::real))
 427                 return conjugate_function(*this).hold();
 428
 429         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
 430         ex newpoint = point.conjugate();
 431
 432         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 433                 return *this;
 434         }
 435
 436         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
 437 }
 438
 439 ex pseries::real_part() const
 440 {
 441         if(!var.info(info_flags::real))
 442                 return real_part_function(*this).hold();
 443         ex newpoint = point.real_part();
 444         if(newpoint != point)
 445                 return real_part_function(*this).hold();
 446
 447         epvector v;
 448         v.reserve(seq.size());
 449         for (auto & it : seq)
 450                 v.push_back(expair((it.rest).real_part(), it.coeff));
 451         return dynallocate<pseries>(var==point, std::move(v));
 452 }
 453
 454 ex pseries::imag_part() const
 455 {
 456         if(!var.info(info_flags::real))
 457                 return imag_part_function(*this).hold();
 458         ex newpoint = point.real_part();
 459         if(newpoint != point)
 460                 return imag_part_function(*this).hold();
 461
 462         epvector v;
 463         v.reserve(seq.size());
 464         for (auto & it : seq)
 465                 v.push_back(expair((it.rest).imag_part(), it.coeff));
 466         return dynallocate<pseries>(var==point, std::move(v));
 467 }
 468
 469 ex pseries::eval_integ() const
 470 {
 471         std::unique_ptr<epvector> newseq(nullptr);
 472         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 473                 if (newseq) {
 474                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 475                         continue;
 476                 }
 477                 ex newterm = i->rest.eval_integ();
 478                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 479                         newseq.reset(new epvector);
 480                         newseq->reserve(seq.size());
 481                         for (auto j=seq.begin(); j!=i; ++j)
 482                                 newseq->push_back(*j);
 483                         newseq->push_back(expair(newterm, i->coeff));
 484                 }
 485         }
 486
 487         ex newpoint = point.eval_integ();
 488         if (newseq || !are_ex_trivially_equal(newpoint, point))
 489                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
 490         return *this;
 491 }
 492
 493 ex pseries::evalm() const
 494 {
 495         // evalm each coefficient
 496         epvector newseq;
 497         bool something_changed = false;
 498         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 499                 if (something_changed) {
 500                         ex newcoeff = i->rest.evalm();
 501                         if (!newcoeff.is_zero())
 502                                 newseq.push_back(expair(newcoeff, i->coeff));
 503                 } else {
 504                         ex newcoeff = i->rest.evalm();
 505                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 506                                 something_changed = true;
 507                                 newseq.reserve(seq.size());
 508                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 509                                 if (!newcoeff.is_zero())
 510                                         newseq.push_back(expair(newcoeff, i->coeff));
 511                         }
 512                 }
 513         }
 514         if (something_changed)
 515                 return dynallocate<pseries>(var==point, std::move(newseq));
 516         else
 517                 return *this;
 518 }
 519
 520 ex pseries::subs(const exmap & m, unsigned options) const
 521 {
 522         // If expansion variable is being substituted, convert the series to a
 523         // polynomial and do the substitution there because the result might
 524         // no longer be a power series
 525         if (m.find(var) != m.end())
 526                 return convert_to_poly(true).subs(m, options);
 527
 528         // Otherwise construct a new series with substituted coefficients and
 529         // expansion point
 530         epvector newseq;
 531         newseq.reserve(seq.size());
 532         for (auto & it : seq)
 533                 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
 534         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
 535 }
 536
 537 /** Implementation of ex::expand() for a power series.  It expands all the
 538  *  terms individually and returns the resulting series as a new pseries. */
 539 ex pseries::expand(unsigned options) const
 540 {
 541         epvector newseq;
 542         for (auto & it : seq) {
 543                 ex restexp = it.rest.expand();
 544                 if (!restexp.is_zero())
 545                         newseq.push_back(expair(restexp, it.coeff));
 546         }
 547         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
 548 }
 549
 550 /** Implementation of ex::diff() for a power series.
 551  *  @see ex::diff */
 552 ex pseries::derivative(const symbol & s) const
 553 {
 554         epvector new_seq;
 555
 556         if (s == var) {
 557
 558                 // FIXME: coeff might depend on var
 559                 for (auto & it : seq) {
 560                         if (is_order_function(it.rest)) {
 561                                 new_seq.push_back(expair(it.rest, it.coeff - 1));
 562                         } else {
 563                                 ex c = it.rest * it.coeff;
 564                                 if (!c.is_zero())
 565                                         new_seq.push_back(expair(c, it.coeff - 1));
 566                         }
 567                 }
 568
 569         } else {
 570
 571                 for (auto & it : seq) {
 572                         if (is_order_function(it.rest)) {
 573                                 new_seq.push_back(it);
 574                         } else {
 575                                 ex c = it.rest.diff(s);
 576                                 if (!c.is_zero())
 577                                         new_seq.push_back(expair(c, it.coeff));
 578                         }
 579                 }
 580         }
 581
 582         return pseries(relational(var,point), std::move(new_seq));
 583 }
 584
 585 ex pseries::convert_to_poly(bool no_order) const
 586 {
 587         ex e;
 588         for (auto & it : seq) {
 589                 if (is_order_function(it.rest)) {
 590                         if (!no_order)
 591                                 e += Order(pow(var - point, it.coeff));
 592                 } else
 593                         e += it.rest * pow(var - point, it.coeff);
 594         }
 595         return e;
 596 }
 597
 598 bool pseries::is_terminating() const
 599 {
 600         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 601 }
 602
 603 ex pseries::coeffop(size_t i) const
 604 {
 605         if (i >= nops())
 606                 throw (std::out_of_range("coeffop() out of range"));
 607         return seq[i].rest;
 608 }
 609
 610 ex pseries::exponop(size_t i) const
 611 {
 612         if (i >= nops())
 613                 throw (std::out_of_range("exponop() out of range"));
 614         return seq[i].coeff;
 615 }
 616
 617
 618 /*
 619  *  Implementations of series expansion
 620  */
 621
 622 /** Default implementation of ex::series(). This performs Taylor expansion.
 623  *  @see ex::series */
 624 ex basic::series(const relational & r, int order, unsigned options) const
 625 {
 626         epvector seq;
 627         const symbol &s = ex_to<symbol>(r.lhs());
 628
 629         // default for order-values that make no sense for Taylor expansion
 630         if ((order <= 0) && this->has(s)) {
 631                 seq.push_back(expair(Order(_ex1), order));
 632                 return pseries(r, std::move(seq));
 633         }
 634
 635         // do Taylor expansion
 636         numeric fac = 1;
 637         ex deriv = *this;
 638         ex coeff = deriv.subs(r, subs_options::no_pattern);
 639
 640         if (!coeff.is_zero()) {
 641                 seq.push_back(expair(coeff, _ex0));
 642         }
 643
 644         int n;
 645         for (n=1; n<order; ++n) {
 646                 fac = fac.div(n);
 647                 // We need to test for zero in order to see if the series terminates.
 648                 // The problem is that there is no such thing as a perfect test for
 649                 // zero.  Expanding the term occasionally helps a little...
 650                 deriv = deriv.diff(s).expand();
 651                 if (deriv.is_zero())  // Series terminates
 652                         return pseries(r, std::move(seq));
 653
 654                 coeff = deriv.subs(r, subs_options::no_pattern);
 655                 if (!coeff.is_zero())
 656                         seq.push_back(expair(fac * coeff, n));
 657         }
 658
 659         // Higher-order terms, if present
 660         deriv = deriv.diff(s);
 661         if (!deriv.expand().is_zero())
 662                 seq.push_back(expair(Order(_ex1), n));
 663         return pseries(r, std::move(seq));
 664 }
 665
 666
 667 /** Implementation of ex::series() for symbols.
 668  *  @see ex::series */
 669 ex symbol::series(const relational & r, int order, unsigned options) const
 670 {
 671         epvector seq;
 672         const ex point = r.rhs();
 673         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 674
 675         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 676                 if (order > 0 && !point.is_zero())
 677                         seq.push_back(expair(point, _ex0));
 678                 if (order > 1)
 679                         seq.push_back(expair(_ex1, _ex1));
 680                 else
 681                         seq.push_back(expair(Order(_ex1), numeric(order)));
 682         } else
 683                 seq.push_back(expair(*this, _ex0));
 684         return pseries(r, std::move(seq));
 685 }
 686
 687
 688 /** Add one series object to another, producing a pseries object that
 689  *  represents the sum.
 690  *
 691  *  @param other  pseries object to add with
 692  *  @return the sum as a pseries */
 693 ex pseries::add_series(const pseries &other) const
 694 {
 695         // Adding two series with different variables or expansion points
 696         // results in an empty (constant) series
 697         if (!is_compatible_to(other)) {
 698                 epvector nul { expair(Order(_ex1), _ex0) };
 699                 return pseries(relational(var,point), std::move(nul));
 700         }
 701
 702         // Series addition
 703         epvector new_seq;
 704         auto a = seq.begin(), a_end = seq.end();
 705         auto b = other.seq.begin(), b_end = other.seq.end();
 706         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 707         for (;;) {
 708                 // If a is empty, fill up with elements from b and stop
 709                 if (a == a_end) {
 710                         while (b != b_end) {
 711                                 new_seq.push_back(*b);
 712                                 ++b;
 713                         }
 714                         break;
 715                 } else
 716                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 717
 718                 // If b is empty, fill up with elements from a and stop
 719                 if (b == b_end) {
 720                         while (a != a_end) {
 721                                 new_seq.push_back(*a);
 722                                 ++a;
 723                         }
 724                         break;
 725                 } else
 726                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 727
 728                 // a and b are non-empty, compare powers
 729                 if (pow_a < pow_b) {
 730                         // a has lesser power, get coefficient from a
 731                         new_seq.push_back(*a);
 732                         if (is_order_function((*a).rest))
 733                                 break;
 734                         ++a;
 735                 } else if (pow_b < pow_a) {
 736                         // b has lesser power, get coefficient from b
 737                         new_seq.push_back(*b);
 738                         if (is_order_function((*b).rest))
 739                                 break;
 740                         ++b;
 741                 } else {
 742                         // Add coefficient of a and b
 743                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 744                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 745                                 break;  // Order term ends the sequence
 746                         } else {
 747                                 ex sum = (*a).rest + (*b).rest;
 748                                 if (!(sum.is_zero()))
 749                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 750                                 ++a;
 751                                 ++b;
 752                         }
 753                 }
 754         }
 755         return pseries(relational(var,point), std::move(new_seq));
 756 }
 757
 758
 759 /** Implementation of ex::series() for sums. This performs series addition when
 760  *  adding pseries objects.
 761  *  @see ex::series */
 762 ex add::series(const relational & r, int order, unsigned options) const
 763 {
 764         ex acc; // Series accumulator
 765
 766         // Get first term from overall_coeff
 767         acc = overall_coeff.series(r, order, options);
 768
 769         // Add remaining terms
 770         for (auto & it : seq) {
 771                 ex op;
 772                 if (is_exactly_a<pseries>(it.rest))
 773                         op = it.rest;
 774                 else
 775                         op = it.rest.series(r, order, options);
 776                 if (!it.coeff.is_equal(_ex1))
 777                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
 778
 779                 // Series addition
 780                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 781         }
 782         return acc;
 783 }
 784
 785
 786 /** Multiply a pseries object with a numeric constant, producing a pseries
 787  *  object that represents the product.
 788  *
 789  *  @param other  constant to multiply with
 790  *  @return the product as a pseries */
 791 ex pseries::mul_const(const numeric &other) const
 792 {
 793         epvector new_seq;
 794         new_seq.reserve(seq.size());
 795
 796         for (auto & it : seq) {
 797                 if (!is_order_function(it.rest))
 798                         new_seq.push_back(expair(it.rest * other, it.coeff));
 799                 else
 800                         new_seq.push_back(it);
 801         }
 802         return pseries(relational(var,point), std::move(new_seq));
 803 }
 804
 805
 806 /** Multiply one pseries object to another, producing a pseries object that
 807  *  represents the product.
 808  *
 809  *  @param other  pseries object to multiply with
 810  *  @return the product as a pseries */
 811 ex pseries::mul_series(const pseries &other) const
 812 {
 813         // Multiplying two series with different variables or expansion points
 814         // results in an empty (constant) series
 815         if (!is_compatible_to(other)) {
 816                 epvector nul { expair(Order(_ex1), _ex0) };
 817                 return pseries(relational(var,point), std::move(nul));
 818         }
 819
 820         if (seq.empty() || other.seq.empty()) {
 821                 return dynallocate<pseries>(var==point, epvector());
 822         }
 823
 824         // Series multiplication
 825         epvector new_seq;
 826         int a_max = degree(var);
 827         int b_max = other.degree(var);
 828         int a_min = ldegree(var);
 829         int b_min = other.ldegree(var);
 830         int cdeg_min = a_min + b_min;
 831         int cdeg_max = a_max + b_max;
 832
 833         int higher_order_a = std::numeric_limits<int>::max();
 834         int higher_order_b = std::numeric_limits<int>::max();
 835         if (is_order_function(coeff(var, a_max)))
 836                 higher_order_a = a_max + b_min;
 837         if (is_order_function(other.coeff(var, b_max)))
 838                 higher_order_b = b_max + a_min;
 839         int higher_order_c = std::min(higher_order_a, higher_order_b);
 840         if (cdeg_max >= higher_order_c)
 841                 cdeg_max = higher_order_c - 1;
 842
 843         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 844                 ex co = _ex0;
 845                 // c(i)=a(0)b(i)+...+a(i)b(0)
 846                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 847                         ex a_coeff = coeff(var, i);
 848                         ex b_coeff = other.coeff(var, cdeg-i);
 849                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 850                                 co += a_coeff * b_coeff;
 851                 }
 852                 if (!co.is_zero())
 853                         new_seq.push_back(expair(co, numeric(cdeg)));
 854         }
 855         if (higher_order_c < std::numeric_limits<int>::max())
 856                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 857         return pseries(relational(var, point), std::move(new_seq));
 858 }
 859
 860
 861 /** Implementation of ex::series() for product. This performs series
 862  *  multiplication when multiplying series.
 863  *  @see ex::series */
 864 ex mul::series(const relational & r, int order, unsigned options) const
 865 {
 866         pseries acc; // Series accumulator
 867
 868         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 869         const ex& sym = r.lhs();
 870
 871         // holds ldegrees of the series of individual factors
 872         std::vector<int> ldegrees;
 873         std::vector<bool> ldegree_redo;
 874
 875         // find minimal degrees
 876         // first round: obtain a bound up to which minimal degrees have to be
 877         // considered
 878         for (auto & it : seq) {
 879
 880                 ex expon = it.coeff;
 881                 int factor = 1;
 882                 ex buf;
 883                 if (expon.info(info_flags::integer)) {
 884                         buf = it.rest;
 885                         factor = ex_to<numeric>(expon).to_int();
 886                 } else {
 887                         buf = recombine_pair_to_ex(it);
 888                 }
 889
 890                 int real_ldegree = 0;
 891                 bool flag_redo = false;
 892                 try {
 893                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 894                 } catch (std::runtime_error) {}
 895
 896                 if (real_ldegree == 0) {
 897                         if ( factor < 0 ) {
 898                                 // This case must terminate, otherwise we would have division by
 899                                 // zero.
 900                                 int orderloop = 0;
 901                                 do {
 902                                         orderloop++;
 903                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 904                                 } while (real_ldegree == orderloop);
 905                         } else {
 906                                 // Here it is possible that buf does not have a ldegree, therefore
 907                                 // check only if ldegree is negative, otherwise reconsider the case
 908                                 // in the second round.
 909                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 910                                 if (real_ldegree == 0)
 911                                         flag_redo = true;
 912                         }
 913                 }
 914
 915                 ldegrees.push_back(factor * real_ldegree);
 916                 ldegree_redo.push_back(flag_redo);
 917         }
 918
 919         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 920         // Second round: determine the remaining positive ldegrees by the series
 921         // method.
 922         // here we can ignore ldegrees larger than degbound
 923         size_t j = 0;
 924         for (auto & it : seq) {
 925                 if ( ldegree_redo[j] ) {
 926                         ex expon = it.coeff;
 927                         int factor = 1;
 928                         ex buf;
 929                         if (expon.info(info_flags::integer)) {
 930                                 buf = it.rest;
 931                                 factor = ex_to<numeric>(expon).to_int();
 932                         } else {
 933                                 buf = recombine_pair_to_ex(it);
 934                         }
 935                         int real_ldegree = 0;
 936                         int orderloop = 0;
 937                         do {
 938                                 orderloop++;
 939                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 940                         } while ((real_ldegree == orderloop)
 941                               && (factor*real_ldegree < degbound));
 942                         ldegrees[j] = factor * real_ldegree;
 943                         degbound -= factor * real_ldegree;
 944                 }
 945                 j++;
 946         }
 947
 948         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 949
 950         if (degsum >= order) {
 951                 epvector epv { expair(Order(_ex1), order) };
 952                 return dynallocate<pseries>(r, std::move(epv));
 953         }
 954
 955         // Multiply with remaining terms
 956         auto itd = ldegrees.begin();
 957         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 958
 959                 // do series expansion with adjusted order
 960                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 961
 962                 // Series multiplication
 963                 if (it == seq.begin())
 964                         acc = ex_to<pseries>(op);
 965                 else
 966                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 967         }
 968
 969         return acc.mul_const(ex_to<numeric>(overall_coeff));
 970 }
 971
 972
 973 /** Compute the p-th power of a series.
 974  *
 975  *  @param p  power to compute
 976  *  @param deg  truncation order of series calculation */
 977 ex pseries::power_const(const numeric &p, int deg) const
 978 {
 979         // method:
 980         // (due to Leonhard Euler)
 981         // let A(x) be this series and for the time being let it start with a
 982         // constant (later we'll generalize):
 983         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 984         // We want to compute
 985         //     C(x) = A(x)^p
 986         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 987         // Taking the derivative on both sides and multiplying with A(x) one
 988         // immediately arrives at
 989         //     C'(x)*A(x) = p*C(x)*A'(x)
 990         // Multiplying this out and comparing coefficients we get the recurrence
 991         // formula
 992         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 993         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 994         // which can easily be solved given the starting value c_0 = (a_0)^p.
 995         // For the more general case where the leading coefficient of A(x) is not
 996         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 997         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 998         // then of course x^(p*m) but the recurrence formula still holds.
 999
1000         if (seq.empty()) {
1001                 // as a special case, handle the empty (zero) series honoring the
1002                 // usual power laws such as implemented in power::eval()
1003                 if (p.real().is_zero())
1004                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1005                 else if (p.real().is_negative())
1006                         throw pole_error("pseries::power_const(): division by zero",1);
1007                 else
1008                         return *this;
1009         }
1010
1011         const int ldeg = ldegree(var);
1012         if (!(p*ldeg).is_integer())
1013                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1014
1015         // adjust number of coefficients
1016         int numcoeff = deg - (p*ldeg).to_int();
1017         if (numcoeff <= 0) {
1018                 epvector epv { expair(Order(_ex1), deg) };
1019                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1020         }
1021
1022         // O(x^n)^(-m) is undefined
1023         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1024                 throw pole_error("pseries::power_const(): division by zero",1);
1025
1026         // Compute coefficients of the powered series
1027         exvector co;
1028         co.reserve(numcoeff);
1029         co.push_back(pow(coeff(var, ldeg), p));
1030         for (int i=1; i<numcoeff; ++i) {
1031                 ex sum = _ex0;
1032                 for (int j=1; j<=i; ++j) {
1033                         ex c = coeff(var, j + ldeg);
1034                         if (is_order_function(c)) {
1035                                 co.push_back(Order(_ex1));
1036                                 break;
1037                         } else
1038                                 sum += (p * j - (i - j)) * co[i - j] * c;
1039                 }
1040                 co.push_back(sum / coeff(var, ldeg) / i);
1041         }
1042
1043         // Construct new series (of non-zero coefficients)
1044         epvector new_seq;
1045         bool higher_order = false;
1046         for (int i=0; i<numcoeff; ++i) {
1047                 if (!co[i].is_zero())
1048                         new_seq.push_back(expair(co[i], p * ldeg + i));
1049                 if (is_order_function(co[i])) {
1050                         higher_order = true;
1051                         break;
1052                 }
1053         }
1054         if (!higher_order)
1055                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1056
1057         return pseries(relational(var,point), std::move(new_seq));
1058 }
1059
1060
1061 /** Return a new pseries object with the powers shifted by deg. */
1062 pseries pseries::shift_exponents(int deg) const
1063 {
1064         epvector newseq = seq;
1065         for (auto & it : newseq)
1066                 it.coeff += deg;
1067         return pseries(relational(var, point), std::move(newseq));
1068 }
1069
1070
1071 /** Implementation of ex::series() for powers. This performs Laurent expansion
1072  *  of reciprocals of series at singularities.
1073  *  @see ex::series */
1074 ex power::series(const relational & r, int order, unsigned options) const
1075 {
1076         // If basis is already a series, just power it
1077         if (is_exactly_a<pseries>(basis))
1078                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1079
1080         // Basis is not a series, may there be a singularity?
1081         bool must_expand_basis = false;
1082         try {
1083                 basis.subs(r, subs_options::no_pattern);
1084         } catch (pole_error) {
1085                 must_expand_basis = true;
1086         }
1087
1088         bool exponent_is_regular = true;
1089         try {
1090                 exponent.subs(r, subs_options::no_pattern);
1091         } catch (pole_error) {
1092                 exponent_is_regular = false;
1093         }
1094
1095         if (!exponent_is_regular) {
1096                 ex l = exponent*log(basis);
1097                 // this == exp(l);
1098                 ex le = l.series(r, order, options);
1099                 // Note: expanding exp(l) won't help, since that will attempt
1100                 // Taylor expansion, and fail (because exponent is "singular")
1101                 // Still l itself might be expanded in Taylor series.
1102                 // Examples:
1103                 // sin(x)/x*log(cos(x))
1104                 // 1/x*log(1 + x)
1105                 return exp(le).series(r, order, options);
1106                 // Note: if l happens to have a Laurent expansion (with
1107                 // negative powers of (var - point)), expanding exp(le)
1108                 // will barf (which is The Right Thing).
1109         }
1110
1111         // Is the expression of type something^(-int)?
1112         if (!must_expand_basis && !exponent.info(info_flags::negint)
1113          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1114                 return basic::series(r, order, options);
1115
1116         // Is the expression of type 0^something?
1117         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1118          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1119                 return basic::series(r, order, options);
1120
1121         // Singularity encountered, is the basis equal to (var - point)?
1122         if (basis.is_equal(r.lhs() - r.rhs())) {
1123                 epvector new_seq;
1124                 if (ex_to<numeric>(exponent).to_int() < order)
1125                         new_seq.push_back(expair(_ex1, exponent));
1126                 else
1127                         new_seq.push_back(expair(Order(_ex1), exponent));
1128                 return pseries(r, std::move(new_seq));
1129         }
1130
1131         // No, expand basis into series
1132
1133         numeric numexp;
1134         if (is_a<numeric>(exponent)) {
1135                 numexp = ex_to<numeric>(exponent);
1136         } else {
1137                 numexp = 0;
1138         }
1139         const ex& sym = r.lhs();
1140         // find existing minimal degree
1141         ex eb = basis.expand();
1142         int real_ldegree = 0;
1143         if (eb.info(info_flags::rational_function))
1144                 real_ldegree = eb.ldegree(sym-r.rhs());
1145         if (real_ldegree == 0) {
1146                 int orderloop = 0;
1147                 do {
1148                         orderloop++;
1149                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1150                 } while (real_ldegree == orderloop);
1151         }
1152
1153         if (!(real_ldegree*numexp).is_integer())
1154                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1155         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1156
1157         ex result;
1158         try {
1159                 result = ex_to<pseries>(e).power_const(numexp, order);
1160         } catch (pole_error) {
1161                 epvector ser { expair(Order(_ex1), order) };
1162                 result = pseries(r, std::move(ser));
1163         }
1164
1165         return result;
1166 }
1167
1168
1169 /** Re-expansion of a pseries object. */
1170 ex pseries::series(const relational & r, int order, unsigned options) const
1171 {
1172         const ex p = r.rhs();
1173         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1174         const symbol &s = ex_to<symbol>(r.lhs());
1175
1176         if (var.is_equal(s) && point.is_equal(p)) {
1177                 if (order > degree(s))
1178                         return *this;
1179                 else {
1180                         epvector new_seq;
1181                         for (auto & it : seq) {
1182                                 int o = ex_to<numeric>(it.coeff).to_int();
1183                                 if (o >= order) {
1184                                         new_seq.push_back(expair(Order(_ex1), o));
1185                                         break;
1186                                 }
1187                                 new_seq.push_back(it);
1188                         }
1189                         return pseries(r, std::move(new_seq));
1190                 }
1191         } else
1192                 return convert_to_poly().series(r, order, options);
1193 }
1194
1195 ex integral::series(const relational & r, int order, unsigned options) const
1196 {
1197         if (x.subs(r) != x)
1198                 throw std::logic_error("Cannot series expand wrt dummy variable");
1199
1200         // Expanding integrand with r substituted taken in boundaries.
1201         ex fseries = f.series(r, order, options);
1202         epvector fexpansion;
1203         fexpansion.reserve(fseries.nops());
1204         for (size_t i=0; i<fseries.nops(); ++i) {
1205                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1206                 currcoeff = (currcoeff == Order(_ex1))
1207                         ? currcoeff
1208                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1209                 if (currcoeff != 0)
1210                         fexpansion.push_back(
1211                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1212         }
1213
1214         // Expanding lower boundary
1215         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1216         ex aseries = (a-a.subs(r)).series(r, order, options);
1217         fseries = f.series(x == (a.subs(r)), order, options);
1218         for (size_t i=0; i<fseries.nops(); ++i) {
1219                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1220                 if (is_order_function(currcoeff))
1221                         break;
1222                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1223                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1224                 currcoeff = currcoeff.series(r, orderforf);
1225                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1226                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1227                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1228                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1229         }
1230
1231         // Expanding upper boundary
1232         ex bseries = (b-b.subs(r)).series(r, order, options);
1233         fseries = f.series(x == (b.subs(r)), order, options);
1234         for (size_t i=0; i<fseries.nops(); ++i) {
1235                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1236                 if (is_order_function(currcoeff))
1237                         break;
1238                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1239                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1240                 currcoeff = currcoeff.series(r, orderforf);
1241                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1242                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1243                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1244                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1245         }
1246
1247         return result;
1248 }
1249
1250
1251 /** Compute the truncated series expansion of an expression.
1252  *  This function returns an expression containing an object of class pseries
1253  *  to represent the series. If the series does not terminate within the given
1254  *  truncation order, the last term of the series will be an order term.
1255  *
1256  *  @param r  expansion relation, lhs holds variable and rhs holds point
1257  *  @param order  truncation order of series calculations
1258  *  @param options  of class series_options
1259  *  @return an expression holding a pseries object */
1260 ex ex::series(const ex & r, int order, unsigned options) const
1261 {
1262         ex e;
1263         relational rel_;
1264
1265         if (is_a<relational>(r))
1266                 rel_ = ex_to<relational>(r);
1267         else if (is_a<symbol>(r))
1268                 rel_ = relational(r,_ex0);
1269         else
1270                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1271
1272         e = bp->series(rel_, order, options);
1273         return e;
1274 }
1275
1276 GINAC_BIND_UNARCHIVER(pseries);
1277
1278 } // namespace GiNaC