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) 1999-2024 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 pseries::pseries(const ex &rel_, const epvector &ops_)
  71   : seq(ops_)
  72 {
  73 #ifdef DO_GINAC_ASSERT
  74         auto i = seq.begin();
  75         while (i != seq.end()) {
  76                 auto ip1 = i+1;
  77                 if (ip1 != seq.end())
  78                         GINAC_ASSERT(!is_order_function(i->rest));
  79                 else
  80                         break;
  81                 GINAC_ASSERT(is_a<numeric>(i->coeff));
  82                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
  83                 ++i;
  84         }
  85 #endif // def DO_GINAC_ASSERT
  86         GINAC_ASSERT(is_a<relational>(rel_));
  87         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  88         point = rel_.rhs();
  89         var = rel_.lhs();
  90 }
  91 pseries::pseries(const ex &rel_, epvector &&ops_)
  92   : seq(std::move(ops_))
  93 {
  94 #ifdef DO_GINAC_ASSERT
  95         auto i = seq.begin();
  96         while (i != seq.end()) {
  97                 auto ip1 = i+1;
  98                 if (ip1 != seq.end())
  99                         GINAC_ASSERT(!is_order_function(i->rest));
 100                 else
 101                         break;
 102                 GINAC_ASSERT(is_a<numeric>(i->coeff));
 103                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
 104                 ++i;
 105         }
 106 #endif // def DO_GINAC_ASSERT
 107         GINAC_ASSERT(is_a<relational>(rel_));
 108         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
 109         point = rel_.rhs();
 110         var = rel_.lhs();
 111 }
 112
 113
 114 /*
 115  *  Archiving
 116  */
 117
 118 void pseries::read_archive(const archive_node &n, lst &sym_lst)
 119 {
 120         inherited::read_archive(n, sym_lst);
 121         auto range = n.find_property_range("coeff", "power");
 122         seq.reserve((range.end-range.begin)/2);
 123
 124         for (auto loc = range.begin; loc < range.end;) {
 125                 ex rest;
 126                 ex coeff;
 127                 n.find_ex_by_loc(loc++, rest, sym_lst);
 128                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 129                 seq.emplace_back(expair(rest, coeff));
 130         }
 131
 132         n.find_ex("var", var, sym_lst);
 133         n.find_ex("point", point, sym_lst);
 134 }
 135
 136 void pseries::archive(archive_node &n) const
 137 {
 138         inherited::archive(n);
 139         for (auto & it : seq) {
 140                 n.add_ex("coeff", it.rest);
 141                 n.add_ex("power", it.coeff);
 142         }
 143         n.add_ex("var", var);
 144         n.add_ex("point", point);
 145 }
 146
 147
 148 //////////
 149 // functions overriding virtual functions from base classes
 150 //////////
 151
 152 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
 153 {
 154         if (precedence() <= level)
 155                 c.s << '(';
 156
 157         // objects of type pseries must not have any zero entries, so the
 158         // trivial (zero) pseries needs a special treatment here:
 159         if (seq.empty())
 160                 c.s << '0';
 161
 162         auto i = seq.begin(), end = seq.end();
 163         while (i != end) {
 164
 165                 // print a sign, if needed
 166                 if (i != seq.begin())
 167                         c.s << '+';
 168
 169                 if (!is_order_function(i->rest)) {
 170
 171                         // print 'rest', i.e. the expansion coefficient
 172                         if (i->rest.info(info_flags::numeric) &&
 173                                 i->rest.info(info_flags::positive)) {
 174                                 i->rest.print(c);
 175                         } else {
 176                                 c.s << openbrace << '(';
 177                                 i->rest.print(c);
 178                                 c.s << ')' << closebrace;
 179                         }
 180
 181                         // print 'coeff', something like (x-1)^42
 182                         if (!i->coeff.is_zero()) {
 183                                 c.s << mul_sym;
 184                                 if (!point.is_zero()) {
 185                                         c.s << openbrace << '(';
 186                                         (var-point).print(c);
 187                                         c.s << ')' << closebrace;
 188                                 } else
 189                                         var.print(c);
 190                                 if (i->coeff.compare(_ex1)) {
 191                                         c.s << pow_sym;
 192                                         c.s << openbrace;
 193                                         if (i->coeff.info(info_flags::negative)) {
 194                                                 c.s << '(';
 195                                                 i->coeff.print(c);
 196                                                 c.s << ')';
 197                                         } else
 198                                                 i->coeff.print(c);
 199                                         c.s << closebrace;
 200                                 }
 201                         }
 202                 } else
 203                         Order(pow(var - point, i->coeff)).print(c);
 204                 ++i;
 205         }
 206
 207         if (precedence() <= level)
 208                 c.s << ')';
 209 }
 210
 211 void pseries::do_print(const print_context & c, unsigned level) const
 212 {
 213         print_series(c, "", "", "*", "^", level);
 214 }
 215
 216 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 217 {
 218         print_series(c, "{", "}", " ", "^", level);
 219 }
 220
 221 void pseries::do_print_python(const print_python & c, unsigned level) const
 222 {
 223         print_series(c, "", "", "*", "**", level);
 224 }
 225
 226 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 227 {
 228         c.s << std::string(level, ' ') << class_name() << " @" << this
 229             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 230             << std::endl;
 231         size_t num = seq.size();
 232         for (size_t i=0; i<num; ++i) {
 233                 seq[i].rest.print(c, level + c.delta_indent);
 234                 seq[i].coeff.print(c, level + c.delta_indent);
 235                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 236         }
 237         var.print(c, level + c.delta_indent);
 238         point.print(c, level + c.delta_indent);
 239 }
 240
 241 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 242 {
 243         c.s << class_name() << "(relational(";
 244         var.print(c);
 245         c.s << ',';
 246         point.print(c);
 247         c.s << "),[";
 248         size_t num = seq.size();
 249         for (size_t i=0; i<num; ++i) {
 250                 if (i)
 251                         c.s << ',';
 252                 c.s << '(';
 253                 seq[i].rest.print(c);
 254                 c.s << ',';
 255                 seq[i].coeff.print(c);
 256                 c.s << ')';
 257         }
 258         c.s << "])";
 259 }
 260
 261 int pseries::compare_same_type(const basic & other) const
 262 {
 263         GINAC_ASSERT(is_a<pseries>(other));
 264         const pseries &o = static_cast<const pseries &>(other);
 265
 266         // first compare the lengths of the series...
 267         if (seq.size()>o.seq.size())
 268                 return 1;
 269         if (seq.size()<o.seq.size())
 270                 return -1;
 271
 272         // ...then the expansion point...
 273         int cmpval = var.compare(o.var);
 274         if (cmpval)
 275                 return cmpval;
 276         cmpval = point.compare(o.point);
 277         if (cmpval)
 278                 return cmpval;
 279
 280         // ...and if that failed the individual elements
 281         auto it = seq.begin(), o_it = o.seq.begin();
 282         while (it!=seq.end() && o_it!=o.seq.end()) {
 283                 cmpval = it->compare(*o_it);
 284                 if (cmpval)
 285                         return cmpval;
 286                 ++it;
 287                 ++o_it;
 288         }
 289
 290         // so they are equal.
 291         return 0;
 292 }
 293
 294 /** Return the number of operands including a possible order term. */
 295 size_t pseries::nops() const
 296 {
 297         return seq.size();
 298 }
 299
 300 /** Return the ith term in the series when represented as a sum. */
 301 ex pseries::op(size_t i) const
 302 {
 303         if (i >= seq.size())
 304                 throw (std::out_of_range("op() out of range"));
 305
 306         if (is_order_function(seq[i].rest))
 307                 return Order(pow(var-point, seq[i].coeff));
 308         return seq[i].rest * pow(var - point, seq[i].coeff);
 309 }
 310
 311 /** Return degree of highest power of the series.  This is usually the exponent
 312  *  of the Order term.  If s is not the expansion variable of the series, the
 313  *  series is examined termwise. */
 314 int pseries::degree(const ex &s) const
 315 {
 316         if (seq.empty())
 317                 return 0;
 318
 319         if (var.is_equal(s))
 320                 // Return last/greatest exponent
 321                 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 322
 323         int max_pow = std::numeric_limits<int>::min();
 324         for (auto & it : seq)
 325                 max_pow = std::max(max_pow, it.rest.degree(s));
 326         return max_pow;
 327 }
 328
 329 /** Return degree of lowest power of the series.  This is usually the exponent
 330  *  of the leading term.  If s is not the expansion variable of the series, the
 331  *  series is examined termwise.  If s is the expansion variable but the
 332  *  expansion point is not zero the series is not expanded to find the degree.
 333  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 334 int pseries::ldegree(const ex &s) const
 335 {
 336         if (seq.empty())
 337                 return 0;
 338
 339         if (var.is_equal(s))
 340                 // Return first/smallest exponent
 341                 return ex_to<numeric>((seq.begin())->coeff).to_int();
 342
 343         int min_pow = std::numeric_limits<int>::max();
 344         for (auto & it : seq)
 345                 min_pow = std::min(min_pow, it.rest.degree(s));
 346         return min_pow;
 347 }
 348
 349 /** Return coefficient of degree n in power series if s is the expansion
 350  *  variable.  If the expansion point is nonzero, by definition the n=1
 351  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 352  *  the expansion took place in the s in the first place).
 353  *  If s is not the expansion variable, an attempt is made to convert the
 354  *  series to a polynomial and return the corresponding coefficient from
 355  *  there. */
 356 ex pseries::coeff(const ex &s, int n) const
 357 {
 358         if (var.is_equal(s)) {
 359                 if (seq.empty())
 360                         return _ex0;
 361
 362                 // Binary search in sequence for given power
 363                 numeric looking_for = numeric(n);
 364                 int lo = 0, hi = seq.size() - 1;
 365                 while (lo <= hi) {
 366                         int mid = (lo + hi) / 2;
 367                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 368                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 369                         switch (cmp) {
 370                                 case -1:
 371                                         lo = mid + 1;
 372                                         break;
 373                                 case 0:
 374                                         return seq[mid].rest;
 375                                 case 1:
 376                                         hi = mid - 1;
 377                                         break;
 378                                 default:
 379                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 380                         }
 381                 }
 382                 return _ex0;
 383         } else
 384                 return convert_to_poly().coeff(s, n);
 385 }
 386
 387 /** Does nothing. */
 388 ex pseries::collect(const ex &s, bool distributed) const
 389 {
 390         return *this;
 391 }
 392
 393 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 394 ex pseries::eval() const
 395 {
 396         return hold();
 397 }
 398
 399 /** Evaluate coefficients numerically. */
 400 ex pseries::evalf() const
 401 {
 402         // Construct a new series with evaluated coefficients
 403         epvector new_seq;
 404         new_seq.reserve(seq.size());
 405         for (auto & it : seq)
 406                 new_seq.emplace_back(expair(it.rest.evalf(), it.coeff));
 407
 408         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 409 }
 410
 411 ex pseries::conjugate() const
 412 {
 413         if(!var.info(info_flags::real))
 414                 return conjugate_function(*this).hold();
 415
 416         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
 417         ex newpoint = point.conjugate();
 418
 419         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 420                 return *this;
 421         }
 422
 423         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
 424 }
 425
 426 ex pseries::real_part() const
 427 {
 428         if(!var.info(info_flags::real))
 429                 return real_part_function(*this).hold();
 430         ex newpoint = point.real_part();
 431         if(newpoint != point)
 432                 return real_part_function(*this).hold();
 433
 434         epvector v;
 435         v.reserve(seq.size());
 436         for (auto & it : seq)
 437                 v.emplace_back(expair(it.rest.real_part(), it.coeff));
 438         return dynallocate<pseries>(var==point, std::move(v));
 439 }
 440
 441 ex pseries::imag_part() const
 442 {
 443         if(!var.info(info_flags::real))
 444                 return imag_part_function(*this).hold();
 445         ex newpoint = point.real_part();
 446         if(newpoint != point)
 447                 return imag_part_function(*this).hold();
 448
 449         epvector v;
 450         v.reserve(seq.size());
 451         for (auto & it : seq)
 452                 v.emplace_back(expair(it.rest.imag_part(), it.coeff));
 453         return dynallocate<pseries>(var==point, std::move(v));
 454 }
 455
 456 ex pseries::eval_integ() const
 457 {
 458         std::unique_ptr<epvector> newseq(nullptr);
 459         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 460                 if (newseq) {
 461                         newseq->emplace_back(expair(i->rest.eval_integ(), i->coeff));
 462                         continue;
 463                 }
 464                 ex newterm = i->rest.eval_integ();
 465                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 466                         newseq.reset(new epvector);
 467                         newseq->reserve(seq.size());
 468                         for (auto j=seq.begin(); j!=i; ++j)
 469                                 newseq->push_back(*j);
 470                         newseq->emplace_back(expair(newterm, i->coeff));
 471                 }
 472         }
 473
 474         ex newpoint = point.eval_integ();
 475         if (newseq || !are_ex_trivially_equal(newpoint, point))
 476                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
 477         return *this;
 478 }
 479
 480 ex pseries::evalm() const
 481 {
 482         // evalm each coefficient
 483         epvector newseq;
 484         bool something_changed = false;
 485         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 486                 if (something_changed) {
 487                         ex newcoeff = i->rest.evalm();
 488                         if (!newcoeff.is_zero())
 489                                 newseq.emplace_back(expair(newcoeff, i->coeff));
 490                 } else {
 491                         ex newcoeff = i->rest.evalm();
 492                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 493                                 something_changed = true;
 494                                 newseq.reserve(seq.size());
 495                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 496                                 if (!newcoeff.is_zero())
 497                                         newseq.emplace_back(expair(newcoeff, i->coeff));
 498                         }
 499                 }
 500         }
 501         if (something_changed)
 502                 return dynallocate<pseries>(var==point, std::move(newseq));
 503         else
 504                 return *this;
 505 }
 506
 507 ex pseries::subs(const exmap & m, unsigned options) const
 508 {
 509         // If expansion variable is being substituted, convert the series to a
 510         // polynomial and do the substitution there because the result might
 511         // no longer be a power series
 512         if (m.find(var) != m.end())
 513                 return convert_to_poly(true).subs(m, options);
 514
 515         // Otherwise construct a new series with substituted coefficients and
 516         // expansion point
 517         epvector newseq;
 518         newseq.reserve(seq.size());
 519         for (auto & it : seq)
 520                 newseq.emplace_back(expair(it.rest.subs(m, options), it.coeff));
 521         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
 522 }
 523
 524 /** Implementation of ex::expand() for a power series.  It expands all the
 525  *  terms individually and returns the resulting series as a new pseries. */
 526 ex pseries::expand(unsigned options) const
 527 {
 528         epvector newseq;
 529         for (auto & it : seq) {
 530                 ex restexp = it.rest.expand();
 531                 if (!restexp.is_zero())
 532                         newseq.emplace_back(expair(restexp, it.coeff));
 533         }
 534         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
 535 }
 536
 537 /** Implementation of ex::diff() for a power series.
 538  *  @see ex::diff */
 539 ex pseries::derivative(const symbol & s) const
 540 {
 541         epvector new_seq;
 542
 543         if (s == var) {
 544
 545                 // FIXME: coeff might depend on var
 546                 for (auto & it : seq) {
 547                         if (is_order_function(it.rest)) {
 548                                 new_seq.emplace_back(expair(it.rest, it.coeff - 1));
 549                         } else {
 550                                 ex c = it.rest * it.coeff;
 551                                 if (!c.is_zero())
 552                                         new_seq.emplace_back(expair(c, it.coeff - 1));
 553                         }
 554                 }
 555
 556         } else {
 557
 558                 for (auto & it : seq) {
 559                         if (is_order_function(it.rest)) {
 560                                 new_seq.push_back(it);
 561                         } else {
 562                                 ex c = it.rest.diff(s);
 563                                 if (!c.is_zero())
 564                                         new_seq.emplace_back(expair(c, it.coeff));
 565                         }
 566                 }
 567         }
 568
 569         return pseries(relational(var,point), std::move(new_seq));
 570 }
 571
 572 ex pseries::convert_to_poly(bool no_order) const
 573 {
 574         ex e;
 575         for (auto & it : seq) {
 576                 if (is_order_function(it.rest)) {
 577                         if (!no_order)
 578                                 e += Order(pow(var - point, it.coeff));
 579                 } else
 580                         e += it.rest * pow(var - point, it.coeff);
 581         }
 582         return e;
 583 }
 584
 585 bool pseries::is_terminating() const
 586 {
 587         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 588 }
 589
 590 ex pseries::coeffop(size_t i) const
 591 {
 592         if (i >= nops())
 593                 throw (std::out_of_range("coeffop() out of range"));
 594         return seq[i].rest;
 595 }
 596
 597 ex pseries::exponop(size_t i) const
 598 {
 599         if (i >= nops())
 600                 throw (std::out_of_range("exponop() out of range"));
 601         return seq[i].coeff;
 602 }
 603
 604
 605 /*
 606  *  Implementations of series expansion
 607  */
 608
 609 /** Default implementation of ex::series(). This performs Taylor expansion.
 610  *  @see ex::series */
 611 ex basic::series(const relational & r, int order, unsigned options) const
 612 {
 613         epvector seq;
 614         const symbol &s = ex_to<symbol>(r.lhs());
 615
 616         // default for order-values that make no sense for Taylor expansion
 617         if ((order <= 0) && this->has(s)) {
 618                 seq.emplace_back(expair(Order(_ex1), order));
 619                 return pseries(r, std::move(seq));
 620         }
 621
 622         // do Taylor expansion
 623         numeric fac = 1;
 624         ex deriv = *this;
 625         ex coeff = deriv.subs(r, subs_options::no_pattern);
 626
 627         if (!coeff.is_zero()) {
 628                 seq.emplace_back(expair(coeff, _ex0));
 629         }
 630
 631         int n;
 632         for (n=1; n<order; ++n) {
 633                 fac = fac.div(n);
 634                 // We need to test for zero in order to see if the series terminates.
 635                 // The problem is that there is no such thing as a perfect test for
 636                 // zero.  Expanding the term occasionally helps a little...
 637                 deriv = deriv.diff(s).expand();
 638                 if (deriv.is_zero())  // Series terminates
 639                         return pseries(r, std::move(seq));
 640
 641                 coeff = deriv.subs(r, subs_options::no_pattern);
 642                 if (!coeff.is_zero())
 643                         seq.emplace_back(expair(fac * coeff, n));
 644         }
 645
 646         // Higher-order terms, if present
 647         deriv = deriv.diff(s);
 648         if (!deriv.expand().is_zero())
 649                 seq.emplace_back(expair(Order(_ex1), n));
 650         return pseries(r, std::move(seq));
 651 }
 652
 653
 654 /** Implementation of ex::series() for symbols.
 655  *  @see ex::series */
 656 ex symbol::series(const relational & r, int order, unsigned options) const
 657 {
 658         epvector seq;
 659         const ex point = r.rhs();
 660         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 661
 662         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 663                 if (order > 0 && !point.is_zero())
 664                         seq.emplace_back(expair(point, _ex0));
 665                 if (order > 1)
 666                         seq.emplace_back(expair(_ex1, _ex1));
 667                 else
 668                         seq.emplace_back(expair(Order(_ex1), numeric(order)));
 669         } else
 670                 seq.emplace_back(expair(*this, _ex0));
 671         return pseries(r, std::move(seq));
 672 }
 673
 674
 675 /** Add one series object to another, producing a pseries object that
 676  *  represents the sum.
 677  *
 678  *  @param other  pseries object to add with
 679  *  @return the sum as a pseries */
 680 ex pseries::add_series(const pseries &other) const
 681 {
 682         // Adding two series with different variables or expansion points
 683         // results in an empty (constant) series
 684         if (!is_compatible_to(other)) {
 685                 epvector nul { expair(Order(_ex1), _ex0) };
 686                 return pseries(relational(var,point), std::move(nul));
 687         }
 688
 689         // Series addition
 690         epvector new_seq;
 691         auto a = seq.begin(), a_end = seq.end();
 692         auto b = other.seq.begin(), b_end = other.seq.end();
 693         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 694         for (;;) {
 695                 // If a is empty, fill up with elements from b and stop
 696                 if (a == a_end) {
 697                         while (b != b_end) {
 698                                 new_seq.push_back(*b);
 699                                 ++b;
 700                         }
 701                         break;
 702                 } else
 703                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 704
 705                 // If b is empty, fill up with elements from a and stop
 706                 if (b == b_end) {
 707                         while (a != a_end) {
 708                                 new_seq.push_back(*a);
 709                                 ++a;
 710                         }
 711                         break;
 712                 } else
 713                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 714
 715                 // a and b are non-empty, compare powers
 716                 if (pow_a < pow_b) {
 717                         // a has lesser power, get coefficient from a
 718                         new_seq.push_back(*a);
 719                         if (is_order_function((*a).rest))
 720                                 break;
 721                         ++a;
 722                 } else if (pow_b < pow_a) {
 723                         // b has lesser power, get coefficient from b
 724                         new_seq.push_back(*b);
 725                         if (is_order_function((*b).rest))
 726                                 break;
 727                         ++b;
 728                 } else {
 729                         // Add coefficient of a and b
 730                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 731                                 new_seq.emplace_back(expair(Order(_ex1), (*a).coeff));
 732                                 break;  // Order term ends the sequence
 733                         } else {
 734                                 ex sum = (*a).rest + (*b).rest;
 735                                 if (!(sum.is_zero()))
 736                                         new_seq.emplace_back(expair(sum, numeric(pow_a)));
 737                                 ++a;
 738                                 ++b;
 739                         }
 740                 }
 741         }
 742         return pseries(relational(var,point), std::move(new_seq));
 743 }
 744
 745
 746 /** Implementation of ex::series() for sums. This performs series addition when
 747  *  adding pseries objects.
 748  *  @see ex::series */
 749 ex add::series(const relational & r, int order, unsigned options) const
 750 {
 751         ex acc; // Series accumulator
 752
 753         // Get first term from overall_coeff
 754         acc = overall_coeff.series(r, order, options);
 755
 756         // Add remaining terms
 757         for (auto & it : seq) {
 758                 ex op;
 759                 if (is_exactly_a<pseries>(it.rest))
 760                         op = it.rest;
 761                 else
 762                         op = it.rest.series(r, order, options);
 763                 if (!it.coeff.is_equal(_ex1))
 764                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
 765
 766                 // Series addition
 767                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 768         }
 769         return acc;
 770 }
 771
 772
 773 /** Multiply a pseries object with a numeric constant, producing a pseries
 774  *  object that represents the product.
 775  *
 776  *  @param other  constant to multiply with
 777  *  @return the product as a pseries */
 778 ex pseries::mul_const(const numeric &other) const
 779 {
 780         epvector new_seq;
 781         new_seq.reserve(seq.size());
 782
 783         for (auto & it : seq) {
 784                 if (!is_order_function(it.rest))
 785                         new_seq.emplace_back(expair(it.rest * other, it.coeff));
 786                 else
 787                         new_seq.push_back(it);
 788         }
 789         return pseries(relational(var,point), std::move(new_seq));
 790 }
 791
 792
 793 /** Multiply one pseries object to another, producing a pseries object that
 794  *  represents the product.
 795  *
 796  *  @param other  pseries object to multiply with
 797  *  @return the product as a pseries */
 798 ex pseries::mul_series(const pseries &other) const
 799 {
 800         // Multiplying two series with different variables or expansion points
 801         // results in an empty (constant) series
 802         if (!is_compatible_to(other)) {
 803                 epvector nul { expair(Order(_ex1), _ex0) };
 804                 return pseries(relational(var,point), std::move(nul));
 805         }
 806
 807         if (seq.empty() || other.seq.empty()) {
 808                 return dynallocate<pseries>(var==point, epvector());
 809         }
 810
 811         // Series multiplication
 812         epvector new_seq;
 813         const int a_max = degree(var);
 814         const int b_max = other.degree(var);
 815         const int a_min = ldegree(var);
 816         const int b_min = other.ldegree(var);
 817         const int cdeg_min = a_min + b_min;
 818         int cdeg_max = a_max + b_max;
 819
 820         int higher_order_a = std::numeric_limits<int>::max();
 821         int higher_order_b = std::numeric_limits<int>::max();
 822         if (is_order_function(coeff(var, a_max)))
 823                 higher_order_a = a_max + b_min;
 824         if (is_order_function(other.coeff(var, b_max)))
 825                 higher_order_b = b_max + a_min;
 826         const int higher_order_c = std::min(higher_order_a, higher_order_b);
 827         if (cdeg_max >= higher_order_c)
 828                 cdeg_max = higher_order_c - 1;
 829
 830         std::map<int, ex> rest_map_a, rest_map_b;
 831         for (const auto& it : seq)
 832                 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 833
 834         if (other.var.is_equal(var))
 835                 for (const auto& it : other.seq)
 836                         rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 837
 838         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 839                 ex co = _ex0;
 840                 // c(i)=a(0)b(i)+...+a(i)b(0)
 841                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 842                         const auto& ita = rest_map_a.find(i);
 843                         if (ita == rest_map_a.end())
 844                                 continue;
 845                         const auto& itb = rest_map_b.find(cdeg-i);
 846                         if (itb == rest_map_b.end())
 847                                 continue;
 848                         if (!is_order_function(ita->second) && !is_order_function(itb->second))
 849                                 co += ita->second * itb->second;
 850                 }
 851                 if (!co.is_zero())
 852                         new_seq.emplace_back(expair(co, numeric(cdeg)));
 853         }
 854         if (higher_order_c < std::numeric_limits<int>::max())
 855                 new_seq.emplace_back(expair(Order(_ex1), numeric(higher_order_c)));
 856         return pseries(relational(var, point), std::move(new_seq));
 857 }
 858
 859
 860 /** Implementation of ex::series() for product. This performs series
 861  *  multiplication when multiplying series.
 862  *  @see ex::series */
 863 ex mul::series(const relational & r, int order, unsigned options) const
 864 {
 865         pseries acc; // Series accumulator
 866
 867         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 868         const ex& sym = r.lhs();
 869
 870         // holds ldegrees of the series of individual factors
 871         std::vector<int> ldegrees;
 872         std::vector<bool> ldegree_redo;
 873
 874         // find minimal degrees
 875         // first round: obtain a bound up to which minimal degrees have to be
 876         // considered
 877         for (auto & it : seq) {
 878
 879                 ex expon = it.coeff;
 880                 int factor = 1;
 881                 ex buf;
 882                 if (expon.info(info_flags::integer)) {
 883                         buf = it.rest;
 884                         factor = ex_to<numeric>(expon).to_int();
 885                 } else {
 886                         buf = recombine_pair_to_ex(it);
 887                 }
 888
 889                 int real_ldegree = 0;
 890                 bool flag_redo = false;
 891                 try {
 892                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 893                 } catch (std::runtime_error &) {}
 894
 895                 if (real_ldegree == 0) {
 896                         if ( factor < 0 ) {
 897                                 // This case must terminate, otherwise we would have division by
 898                                 // zero.
 899                                 int orderloop = 0;
 900                                 do {
 901                                         orderloop++;
 902                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 903                                 } while (real_ldegree == orderloop);
 904                         } else {
 905                                 // Here it is possible that buf does not have a ldegree, therefore
 906                                 // check only if ldegree is negative, otherwise reconsider the case
 907                                 // in the second round.
 908                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 909                                 if (real_ldegree == 0)
 910                                         flag_redo = true;
 911                         }
 912                 }
 913
 914                 ldegrees.push_back(factor * real_ldegree);
 915                 ldegree_redo.push_back(flag_redo);
 916         }
 917
 918         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 919         // Second round: determine the remaining positive ldegrees by the series
 920         // method.
 921         // here we can ignore ldegrees larger than degbound
 922         size_t j = 0;
 923         for (auto & it : seq) {
 924                 if ( ldegree_redo[j] ) {
 925                         ex expon = it.coeff;
 926                         int factor = 1;
 927                         ex buf;
 928                         if (expon.info(info_flags::integer)) {
 929                                 buf = it.rest;
 930                                 factor = ex_to<numeric>(expon).to_int();
 931                         } else {
 932                                 buf = recombine_pair_to_ex(it);
 933                         }
 934                         int real_ldegree = 0;
 935                         int orderloop = 0;
 936                         do {
 937                                 orderloop++;
 938                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 939                         } while ((real_ldegree == orderloop)
 940                               && (factor*real_ldegree < degbound));
 941                         ldegrees[j] = factor * real_ldegree;
 942                         degbound -= factor * real_ldegree;
 943                 }
 944                 j++;
 945         }
 946
 947         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 948
 949         if (degsum > order) {
 950                 return dynallocate<pseries>(r, epvector{{Order(_ex1), order}});
 951         }
 952
 953         // Multiply with remaining terms
 954         auto itd = ldegrees.begin();
 955         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 956
 957                 // do series expansion with adjusted order
 958                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 959
 960                 // Series multiplication
 961                 if (it == seq.begin())
 962                         acc = ex_to<pseries>(op);
 963                 else
 964                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 965         }
 966
 967         return acc.mul_const(ex_to<numeric>(overall_coeff));
 968 }
 969
 970
 971 /** Compute the p-th power of a series.
 972  *
 973  *  @param p  power to compute
 974  *  @param deg  truncation order of series calculation */
 975 ex pseries::power_const(const numeric &p, int deg) const
 976 {
 977         // method:
 978         // (due to Leonhard Euler)
 979         // let A(x) be this series and for the time being let it start with a
 980         // constant (later we'll generalize):
 981         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 982         // We want to compute
 983         //     C(x) = A(x)^p
 984         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 985         // Taking the derivative on both sides and multiplying with A(x) one
 986         // immediately arrives at
 987         //     C'(x)*A(x) = p*C(x)*A'(x)
 988         // Multiplying this out and comparing coefficients we get the recurrence
 989         // formula
 990         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 991         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 992         // which can easily be solved given the starting value c_0 = (a_0)^p.
 993         // For the more general case where the leading coefficient of A(x) is not
 994         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 995         // repeat the above derivation.  The leading power of C2(x) = A2(x)^p is
 996         // then of course a_0^p*x^(p*m) but the recurrence formula still holds.
 997
 998         if (seq.empty()) {
 999                 // as a special case, handle the empty (zero) series honoring the
1000                 // usual power laws such as implemented in power::eval()
1001                 if (p.real().is_zero())
1002                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1003                 else if (p.real().is_negative())
1004                         throw pole_error("pseries::power_const(): division by zero",1);
1005                 else
1006                         return *this;
1007         }
1008
1009         const int base_ldeg = ldegree(var);
1010         if (!(p*base_ldeg).is_integer())
1011                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1012         int new_ldeg = (p*base_ldeg).to_int();
1013
1014         const int base_deg = degree(var);
1015         int new_deg = deg;
1016         if (p.is_pos_integer()) {
1017                 // No need to compute beyond p*base_deg.
1018                 new_deg = std::min((p*base_deg).to_int(), deg);
1019         }
1020
1021         // adjust number of coefficients
1022         int numcoeff = new_deg - new_ldeg;
1023         if (new_deg < deg)
1024                 numcoeff += 1;
1025
1026         if (numcoeff <= 0) {
1027                 return dynallocate<pseries>(relational(var, point),
1028                                             epvector{{Order(_ex1), deg}});
1029         }
1030
1031         // O(x^n)^(-m) is undefined
1032         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1033                 throw pole_error("pseries::power_const(): division by zero",1);
1034
1035         // Compute coefficients of the powered series
1036         exvector co;
1037         co.reserve(numcoeff);
1038         co.push_back(pow(coeff(var, base_ldeg), p));
1039         for (int i=1; i<numcoeff; ++i) {
1040                 ex sum = _ex0;
1041                 for (int j=1; j<=i; ++j) {
1042                         ex c = coeff(var, j + base_ldeg);
1043                         if (is_order_function(c)) {
1044                                 co.push_back(Order(_ex1));
1045                                 break;
1046                         } else
1047                                 sum += (p * j - (i - j)) * co[i - j] * c;
1048                 }
1049                 co.push_back(sum / coeff(var, base_ldeg) / i);
1050         }
1051
1052         // Construct new series (of non-zero coefficients)
1053         epvector new_seq;
1054         bool higher_order = false;
1055         for (int i=0; i<numcoeff; ++i) {
1056                 if (!co[i].is_zero()) {
1057                         new_seq.emplace_back(expair(co[i], new_ldeg + i));
1058                 }
1059                 if (is_order_function(co[i])) {
1060                         higher_order = true;
1061                         break;
1062                 }
1063         }
1064         if (!higher_order && new_deg == deg) {
1065                 new_seq.emplace_back(expair{Order(_ex1), new_deg});
1066         }
1067
1068         return pseries(relational(var,point), std::move(new_seq));
1069 }
1070
1071
1072 /** Return a new pseries object with the powers shifted by deg. */
1073 pseries pseries::shift_exponents(int deg) const
1074 {
1075         epvector newseq = seq;
1076         for (auto & it : newseq)
1077                 it.coeff += deg;
1078         return pseries(relational(var, point), std::move(newseq));
1079 }
1080
1081
1082 /** Implementation of ex::series() for powers. This performs Laurent expansion
1083  *  of reciprocals of series at singularities.
1084  *  @see ex::series */
1085 ex power::series(const relational & r, int order, unsigned options) const
1086 {
1087         // If basis is already a series, just power it
1088         if (is_exactly_a<pseries>(basis))
1089                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1090
1091         // Basis is not a series, may there be a singularity?
1092         bool must_expand_basis = false;
1093         try {
1094                 basis.subs(r, subs_options::no_pattern);
1095         } catch (pole_error &) {
1096                 must_expand_basis = true;
1097         }
1098
1099         bool exponent_is_regular = true;
1100         try {
1101                 exponent.subs(r, subs_options::no_pattern);
1102         } catch (pole_error &) {
1103                 exponent_is_regular = false;
1104         }
1105
1106         if (!exponent_is_regular) {
1107                 ex l = exponent*log(basis);
1108                 // this == exp(l);
1109                 ex le = l.series(r, order, options);
1110                 // Note: expanding exp(l) won't help, since that will attempt
1111                 // Taylor expansion, and fail (because exponent is "singular")
1112                 // Still l itself might be expanded in Taylor series.
1113                 // Examples:
1114                 // sin(x)/x*log(cos(x))
1115                 // 1/x*log(1 + x)
1116                 return exp(le).series(r, order, options);
1117                 // Note: if l happens to have a Laurent expansion (with
1118                 // negative powers of (var - point)), expanding exp(le)
1119                 // will barf (which is The Right Thing).
1120         }
1121
1122         // Is the expression of type something^(-int)?
1123         if (!must_expand_basis && !exponent.info(info_flags::negint)
1124          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1125                 return basic::series(r, order, options);
1126
1127         // Is the expression of type 0^something?
1128         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1129          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1130                 return basic::series(r, order, options);
1131
1132         // Singularity encountered, is the basis equal to (var - point)?
1133         if (basis.is_equal(r.lhs() - r.rhs())) {
1134                 epvector new_seq;
1135                 if (ex_to<numeric>(exponent).to_int() < order)
1136                         new_seq.emplace_back(expair(_ex1, exponent));
1137                 else
1138                         new_seq.emplace_back(expair(Order(_ex1), exponent));
1139                 return pseries(r, std::move(new_seq));
1140         }
1141
1142         // No, expand basis into series
1143
1144         numeric numexp;
1145         if (is_a<numeric>(exponent)) {
1146                 numexp = ex_to<numeric>(exponent);
1147         } else {
1148                 numexp = 0;
1149         }
1150         const ex& sym = r.lhs();
1151         // find existing minimal degree
1152         ex eb = basis.expand();
1153         int real_ldegree = 0;
1154         if (eb.info(info_flags::rational_function))
1155                 real_ldegree = eb.ldegree(sym-r.rhs());
1156         if (real_ldegree == 0) {
1157                 int orderloop = 0;
1158                 do {
1159                         orderloop++;
1160                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1161                 } while (real_ldegree == orderloop);
1162         }
1163
1164         if (!(real_ldegree*numexp).is_integer())
1165                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1166         int extra_terms = (real_ldegree*(1-numexp)).to_int();
1167         ex e = basis.series(r, order + std::max(0, extra_terms), 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.emplace_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.emplace_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