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) 1999-2018 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 #ifdef DO_GINAC_ASSERT
  75         auto i = seq.begin();
  76         while (i != seq.end()) {
  77                 auto ip1 = i+1;
  78                 if (ip1 != seq.end())
  79                         GINAC_ASSERT(!is_order_function(i->rest));
  80                 else
  81                         break;
  82                 GINAC_ASSERT(is_a<numeric>(i->coeff));
  83                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
  84                 ++i;
  85         }
  86 #endif // def DO_GINAC_ASSERT
  87         GINAC_ASSERT(is_a<relational>(rel_));
  88         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  89         point = rel_.rhs();
  90         var = rel_.lhs();
  91 }
  92 pseries::pseries(const ex &rel_, epvector &&ops_)
  93   : seq(std::move(ops_))
  94 {
  95 #ifdef DO_GINAC_ASSERT
  96         auto i = seq.begin();
  97         while (i != seq.end()) {
  98                 auto ip1 = i+1;
  99                 if (ip1 != seq.end())
 100                         GINAC_ASSERT(!is_order_function(i->rest));
 101                 else
 102                         break;
 103                 GINAC_ASSERT(is_a<numeric>(i->coeff));
 104                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
 105                 ++i;
 106         }
 107 #endif // def DO_GINAC_ASSERT
 108         GINAC_ASSERT(is_a<relational>(rel_));
 109         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
 110         point = rel_.rhs();
 111         var = rel_.lhs();
 112 }
 113
 114
 115 /*
 116  *  Archiving
 117  */
 118
 119 void pseries::read_archive(const archive_node &n, lst &sym_lst)
 120 {
 121         inherited::read_archive(n, sym_lst);
 122         auto first = n.find_first("coeff");
 123         auto last = n.find_last("power");
 124         ++last;
 125         seq.reserve((last-first)/2);
 126
 127         for (auto loc = first; loc < last;) {
 128                 ex rest;
 129                 ex coeff;
 130                 n.find_ex_by_loc(loc++, rest, sym_lst);
 131                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 132                 seq.push_back(expair(rest, coeff));
 133         }
 134
 135         n.find_ex("var", var, sym_lst);
 136         n.find_ex("point", point, sym_lst);
 137 }
 138
 139 void pseries::archive(archive_node &n) const
 140 {
 141         inherited::archive(n);
 142         for (auto & it : seq) {
 143                 n.add_ex("coeff", it.rest);
 144                 n.add_ex("power", it.coeff);
 145         }
 146         n.add_ex("var", var);
 147         n.add_ex("point", point);
 148 }
 149
 150
 151 //////////
 152 // functions overriding virtual functions from base classes
 153 //////////
 154
 155 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
 156 {
 157         if (precedence() <= level)
 158                 c.s << '(';
 159
 160         // objects of type pseries must not have any zero entries, so the
 161         // trivial (zero) pseries needs a special treatment here:
 162         if (seq.empty())
 163                 c.s << '0';
 164
 165         auto i = seq.begin(), end = seq.end();
 166         while (i != end) {
 167
 168                 // print a sign, if needed
 169                 if (i != seq.begin())
 170                         c.s << '+';
 171
 172                 if (!is_order_function(i->rest)) {
 173
 174                         // print 'rest', i.e. the expansion coefficient
 175                         if (i->rest.info(info_flags::numeric) &&
 176                                 i->rest.info(info_flags::positive)) {
 177                                 i->rest.print(c);
 178                         } else {
 179                                 c.s << openbrace << '(';
 180                                 i->rest.print(c);
 181                                 c.s << ')' << closebrace;
 182                         }
 183
 184                         // print 'coeff', something like (x-1)^42
 185                         if (!i->coeff.is_zero()) {
 186                                 c.s << mul_sym;
 187                                 if (!point.is_zero()) {
 188                                         c.s << openbrace << '(';
 189                                         (var-point).print(c);
 190                                         c.s << ')' << closebrace;
 191                                 } else
 192                                         var.print(c);
 193                                 if (i->coeff.compare(_ex1)) {
 194                                         c.s << pow_sym;
 195                                         c.s << openbrace;
 196                                         if (i->coeff.info(info_flags::negative)) {
 197                                                 c.s << '(';
 198                                                 i->coeff.print(c);
 199                                                 c.s << ')';
 200                                         } else
 201                                                 i->coeff.print(c);
 202                                         c.s << closebrace;
 203                                 }
 204                         }
 205                 } else
 206                         Order(pow(var - point, i->coeff)).print(c);
 207                 ++i;
 208         }
 209
 210         if (precedence() <= level)
 211                 c.s << ')';
 212 }
 213
 214 void pseries::do_print(const print_context & c, unsigned level) const
 215 {
 216         print_series(c, "", "", "*", "^", level);
 217 }
 218
 219 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 220 {
 221         print_series(c, "{", "}", " ", "^", level);
 222 }
 223
 224 void pseries::do_print_python(const print_python & c, unsigned level) const
 225 {
 226         print_series(c, "", "", "*", "**", level);
 227 }
 228
 229 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 230 {
 231         c.s << std::string(level, ' ') << class_name() << " @" << this
 232             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 233             << std::endl;
 234         size_t num = seq.size();
 235         for (size_t i=0; i<num; ++i) {
 236                 seq[i].rest.print(c, level + c.delta_indent);
 237                 seq[i].coeff.print(c, level + c.delta_indent);
 238                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 239         }
 240         var.print(c, level + c.delta_indent);
 241         point.print(c, level + c.delta_indent);
 242 }
 243
 244 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 245 {
 246         c.s << class_name() << "(relational(";
 247         var.print(c);
 248         c.s << ',';
 249         point.print(c);
 250         c.s << "),[";
 251         size_t num = seq.size();
 252         for (size_t i=0; i<num; ++i) {
 253                 if (i)
 254                         c.s << ',';
 255                 c.s << '(';
 256                 seq[i].rest.print(c);
 257                 c.s << ',';
 258                 seq[i].coeff.print(c);
 259                 c.s << ')';
 260         }
 261         c.s << "])";
 262 }
 263
 264 int pseries::compare_same_type(const basic & other) const
 265 {
 266         GINAC_ASSERT(is_a<pseries>(other));
 267         const pseries &o = static_cast<const pseries &>(other);
 268
 269         // first compare the lengths of the series...
 270         if (seq.size()>o.seq.size())
 271                 return 1;
 272         if (seq.size()<o.seq.size())
 273                 return -1;
 274
 275         // ...then the expansion point...
 276         int cmpval = var.compare(o.var);
 277         if (cmpval)
 278                 return cmpval;
 279         cmpval = point.compare(o.point);
 280         if (cmpval)
 281                 return cmpval;
 282
 283         // ...and if that failed the individual elements
 284         auto it = seq.begin(), o_it = o.seq.begin();
 285         while (it!=seq.end() && o_it!=o.seq.end()) {
 286                 cmpval = it->compare(*o_it);
 287                 if (cmpval)
 288                         return cmpval;
 289                 ++it;
 290                 ++o_it;
 291         }
 292
 293         // so they are equal.
 294         return 0;
 295 }
 296
 297 /** Return the number of operands including a possible order term. */
 298 size_t pseries::nops() const
 299 {
 300         return seq.size();
 301 }
 302
 303 /** Return the ith term in the series when represented as a sum. */
 304 ex pseries::op(size_t i) const
 305 {
 306         if (i >= seq.size())
 307                 throw (std::out_of_range("op() out of range"));
 308
 309         if (is_order_function(seq[i].rest))
 310                 return Order(pow(var-point, seq[i].coeff));
 311         return seq[i].rest * pow(var - point, seq[i].coeff);
 312 }
 313
 314 /** Return degree of highest power of the series.  This is usually the exponent
 315  *  of the Order term.  If s is not the expansion variable of the series, the
 316  *  series is examined termwise. */
 317 int pseries::degree(const ex &s) const
 318 {
 319         if (seq.empty())
 320                 return 0;
 321
 322         if (var.is_equal(s))
 323                 // Return last/greatest exponent
 324                 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 325
 326         int max_pow = std::numeric_limits<int>::min();
 327         for (auto & it : seq)
 328                 max_pow = std::max(max_pow, it.rest.degree(s));
 329         return max_pow;
 330 }
 331
 332 /** Return degree of lowest power of the series.  This is usually the exponent
 333  *  of the leading term.  If s is not the expansion variable of the series, the
 334  *  series is examined termwise.  If s is the expansion variable but the
 335  *  expansion point is not zero the series is not expanded to find the degree.
 336  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 337 int pseries::ldegree(const ex &s) const
 338 {
 339         if (seq.empty())
 340                 return 0;
 341
 342         if (var.is_equal(s))
 343                 // Return first/smallest exponent
 344                 return ex_to<numeric>((seq.begin())->coeff).to_int();
 345
 346         int min_pow = std::numeric_limits<int>::max();
 347         for (auto & it : seq)
 348                 min_pow = std::min(min_pow, it.rest.degree(s));
 349         return min_pow;
 350 }
 351
 352 /** Return coefficient of degree n in power series if s is the expansion
 353  *  variable.  If the expansion point is nonzero, by definition the n=1
 354  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 355  *  the expansion took place in the s in the first place).
 356  *  If s is not the expansion variable, an attempt is made to convert the
 357  *  series to a polynomial and return the corresponding coefficient from
 358  *  there. */
 359 ex pseries::coeff(const ex &s, int n) const
 360 {
 361         if (var.is_equal(s)) {
 362                 if (seq.empty())
 363                         return _ex0;
 364
 365                 // Binary search in sequence for given power
 366                 numeric looking_for = numeric(n);
 367                 int lo = 0, hi = seq.size() - 1;
 368                 while (lo <= hi) {
 369                         int mid = (lo + hi) / 2;
 370                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 371                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 372                         switch (cmp) {
 373                                 case -1:
 374                                         lo = mid + 1;
 375                                         break;
 376                                 case 0:
 377                                         return seq[mid].rest;
 378                                 case 1:
 379                                         hi = mid - 1;
 380                                         break;
 381                                 default:
 382                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 383                         }
 384                 }
 385                 return _ex0;
 386         } else
 387                 return convert_to_poly().coeff(s, n);
 388 }
 389
 390 /** Does nothing. */
 391 ex pseries::collect(const ex &s, bool distributed) const
 392 {
 393         return *this;
 394 }
 395
 396 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 397 ex pseries::eval() const
 398 {
 399         if (flags & status_flags::evaluated) {
 400                 return *this;
 401         }
 402
 403         // Construct a new series with evaluated coefficients
 404         epvector new_seq;
 405         new_seq.reserve(seq.size());
 406         for (auto & it : seq)
 407                 new_seq.push_back(expair(it.rest, it.coeff));
 408
 409         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 410 }
 411
 412 /** Evaluate coefficients numerically. */
 413 ex pseries::evalf() const
 414 {
 415         // Construct a new series with evaluated coefficients
 416         epvector new_seq;
 417         new_seq.reserve(seq.size());
 418         for (auto & it : seq)
 419                 new_seq.push_back(expair(it.rest, it.coeff));
 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         const int a_max = degree(var);
 827         const int b_max = other.degree(var);
 828         const int a_min = ldegree(var);
 829         const int b_min = other.ldegree(var);
 830         const 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         const 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         std::map<int, ex> rest_map_a, rest_map_b;
 844         for (const auto& it : seq)
 845                 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 846
 847         if (other.var.is_equal(var))
 848                 for (const auto& it : other.seq)
 849                         rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 850
 851         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 852                 ex co = _ex0;
 853                 // c(i)=a(0)b(i)+...+a(i)b(0)
 854                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 855                         const auto& ita = rest_map_a.find(i);
 856                         if (ita == rest_map_a.end())
 857                                 continue;
 858                         const auto& itb = rest_map_b.find(cdeg-i);
 859                         if (itb == rest_map_b.end())
 860                                 continue;
 861                         if (!is_order_function(ita->second) && !is_order_function(itb->second))
 862                                 co += ita->second * itb->second;
 863                 }
 864                 if (!co.is_zero())
 865                         new_seq.push_back(expair(co, numeric(cdeg)));
 866         }
 867         if (higher_order_c < std::numeric_limits<int>::max())
 868                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 869         return pseries(relational(var, point), std::move(new_seq));
 870 }
 871
 872
 873 /** Implementation of ex::series() for product. This performs series
 874  *  multiplication when multiplying series.
 875  *  @see ex::series */
 876 ex mul::series(const relational & r, int order, unsigned options) const
 877 {
 878         pseries acc; // Series accumulator
 879
 880         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 881         const ex& sym = r.lhs();
 882
 883         // holds ldegrees of the series of individual factors
 884         std::vector<int> ldegrees;
 885         std::vector<bool> ldegree_redo;
 886
 887         // find minimal degrees
 888         // first round: obtain a bound up to which minimal degrees have to be
 889         // considered
 890         for (auto & it : seq) {
 891
 892                 ex expon = it.coeff;
 893                 int factor = 1;
 894                 ex buf;
 895                 if (expon.info(info_flags::integer)) {
 896                         buf = it.rest;
 897                         factor = ex_to<numeric>(expon).to_int();
 898                 } else {
 899                         buf = recombine_pair_to_ex(it);
 900                 }
 901
 902                 int real_ldegree = 0;
 903                 bool flag_redo = false;
 904                 try {
 905                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 906                 } catch (std::runtime_error) {}
 907
 908                 if (real_ldegree == 0) {
 909                         if ( factor < 0 ) {
 910                                 // This case must terminate, otherwise we would have division by
 911                                 // zero.
 912                                 int orderloop = 0;
 913                                 do {
 914                                         orderloop++;
 915                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 916                                 } while (real_ldegree == orderloop);
 917                         } else {
 918                                 // Here it is possible that buf does not have a ldegree, therefore
 919                                 // check only if ldegree is negative, otherwise reconsider the case
 920                                 // in the second round.
 921                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 922                                 if (real_ldegree == 0)
 923                                         flag_redo = true;
 924                         }
 925                 }
 926
 927                 ldegrees.push_back(factor * real_ldegree);
 928                 ldegree_redo.push_back(flag_redo);
 929         }
 930
 931         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 932         // Second round: determine the remaining positive ldegrees by the series
 933         // method.
 934         // here we can ignore ldegrees larger than degbound
 935         size_t j = 0;
 936         for (auto & it : seq) {
 937                 if ( ldegree_redo[j] ) {
 938                         ex expon = it.coeff;
 939                         int factor = 1;
 940                         ex buf;
 941                         if (expon.info(info_flags::integer)) {
 942                                 buf = it.rest;
 943                                 factor = ex_to<numeric>(expon).to_int();
 944                         } else {
 945                                 buf = recombine_pair_to_ex(it);
 946                         }
 947                         int real_ldegree = 0;
 948                         int orderloop = 0;
 949                         do {
 950                                 orderloop++;
 951                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 952                         } while ((real_ldegree == orderloop)
 953                               && (factor*real_ldegree < degbound));
 954                         ldegrees[j] = factor * real_ldegree;
 955                         degbound -= factor * real_ldegree;
 956                 }
 957                 j++;
 958         }
 959
 960         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 961
 962         if (degsum >= order) {
 963                 epvector epv { expair(Order(_ex1), order) };
 964                 return dynallocate<pseries>(r, std::move(epv));
 965         }
 966
 967         // Multiply with remaining terms
 968         auto itd = ldegrees.begin();
 969         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 970
 971                 // do series expansion with adjusted order
 972                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 973
 974                 // Series multiplication
 975                 if (it == seq.begin())
 976                         acc = ex_to<pseries>(op);
 977                 else
 978                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 979         }
 980
 981         return acc.mul_const(ex_to<numeric>(overall_coeff));
 982 }
 983
 984
 985 /** Compute the p-th power of a series.
 986  *
 987  *  @param p  power to compute
 988  *  @param deg  truncation order of series calculation */
 989 ex pseries::power_const(const numeric &p, int deg) const
 990 {
 991         // method:
 992         // (due to Leonhard Euler)
 993         // let A(x) be this series and for the time being let it start with a
 994         // constant (later we'll generalize):
 995         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 996         // We want to compute
 997         //     C(x) = A(x)^p
 998         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 999         // Taking the derivative on both sides and multiplying with A(x) one
1000         // immediately arrives at
1001         //     C'(x)*A(x) = p*C(x)*A'(x)
1002         // Multiplying this out and comparing coefficients we get the recurrence
1003         // formula
1004         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1005         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1006         // which can easily be solved given the starting value c_0 = (a_0)^p.
1007         // For the more general case where the leading coefficient of A(x) is not
1008         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1009         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
1010         // then of course x^(p*m) but the recurrence formula still holds.
1011
1012         if (seq.empty()) {
1013                 // as a special case, handle the empty (zero) series honoring the
1014                 // usual power laws such as implemented in power::eval()
1015                 if (p.real().is_zero())
1016                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1017                 else if (p.real().is_negative())
1018                         throw pole_error("pseries::power_const(): division by zero",1);
1019                 else
1020                         return *this;
1021         }
1022
1023         const int ldeg = ldegree(var);
1024         if (!(p*ldeg).is_integer())
1025                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1026
1027         // adjust number of coefficients
1028         int numcoeff = deg - (p*ldeg).to_int();
1029         if (numcoeff <= 0) {
1030                 epvector epv { expair(Order(_ex1), deg) };
1031                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1032         }
1033
1034         // O(x^n)^(-m) is undefined
1035         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1036                 throw pole_error("pseries::power_const(): division by zero",1);
1037
1038         // Compute coefficients of the powered series
1039         exvector co;
1040         co.reserve(numcoeff);
1041         co.push_back(pow(coeff(var, ldeg), p));
1042         for (int i=1; i<numcoeff; ++i) {
1043                 ex sum = _ex0;
1044                 for (int j=1; j<=i; ++j) {
1045                         ex c = coeff(var, j + ldeg);
1046                         if (is_order_function(c)) {
1047                                 co.push_back(Order(_ex1));
1048                                 break;
1049                         } else
1050                                 sum += (p * j - (i - j)) * co[i - j] * c;
1051                 }
1052                 co.push_back(sum / coeff(var, ldeg) / i);
1053         }
1054
1055         // Construct new series (of non-zero coefficients)
1056         epvector new_seq;
1057         bool higher_order = false;
1058         for (int i=0; i<numcoeff; ++i) {
1059                 if (!co[i].is_zero())
1060                         new_seq.push_back(expair(co[i], p * ldeg + i));
1061                 if (is_order_function(co[i])) {
1062                         higher_order = true;
1063                         break;
1064                 }
1065         }
1066         if (!higher_order)
1067                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1068
1069         return pseries(relational(var,point), std::move(new_seq));
1070 }
1071
1072
1073 /** Return a new pseries object with the powers shifted by deg. */
1074 pseries pseries::shift_exponents(int deg) const
1075 {
1076         epvector newseq = seq;
1077         for (auto & it : newseq)
1078                 it.coeff += deg;
1079         return pseries(relational(var, point), std::move(newseq));
1080 }
1081
1082
1083 /** Implementation of ex::series() for powers. This performs Laurent expansion
1084  *  of reciprocals of series at singularities.
1085  *  @see ex::series */
1086 ex power::series(const relational & r, int order, unsigned options) const
1087 {
1088         // If basis is already a series, just power it
1089         if (is_exactly_a<pseries>(basis))
1090                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1091
1092         // Basis is not a series, may there be a singularity?
1093         bool must_expand_basis = false;
1094         try {
1095                 basis.subs(r, subs_options::no_pattern);
1096         } catch (pole_error) {
1097                 must_expand_basis = true;
1098         }
1099
1100         bool exponent_is_regular = true;
1101         try {
1102                 exponent.subs(r, subs_options::no_pattern);
1103         } catch (pole_error) {
1104                 exponent_is_regular = false;
1105         }
1106
1107         if (!exponent_is_regular) {
1108                 ex l = exponent*log(basis);
1109                 // this == exp(l);
1110                 ex le = l.series(r, order, options);
1111                 // Note: expanding exp(l) won't help, since that will attempt
1112                 // Taylor expansion, and fail (because exponent is "singular")
1113                 // Still l itself might be expanded in Taylor series.
1114                 // Examples:
1115                 // sin(x)/x*log(cos(x))
1116                 // 1/x*log(1 + x)
1117                 return exp(le).series(r, order, options);
1118                 // Note: if l happens to have a Laurent expansion (with
1119                 // negative powers of (var - point)), expanding exp(le)
1120                 // will barf (which is The Right Thing).
1121         }
1122
1123         // Is the expression of type something^(-int)?
1124         if (!must_expand_basis && !exponent.info(info_flags::negint)
1125          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1126                 return basic::series(r, order, options);
1127
1128         // Is the expression of type 0^something?
1129         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1130          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1131                 return basic::series(r, order, options);
1132
1133         // Singularity encountered, is the basis equal to (var - point)?
1134         if (basis.is_equal(r.lhs() - r.rhs())) {
1135                 epvector new_seq;
1136                 if (ex_to<numeric>(exponent).to_int() < order)
1137                         new_seq.push_back(expair(_ex1, exponent));
1138                 else
1139                         new_seq.push_back(expair(Order(_ex1), exponent));
1140                 return pseries(r, std::move(new_seq));
1141         }
1142
1143         // No, expand basis into series
1144
1145         numeric numexp;
1146         if (is_a<numeric>(exponent)) {
1147                 numexp = ex_to<numeric>(exponent);
1148         } else {
1149                 numexp = 0;
1150         }
1151         const ex& sym = r.lhs();
1152         // find existing minimal degree
1153         ex eb = basis.expand();
1154         int real_ldegree = 0;
1155         if (eb.info(info_flags::rational_function))
1156                 real_ldegree = eb.ldegree(sym-r.rhs());
1157         if (real_ldegree == 0) {
1158                 int orderloop = 0;
1159                 do {
1160                         orderloop++;
1161                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1162                 } while (real_ldegree == orderloop);
1163         }
1164
1165         if (!(real_ldegree*numexp).is_integer())
1166                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1167         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1168
1169         ex result;
1170         try {
1171                 result = ex_to<pseries>(e).power_const(numexp, order);
1172         } catch (pole_error) {
1173                 epvector ser { expair(Order(_ex1), order) };
1174                 result = pseries(r, std::move(ser));
1175         }
1176
1177         return result;
1178 }
1179
1180
1181 /** Re-expansion of a pseries object. */
1182 ex pseries::series(const relational & r, int order, unsigned options) const
1183 {
1184         const ex p = r.rhs();
1185         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1186         const symbol &s = ex_to<symbol>(r.lhs());
1187
1188         if (var.is_equal(s) && point.is_equal(p)) {
1189                 if (order > degree(s))
1190                         return *this;
1191                 else {
1192                         epvector new_seq;
1193                         for (auto & it : seq) {
1194                                 int o = ex_to<numeric>(it.coeff).to_int();
1195                                 if (o >= order) {
1196                                         new_seq.push_back(expair(Order(_ex1), o));
1197                                         break;
1198                                 }
1199                                 new_seq.push_back(it);
1200                         }
1201                         return pseries(r, std::move(new_seq));
1202                 }
1203         } else
1204                 return convert_to_poly().series(r, order, options);
1205 }
1206
1207 ex integral::series(const relational & r, int order, unsigned options) const
1208 {
1209         if (x.subs(r) != x)
1210                 throw std::logic_error("Cannot series expand wrt dummy variable");
1211
1212         // Expanding integrand with r substituted taken in boundaries.
1213         ex fseries = f.series(r, order, options);
1214         epvector fexpansion;
1215         fexpansion.reserve(fseries.nops());
1216         for (size_t i=0; i<fseries.nops(); ++i) {
1217                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1218                 currcoeff = (currcoeff == Order(_ex1))
1219                         ? currcoeff
1220                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1221                 if (currcoeff != 0)
1222                         fexpansion.push_back(
1223                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1224         }
1225
1226         // Expanding lower boundary
1227         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1228         ex aseries = (a-a.subs(r)).series(r, order, options);
1229         fseries = f.series(x == (a.subs(r)), order, options);
1230         for (size_t i=0; i<fseries.nops(); ++i) {
1231                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1232                 if (is_order_function(currcoeff))
1233                         break;
1234                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1235                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1236                 currcoeff = currcoeff.series(r, orderforf);
1237                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1238                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1239                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1240                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1241         }
1242
1243         // Expanding upper boundary
1244         ex bseries = (b-b.subs(r)).series(r, order, options);
1245         fseries = f.series(x == (b.subs(r)), order, options);
1246         for (size_t i=0; i<fseries.nops(); ++i) {
1247                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1248                 if (is_order_function(currcoeff))
1249                         break;
1250                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1251                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1252                 currcoeff = currcoeff.series(r, orderforf);
1253                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1254                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1255                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1256                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1257         }
1258
1259         return result;
1260 }
1261
1262
1263 /** Compute the truncated series expansion of an expression.
1264  *  This function returns an expression containing an object of class pseries
1265  *  to represent the series. If the series does not terminate within the given
1266  *  truncation order, the last term of the series will be an order term.
1267  *
1268  *  @param r  expansion relation, lhs holds variable and rhs holds point
1269  *  @param order  truncation order of series calculations
1270  *  @param options  of class series_options
1271  *  @return an expression holding a pseries object */
1272 ex ex::series(const ex & r, int order, unsigned options) const
1273 {
1274         ex e;
1275         relational rel_;
1276
1277         if (is_a<relational>(r))
1278                 rel_ = ex_to<relational>(r);
1279         else if (is_a<symbol>(r))
1280                 rel_ = relational(r,_ex0);
1281         else
1282                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1283
1284         e = bp->series(rel_, order, options);
1285         return e;
1286 }
1287
1288 GINAC_BIND_UNARCHIVER(pseries);
1289
1290 } // namespace GiNaC