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(power(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(power(var-point, seq[i].coeff));
 285         return seq[i].rest * power(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(int level) const
 388 {
 389         if (level == 1)
 390                 return this->hold();
 391
 392         if (level == -max_recursion_level)
 393                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 394
 395         // Construct a new series with evaluated coefficients
 396         epvector new_seq;
 397         new_seq.reserve(seq.size());
 398         epvector::const_iterator it = seq.begin(), itend = seq.end();
 399         while (it != itend) {
 400                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 401                 ++it;
 402         }
 403         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 404 }
 405
 406 /** Evaluate coefficients numerically. */
 407 ex pseries::evalf(int level) const
 408 {
 409         if (level == 1)
 410                 return *this;
 411
 412         if (level == -max_recursion_level)
 413                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 414
 415         // Construct a new series with evaluated coefficients
 416         epvector new_seq;
 417         new_seq.reserve(seq.size());
 418         epvector::const_iterator it = seq.begin(), itend = seq.end();
 419         while (it != itend) {
 420                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 421                 ++it;
 422         }
 423         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 424 }
 425
 426 ex pseries::conjugate() const
 427 {
 428         if(!var.info(info_flags::real))
 429                 return conjugate_function(*this).hold();
 430
 431         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
 432         ex newpoint = point.conjugate();
 433
 434         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 435                 return *this;
 436         }
 437
 438         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
 439 }
 440
 441 ex pseries::real_part() const
 442 {
 443         if(!var.info(info_flags::real))
 444                 return real_part_function(*this).hold();
 445         ex newpoint = point.real_part();
 446         if(newpoint != point)
 447                 return real_part_function(*this).hold();
 448
 449         epvector v;
 450         v.reserve(seq.size());
 451         for (auto & it : seq)
 452                 v.push_back(expair((it.rest).real_part(), it.coeff));
 453         return dynallocate<pseries>(var==point, std::move(v));
 454 }
 455
 456 ex pseries::imag_part() const
 457 {
 458         if(!var.info(info_flags::real))
 459                 return imag_part_function(*this).hold();
 460         ex newpoint = point.real_part();
 461         if(newpoint != point)
 462                 return imag_part_function(*this).hold();
 463
 464         epvector v;
 465         v.reserve(seq.size());
 466         for (auto & it : seq)
 467                 v.push_back(expair((it.rest).imag_part(), it.coeff));
 468         return dynallocate<pseries>(var==point, std::move(v));
 469 }
 470
 471 ex pseries::eval_integ() const
 472 {
 473         std::unique_ptr<epvector> newseq(nullptr);
 474         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 475                 if (newseq) {
 476                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 477                         continue;
 478                 }
 479                 ex newterm = i->rest.eval_integ();
 480                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 481                         newseq.reset(new epvector);
 482                         newseq->reserve(seq.size());
 483                         for (auto j=seq.begin(); j!=i; ++j)
 484                                 newseq->push_back(*j);
 485                         newseq->push_back(expair(newterm, i->coeff));
 486                 }
 487         }
 488
 489         ex newpoint = point.eval_integ();
 490         if (newseq || !are_ex_trivially_equal(newpoint, point))
 491                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
 492         return *this;
 493 }
 494
 495 ex pseries::evalm() const
 496 {
 497         // evalm each coefficient
 498         epvector newseq;
 499         bool something_changed = false;
 500         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 501                 if (something_changed) {
 502                         ex newcoeff = i->rest.evalm();
 503                         if (!newcoeff.is_zero())
 504                                 newseq.push_back(expair(newcoeff, i->coeff));
 505                 }
 506                 else {
 507                         ex newcoeff = i->rest.evalm();
 508                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 509                                 something_changed = true;
 510                                 newseq.reserve(seq.size());
 511                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 512                                 if (!newcoeff.is_zero())
 513                                         newseq.push_back(expair(newcoeff, i->coeff));
 514                         }
 515                 }
 516         }
 517         if (something_changed)
 518                 return dynallocate<pseries>(var==point, std::move(newseq));
 519         else
 520                 return *this;
 521 }
 522
 523 ex pseries::subs(const exmap & m, unsigned options) const
 524 {
 525         // If expansion variable is being substituted, convert the series to a
 526         // polynomial and do the substitution there because the result might
 527         // no longer be a power series
 528         if (m.find(var) != m.end())
 529                 return convert_to_poly(true).subs(m, options);
 530
 531         // Otherwise construct a new series with substituted coefficients and
 532         // expansion point
 533         epvector newseq;
 534         newseq.reserve(seq.size());
 535         for (auto & it : seq)
 536                 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
 537         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
 538 }
 539
 540 /** Implementation of ex::expand() for a power series.  It expands all the
 541  *  terms individually and returns the resulting series as a new pseries. */
 542 ex pseries::expand(unsigned options) const
 543 {
 544         epvector newseq;
 545         for (auto & it : seq) {
 546                 ex restexp = it.rest.expand();
 547                 if (!restexp.is_zero())
 548                         newseq.push_back(expair(restexp, it.coeff));
 549         }
 550         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
 551 }
 552
 553 /** Implementation of ex::diff() for a power series.
 554  *  @see ex::diff */
 555 ex pseries::derivative(const symbol & s) const
 556 {
 557         epvector new_seq;
 558
 559         if (s == var) {
 560
 561                 // FIXME: coeff might depend on var
 562                 for (auto & it : seq) {
 563                         if (is_order_function(it.rest)) {
 564                                 new_seq.push_back(expair(it.rest, it.coeff - 1));
 565                         } else {
 566                                 ex c = it.rest * it.coeff;
 567                                 if (!c.is_zero())
 568                                         new_seq.push_back(expair(c, it.coeff - 1));
 569                         }
 570                 }
 571
 572         } else {
 573
 574                 for (auto & it : seq) {
 575                         if (is_order_function(it.rest)) {
 576                                 new_seq.push_back(it);
 577                         } else {
 578                                 ex c = it.rest.diff(s);
 579                                 if (!c.is_zero())
 580                                         new_seq.push_back(expair(c, it.coeff));
 581                         }
 582                 }
 583         }
 584
 585         return pseries(relational(var,point), std::move(new_seq));
 586 }
 587
 588 ex pseries::convert_to_poly(bool no_order) const
 589 {
 590         ex e;
 591         for (auto & it : seq) {
 592                 if (is_order_function(it.rest)) {
 593                         if (!no_order)
 594                                 e += Order(power(var - point, it.coeff));
 595                 } else
 596                         e += it.rest * power(var - point, it.coeff);
 597         }
 598         return e;
 599 }
 600
 601 bool pseries::is_terminating() const
 602 {
 603         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 604 }
 605
 606 ex pseries::coeffop(size_t i) const
 607 {
 608         if (i >= nops())
 609                 throw (std::out_of_range("coeffop() out of range"));
 610         return seq[i].rest;
 611 }
 612
 613 ex pseries::exponop(size_t i) const
 614 {
 615         if (i >= nops())
 616                 throw (std::out_of_range("exponop() out of range"));
 617         return seq[i].coeff;
 618 }
 619
 620
 621 /*
 622  *  Implementations of series expansion
 623  */
 624
 625 /** Default implementation of ex::series(). This performs Taylor expansion.
 626  *  @see ex::series */
 627 ex basic::series(const relational & r, int order, unsigned options) const
 628 {
 629         epvector seq;
 630         const symbol &s = ex_to<symbol>(r.lhs());
 631
 632         // default for order-values that make no sense for Taylor expansion
 633         if ((order <= 0) && this->has(s)) {
 634                 seq.push_back(expair(Order(_ex1), order));
 635                 return pseries(r, std::move(seq));
 636         }
 637
 638         // do Taylor expansion
 639         numeric fac = 1;
 640         ex deriv = *this;
 641         ex coeff = deriv.subs(r, subs_options::no_pattern);
 642
 643         if (!coeff.is_zero()) {
 644                 seq.push_back(expair(coeff, _ex0));
 645         }
 646
 647         int n;
 648         for (n=1; n<order; ++n) {
 649                 fac = fac.mul(n);
 650                 // We need to test for zero in order to see if the series terminates.
 651                 // The problem is that there is no such thing as a perfect test for
 652                 // zero.  Expanding the term occasionally helps a little...
 653                 deriv = deriv.diff(s).expand();
 654                 if (deriv.is_zero())  // Series terminates
 655                         return pseries(r, std::move(seq));
 656
 657                 coeff = deriv.subs(r, subs_options::no_pattern);
 658                 if (!coeff.is_zero())
 659                         seq.push_back(expair(fac.inverse() * coeff, n));
 660         }
 661
 662         // Higher-order terms, if present
 663         deriv = deriv.diff(s);
 664         if (!deriv.expand().is_zero())
 665                 seq.push_back(expair(Order(_ex1), n));
 666         return pseries(r, std::move(seq));
 667 }
 668
 669
 670 /** Implementation of ex::series() for symbols.
 671  *  @see ex::series */
 672 ex symbol::series(const relational & r, int order, unsigned options) const
 673 {
 674         epvector seq;
 675         const ex point = r.rhs();
 676         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 677
 678         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 679                 if (order > 0 && !point.is_zero())
 680                         seq.push_back(expair(point, _ex0));
 681                 if (order > 1)
 682                         seq.push_back(expair(_ex1, _ex1));
 683                 else
 684                         seq.push_back(expair(Order(_ex1), numeric(order)));
 685         } else
 686                 seq.push_back(expair(*this, _ex0));
 687         return pseries(r, std::move(seq));
 688 }
 689
 690
 691 /** Add one series object to another, producing a pseries object that
 692  *  represents the sum.
 693  *
 694  *  @param other  pseries object to add with
 695  *  @return the sum as a pseries */
 696 ex pseries::add_series(const pseries &other) const
 697 {
 698         // Adding two series with different variables or expansion points
 699         // results in an empty (constant) series
 700         if (!is_compatible_to(other)) {
 701                 epvector nul { expair(Order(_ex1), _ex0) };
 702                 return pseries(relational(var,point), std::move(nul));
 703         }
 704
 705         // Series addition
 706         epvector new_seq;
 707         auto a = seq.begin(), a_end = seq.end();
 708         auto b = other.seq.begin(), b_end = other.seq.end();
 709         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 710         for (;;) {
 711                 // If a is empty, fill up with elements from b and stop
 712                 if (a == a_end) {
 713                         while (b != b_end) {
 714                                 new_seq.push_back(*b);
 715                                 ++b;
 716                         }
 717                         break;
 718                 } else
 719                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 720
 721                 // If b is empty, fill up with elements from a and stop
 722                 if (b == b_end) {
 723                         while (a != a_end) {
 724                                 new_seq.push_back(*a);
 725                                 ++a;
 726                         }
 727                         break;
 728                 } else
 729                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 730
 731                 // a and b are non-empty, compare powers
 732                 if (pow_a < pow_b) {
 733                         // a has lesser power, get coefficient from a
 734                         new_seq.push_back(*a);
 735                         if (is_order_function((*a).rest))
 736                                 break;
 737                         ++a;
 738                 } else if (pow_b < pow_a) {
 739                         // b has lesser power, get coefficient from b
 740                         new_seq.push_back(*b);
 741                         if (is_order_function((*b).rest))
 742                                 break;
 743                         ++b;
 744                 } else {
 745                         // Add coefficient of a and b
 746                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 747                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 748                                 break;  // Order term ends the sequence
 749                         } else {
 750                                 ex sum = (*a).rest + (*b).rest;
 751                                 if (!(sum.is_zero()))
 752                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 753                                 ++a;
 754                                 ++b;
 755                         }
 756                 }
 757         }
 758         return pseries(relational(var,point), std::move(new_seq));
 759 }
 760
 761
 762 /** Implementation of ex::series() for sums. This performs series addition when
 763  *  adding pseries objects.
 764  *  @see ex::series */
 765 ex add::series(const relational & r, int order, unsigned options) const
 766 {
 767         ex acc; // Series accumulator
 768
 769         // Get first term from overall_coeff
 770         acc = overall_coeff.series(r, order, options);
 771
 772         // Add remaining terms
 773         for (auto & it : seq) {
 774                 ex op;
 775                 if (is_exactly_a<pseries>(it.rest))
 776                         op = it.rest;
 777                 else
 778                         op = it.rest.series(r, order, options);
 779                 if (!it.coeff.is_equal(_ex1))
 780                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
 781
 782                 // Series addition
 783                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 784         }
 785         return acc;
 786 }
 787
 788
 789 /** Multiply a pseries object with a numeric constant, producing a pseries
 790  *  object that represents the product.
 791  *
 792  *  @param other  constant to multiply with
 793  *  @return the product as a pseries */
 794 ex pseries::mul_const(const numeric &other) const
 795 {
 796         epvector new_seq;
 797         new_seq.reserve(seq.size());
 798
 799         for (auto & it : seq) {
 800                 if (!is_order_function(it.rest))
 801                         new_seq.push_back(expair(it.rest * other, it.coeff));
 802                 else
 803                         new_seq.push_back(it);
 804         }
 805         return pseries(relational(var,point), std::move(new_seq));
 806 }
 807
 808
 809 /** Multiply one pseries object to another, producing a pseries object that
 810  *  represents the product.
 811  *
 812  *  @param other  pseries object to multiply with
 813  *  @return the product as a pseries */
 814 ex pseries::mul_series(const pseries &other) const
 815 {
 816         // Multiplying two series with different variables or expansion points
 817         // results in an empty (constant) series
 818         if (!is_compatible_to(other)) {
 819                 epvector nul { expair(Order(_ex1), _ex0) };
 820                 return pseries(relational(var,point), std::move(nul));
 821         }
 822
 823         if (seq.empty() || other.seq.empty()) {
 824                 return dynallocate<pseries>(var==point, epvector());
 825         }
 826
 827         // Series multiplication
 828         epvector new_seq;
 829         int a_max = degree(var);
 830         int b_max = other.degree(var);
 831         int a_min = ldegree(var);
 832         int b_min = other.ldegree(var);
 833         int cdeg_min = a_min + b_min;
 834         int cdeg_max = a_max + b_max;
 835
 836         int higher_order_a = std::numeric_limits<int>::max();
 837         int higher_order_b = std::numeric_limits<int>::max();
 838         if (is_order_function(coeff(var, a_max)))
 839                 higher_order_a = a_max + b_min;
 840         if (is_order_function(other.coeff(var, b_max)))
 841                 higher_order_b = b_max + a_min;
 842         int higher_order_c = std::min(higher_order_a, higher_order_b);
 843         if (cdeg_max >= higher_order_c)
 844                 cdeg_max = higher_order_c - 1;
 845
 846         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 847                 ex co = _ex0;
 848                 // c(i)=a(0)b(i)+...+a(i)b(0)
 849                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 850                         ex a_coeff = coeff(var, i);
 851                         ex b_coeff = other.coeff(var, cdeg-i);
 852                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 853                                 co += a_coeff * b_coeff;
 854                 }
 855                 if (!co.is_zero())
 856                         new_seq.push_back(expair(co, numeric(cdeg)));
 857         }
 858         if (higher_order_c < std::numeric_limits<int>::max())
 859                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 860         return pseries(relational(var, point), std::move(new_seq));
 861 }
 862
 863
 864 /** Implementation of ex::series() for product. This performs series
 865  *  multiplication when multiplying series.
 866  *  @see ex::series */
 867 ex mul::series(const relational & r, int order, unsigned options) const
 868 {
 869         pseries acc; // Series accumulator
 870
 871         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 872         const ex& sym = r.lhs();
 873
 874         // holds ldegrees of the series of individual factors
 875         std::vector<int> ldegrees;
 876         std::vector<bool> ldegree_redo;
 877
 878         // find minimal degrees
 879         // first round: obtain a bound up to which minimal degrees have to be
 880         // considered
 881         for (auto & it : seq) {
 882
 883                 ex expon = it.coeff;
 884                 int factor = 1;
 885                 ex buf;
 886                 if (expon.info(info_flags::integer)) {
 887                         buf = it.rest;
 888                         factor = ex_to<numeric>(expon).to_int();
 889                 } else {
 890                         buf = recombine_pair_to_ex(it);
 891                 }
 892
 893                 int real_ldegree = 0;
 894                 bool flag_redo = false;
 895                 try {
 896                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 897                 } catch (std::runtime_error) {}
 898
 899                 if (real_ldegree == 0) {
 900                         if ( factor < 0 ) {
 901                                 // This case must terminate, otherwise we would have division by
 902                                 // zero.
 903                                 int orderloop = 0;
 904                                 do {
 905                                         orderloop++;
 906                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 907                                 } while (real_ldegree == orderloop);
 908                         } else {
 909                                 // Here it is possible that buf does not have a ldegree, therefore
 910                                 // check only if ldegree is negative, otherwise reconsider the case
 911                                 // in the second round.
 912                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 913                                 if (real_ldegree == 0)
 914                                         flag_redo = true;
 915                         }
 916                 }
 917
 918                 ldegrees.push_back(factor * real_ldegree);
 919                 ldegree_redo.push_back(flag_redo);
 920         }
 921
 922         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 923         // Second round: determine the remaining positive ldegrees by the series
 924         // method.
 925         // here we can ignore ldegrees larger than degbound
 926         size_t j = 0;
 927         for (auto & it : seq) {
 928                 if ( ldegree_redo[j] ) {
 929                         ex expon = it.coeff;
 930                         int factor = 1;
 931                         ex buf;
 932                         if (expon.info(info_flags::integer)) {
 933                                 buf = it.rest;
 934                                 factor = ex_to<numeric>(expon).to_int();
 935                         } else {
 936                                 buf = recombine_pair_to_ex(it);
 937                         }
 938                         int real_ldegree = 0;
 939                         int orderloop = 0;
 940                         do {
 941                                 orderloop++;
 942                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 943                         } while ((real_ldegree == orderloop)
 944                               && (factor*real_ldegree < degbound));
 945                         ldegrees[j] = factor * real_ldegree;
 946                         degbound -= factor * real_ldegree;
 947                 }
 948                 j++;
 949         }
 950
 951         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 952
 953         if (degsum >= order) {
 954                 epvector epv { expair(Order(_ex1), order) };
 955                 return dynallocate<pseries>(r, std::move(epv));
 956         }
 957
 958         // Multiply with remaining terms
 959         auto itd = ldegrees.begin();
 960         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 961
 962                 // do series expansion with adjusted order
 963                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 964
 965                 // Series multiplication
 966                 if (it == seq.begin())
 967                         acc = ex_to<pseries>(op);
 968                 else
 969                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 970         }
 971
 972         return acc.mul_const(ex_to<numeric>(overall_coeff));
 973 }
 974
 975
 976 /** Compute the p-th power of a series.
 977  *
 978  *  @param p  power to compute
 979  *  @param deg  truncation order of series calculation */
 980 ex pseries::power_const(const numeric &p, int deg) const
 981 {
 982         // method:
 983         // (due to Leonhard Euler)
 984         // let A(x) be this series and for the time being let it start with a
 985         // constant (later we'll generalize):
 986         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 987         // We want to compute
 988         //     C(x) = A(x)^p
 989         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 990         // Taking the derivative on both sides and multiplying with A(x) one
 991         // immediately arrives at
 992         //     C'(x)*A(x) = p*C(x)*A'(x)
 993         // Multiplying this out and comparing coefficients we get the recurrence
 994         // formula
 995         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 996         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 997         // which can easily be solved given the starting value c_0 = (a_0)^p.
 998         // For the more general case where the leading coefficient of A(x) is not
 999         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1000         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
1001         // then of course x^(p*m) but the recurrence formula still holds.
1002
1003         if (seq.empty()) {
1004                 // as a special case, handle the empty (zero) series honoring the
1005                 // usual power laws such as implemented in power::eval()
1006                 if (p.real().is_zero())
1007                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1008                 else if (p.real().is_negative())
1009                         throw pole_error("pseries::power_const(): division by zero",1);
1010                 else
1011                         return *this;
1012         }
1013
1014         const int ldeg = ldegree(var);
1015         if (!(p*ldeg).is_integer())
1016                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1017
1018         // adjust number of coefficients
1019         int numcoeff = deg - (p*ldeg).to_int();
1020         if (numcoeff <= 0) {
1021                 epvector epv { expair(Order(_ex1), deg) };
1022                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1023         }
1024
1025         // O(x^n)^(-m) is undefined
1026         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1027                 throw pole_error("pseries::power_const(): division by zero",1);
1028
1029         // Compute coefficients of the powered series
1030         exvector co;
1031         co.reserve(numcoeff);
1032         co.push_back(power(coeff(var, ldeg), p));
1033         for (int i=1; i<numcoeff; ++i) {
1034                 ex sum = _ex0;
1035                 for (int j=1; j<=i; ++j) {
1036                         ex c = coeff(var, j + ldeg);
1037                         if (is_order_function(c)) {
1038                                 co.push_back(Order(_ex1));
1039                                 break;
1040                         } else
1041                                 sum += (p * j - (i - j)) * co[i - j] * c;
1042                 }
1043                 co.push_back(sum / coeff(var, ldeg) / i);
1044         }
1045
1046         // Construct new series (of non-zero coefficients)
1047         epvector new_seq;
1048         bool higher_order = false;
1049         for (int i=0; i<numcoeff; ++i) {
1050                 if (!co[i].is_zero())
1051                         new_seq.push_back(expair(co[i], p * ldeg + i));
1052                 if (is_order_function(co[i])) {
1053                         higher_order = true;
1054                         break;
1055                 }
1056         }
1057         if (!higher_order)
1058                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1059
1060         return pseries(relational(var,point), std::move(new_seq));
1061 }
1062
1063
1064 /** Return a new pseries object with the powers shifted by deg. */
1065 pseries pseries::shift_exponents(int deg) const
1066 {
1067         epvector newseq = seq;
1068         for (auto & it : newseq)
1069                 it.coeff += deg;
1070         return pseries(relational(var, point), std::move(newseq));
1071 }
1072
1073
1074 /** Implementation of ex::series() for powers. This performs Laurent expansion
1075  *  of reciprocals of series at singularities.
1076  *  @see ex::series */
1077 ex power::series(const relational & r, int order, unsigned options) const
1078 {
1079         // If basis is already a series, just power it
1080         if (is_exactly_a<pseries>(basis))
1081                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1082
1083         // Basis is not a series, may there be a singularity?
1084         bool must_expand_basis = false;
1085         try {
1086                 basis.subs(r, subs_options::no_pattern);
1087         } catch (pole_error) {
1088                 must_expand_basis = true;
1089         }
1090
1091         bool exponent_is_regular = true;
1092         try {
1093                 exponent.subs(r, subs_options::no_pattern);
1094         } catch (pole_error) {
1095                 exponent_is_regular = false;
1096         }
1097
1098         if (!exponent_is_regular) {
1099                 ex l = exponent*log(basis);
1100                 // this == exp(l);
1101                 ex le = l.series(r, order, options);
1102                 // Note: expanding exp(l) won't help, since that will attempt
1103                 // Taylor expansion, and fail (because exponent is "singular")
1104                 // Still l itself might be expanded in Taylor series.
1105                 // Examples:
1106                 // sin(x)/x*log(cos(x))
1107                 // 1/x*log(1 + x)
1108                 return exp(le).series(r, order, options);
1109                 // Note: if l happens to have a Laurent expansion (with
1110                 // negative powers of (var - point)), expanding exp(le)
1111                 // will barf (which is The Right Thing).
1112         }
1113
1114         // Is the expression of type something^(-int)?
1115         if (!must_expand_basis && !exponent.info(info_flags::negint)
1116          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1117                 return basic::series(r, order, options);
1118
1119         // Is the expression of type 0^something?
1120         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1121          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1122                 return basic::series(r, order, options);
1123
1124         // Singularity encountered, is the basis equal to (var - point)?
1125         if (basis.is_equal(r.lhs() - r.rhs())) {
1126                 epvector new_seq;
1127                 if (ex_to<numeric>(exponent).to_int() < order)
1128                         new_seq.push_back(expair(_ex1, exponent));
1129                 else
1130                         new_seq.push_back(expair(Order(_ex1), exponent));
1131                 return pseries(r, std::move(new_seq));
1132         }
1133
1134         // No, expand basis into series
1135
1136         numeric numexp;
1137         if (is_a<numeric>(exponent)) {
1138                 numexp = ex_to<numeric>(exponent);
1139         } else {
1140                 numexp = 0;
1141         }
1142         const ex& sym = r.lhs();
1143         // find existing minimal degree
1144         ex eb = basis.expand();
1145         int real_ldegree = 0;
1146         if (eb.info(info_flags::rational_function))
1147                 real_ldegree = eb.ldegree(sym-r.rhs());
1148         if (real_ldegree == 0) {
1149                 int orderloop = 0;
1150                 do {
1151                         orderloop++;
1152                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1153                 } while (real_ldegree == orderloop);
1154         }
1155
1156         if (!(real_ldegree*numexp).is_integer())
1157                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1158         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1159
1160         ex result;
1161         try {
1162                 result = ex_to<pseries>(e).power_const(numexp, order);
1163         } catch (pole_error) {
1164                 epvector ser { expair(Order(_ex1), order) };
1165                 result = pseries(r, std::move(ser));
1166         }
1167
1168         return result;
1169 }
1170
1171
1172 /** Re-expansion of a pseries object. */
1173 ex pseries::series(const relational & r, int order, unsigned options) const
1174 {
1175         const ex p = r.rhs();
1176         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1177         const symbol &s = ex_to<symbol>(r.lhs());
1178
1179         if (var.is_equal(s) && point.is_equal(p)) {
1180                 if (order > degree(s))
1181                         return *this;
1182                 else {
1183                         epvector new_seq;
1184                         for (auto & it : seq) {
1185                                 int o = ex_to<numeric>(it.coeff).to_int();
1186                                 if (o >= order) {
1187                                         new_seq.push_back(expair(Order(_ex1), o));
1188                                         break;
1189                                 }
1190                                 new_seq.push_back(it);
1191                         }
1192                         return pseries(r, std::move(new_seq));
1193                 }
1194         } else
1195                 return convert_to_poly().series(r, order, options);
1196 }
1197
1198 ex integral::series(const relational & r, int order, unsigned options) const
1199 {
1200         if (x.subs(r) != x)
1201                 throw std::logic_error("Cannot series expand wrt dummy variable");
1202
1203         // Expanding integrand with r substituted taken in boundaries.
1204         ex fseries = f.series(r, order, options);
1205         epvector fexpansion;
1206         fexpansion.reserve(fseries.nops());
1207         for (size_t i=0; i<fseries.nops(); ++i) {
1208                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1209                 currcoeff = (currcoeff == Order(_ex1))
1210                         ? currcoeff
1211                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1212                 if (currcoeff != 0)
1213                         fexpansion.push_back(
1214                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1215         }
1216
1217         // Expanding lower boundary
1218         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1219         ex aseries = (a-a.subs(r)).series(r, order, options);
1220         fseries = f.series(x == (a.subs(r)), order, options);
1221         for (size_t i=0; i<fseries.nops(); ++i) {
1222                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1223                 if (is_order_function(currcoeff))
1224                         break;
1225                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1226                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1227                 currcoeff = currcoeff.series(r, orderforf);
1228                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1229                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1230                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1231                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1232         }
1233
1234         // Expanding upper boundary
1235         ex bseries = (b-b.subs(r)).series(r, order, options);
1236         fseries = f.series(x == (b.subs(r)), order, options);
1237         for (size_t i=0; i<fseries.nops(); ++i) {
1238                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1239                 if (is_order_function(currcoeff))
1240                         break;
1241                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1242                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1243                 currcoeff = currcoeff.series(r, orderforf);
1244                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1245                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1246                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1247                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1248         }
1249
1250         return result;
1251 }
1252
1253
1254 /** Compute the truncated series expansion of an expression.
1255  *  This function returns an expression containing an object of class pseries
1256  *  to represent the series. If the series does not terminate within the given
1257  *  truncation order, the last term of the series will be an order term.
1258  *
1259  *  @param r  expansion relation, lhs holds variable and rhs holds point
1260  *  @param order  truncation order of series calculations
1261  *  @param options  of class series_options
1262  *  @return an expression holding a pseries object */
1263 ex ex::series(const ex & r, int order, unsigned options) const
1264 {
1265         ex e;
1266         relational rel_;
1267
1268         if (is_a<relational>(r))
1269                 rel_ = ex_to<relational>(r);
1270         else if (is_a<symbol>(r))
1271                 rel_ = relational(r,_ex0);
1272         else
1273                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1274
1275         e = bp->series(rel_, order, options);
1276         return e;
1277 }
1278
1279 GINAC_BIND_UNARCHIVER(pseries);
1280
1281 } // namespace GiNaC