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) 1999-2021 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 range = n.find_property_range("coeff", "power");
 123         seq.reserve((range.end-range.begin)/2);
 124
 125         for (auto loc = range.begin; loc < range.end;) {
 126                 ex rest;
 127                 ex coeff;
 128                 n.find_ex_by_loc(loc++, rest, sym_lst);
 129                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 130                 seq.emplace_back(expair(rest, coeff));
 131         }
 132
 133         n.find_ex("var", var, sym_lst);
 134         n.find_ex("point", point, sym_lst);
 135 }
 136
 137 void pseries::archive(archive_node &n) const
 138 {
 139         inherited::archive(n);
 140         for (auto & it : seq) {
 141                 n.add_ex("coeff", it.rest);
 142                 n.add_ex("power", it.coeff);
 143         }
 144         n.add_ex("var", var);
 145         n.add_ex("point", point);
 146 }
 147
 148
 149 //////////
 150 // functions overriding virtual functions from base classes
 151 //////////
 152
 153 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
 154 {
 155         if (precedence() <= level)
 156                 c.s << '(';
 157
 158         // objects of type pseries must not have any zero entries, so the
 159         // trivial (zero) pseries needs a special treatment here:
 160         if (seq.empty())
 161                 c.s << '0';
 162
 163         auto i = seq.begin(), end = seq.end();
 164         while (i != end) {
 165
 166                 // print a sign, if needed
 167                 if (i != seq.begin())
 168                         c.s << '+';
 169
 170                 if (!is_order_function(i->rest)) {
 171
 172                         // print 'rest', i.e. the expansion coefficient
 173                         if (i->rest.info(info_flags::numeric) &&
 174                                 i->rest.info(info_flags::positive)) {
 175                                 i->rest.print(c);
 176                         } else {
 177                                 c.s << openbrace << '(';
 178                                 i->rest.print(c);
 179                                 c.s << ')' << closebrace;
 180                         }
 181
 182                         // print 'coeff', something like (x-1)^42
 183                         if (!i->coeff.is_zero()) {
 184                                 c.s << mul_sym;
 185                                 if (!point.is_zero()) {
 186                                         c.s << openbrace << '(';
 187                                         (var-point).print(c);
 188                                         c.s << ')' << closebrace;
 189                                 } else
 190                                         var.print(c);
 191                                 if (i->coeff.compare(_ex1)) {
 192                                         c.s << pow_sym;
 193                                         c.s << openbrace;
 194                                         if (i->coeff.info(info_flags::negative)) {
 195                                                 c.s << '(';
 196                                                 i->coeff.print(c);
 197                                                 c.s << ')';
 198                                         } else
 199                                                 i->coeff.print(c);
 200                                         c.s << closebrace;
 201                                 }
 202                         }
 203                 } else
 204                         Order(pow(var - point, i->coeff)).print(c);
 205                 ++i;
 206         }
 207
 208         if (precedence() <= level)
 209                 c.s << ')';
 210 }
 211
 212 void pseries::do_print(const print_context & c, unsigned level) const
 213 {
 214         print_series(c, "", "", "*", "^", level);
 215 }
 216
 217 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 218 {
 219         print_series(c, "{", "}", " ", "^", level);
 220 }
 221
 222 void pseries::do_print_python(const print_python & c, unsigned level) const
 223 {
 224         print_series(c, "", "", "*", "**", level);
 225 }
 226
 227 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 228 {
 229         c.s << std::string(level, ' ') << class_name() << " @" << this
 230             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 231             << std::endl;
 232         size_t num = seq.size();
 233         for (size_t i=0; i<num; ++i) {
 234                 seq[i].rest.print(c, level + c.delta_indent);
 235                 seq[i].coeff.print(c, level + c.delta_indent);
 236                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 237         }
 238         var.print(c, level + c.delta_indent);
 239         point.print(c, level + c.delta_indent);
 240 }
 241
 242 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 243 {
 244         c.s << class_name() << "(relational(";
 245         var.print(c);
 246         c.s << ',';
 247         point.print(c);
 248         c.s << "),[";
 249         size_t num = seq.size();
 250         for (size_t i=0; i<num; ++i) {
 251                 if (i)
 252                         c.s << ',';
 253                 c.s << '(';
 254                 seq[i].rest.print(c);
 255                 c.s << ',';
 256                 seq[i].coeff.print(c);
 257                 c.s << ')';
 258         }
 259         c.s << "])";
 260 }
 261
 262 int pseries::compare_same_type(const basic & other) const
 263 {
 264         GINAC_ASSERT(is_a<pseries>(other));
 265         const pseries &o = static_cast<const pseries &>(other);
 266
 267         // first compare the lengths of the series...
 268         if (seq.size()>o.seq.size())
 269                 return 1;
 270         if (seq.size()<o.seq.size())
 271                 return -1;
 272
 273         // ...then the expansion point...
 274         int cmpval = var.compare(o.var);
 275         if (cmpval)
 276                 return cmpval;
 277         cmpval = point.compare(o.point);
 278         if (cmpval)
 279                 return cmpval;
 280
 281         // ...and if that failed the individual elements
 282         auto it = seq.begin(), o_it = o.seq.begin();
 283         while (it!=seq.end() && o_it!=o.seq.end()) {
 284                 cmpval = it->compare(*o_it);
 285                 if (cmpval)
 286                         return cmpval;
 287                 ++it;
 288                 ++o_it;
 289         }
 290
 291         // so they are equal.
 292         return 0;
 293 }
 294
 295 /** Return the number of operands including a possible order term. */
 296 size_t pseries::nops() const
 297 {
 298         return seq.size();
 299 }
 300
 301 /** Return the ith term in the series when represented as a sum. */
 302 ex pseries::op(size_t i) const
 303 {
 304         if (i >= seq.size())
 305                 throw (std::out_of_range("op() out of range"));
 306
 307         if (is_order_function(seq[i].rest))
 308                 return Order(pow(var-point, seq[i].coeff));
 309         return seq[i].rest * pow(var - point, seq[i].coeff);
 310 }
 311
 312 /** Return degree of highest power of the series.  This is usually the exponent
 313  *  of the Order term.  If s is not the expansion variable of the series, the
 314  *  series is examined termwise. */
 315 int pseries::degree(const ex &s) const
 316 {
 317         if (seq.empty())
 318                 return 0;
 319
 320         if (var.is_equal(s))
 321                 // Return last/greatest exponent
 322                 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 323
 324         int max_pow = std::numeric_limits<int>::min();
 325         for (auto & it : seq)
 326                 max_pow = std::max(max_pow, it.rest.degree(s));
 327         return max_pow;
 328 }
 329
 330 /** Return degree of lowest power of the series.  This is usually the exponent
 331  *  of the leading term.  If s is not the expansion variable of the series, the
 332  *  series is examined termwise.  If s is the expansion variable but the
 333  *  expansion point is not zero the series is not expanded to find the degree.
 334  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 335 int pseries::ldegree(const ex &s) const
 336 {
 337         if (seq.empty())
 338                 return 0;
 339
 340         if (var.is_equal(s))
 341                 // Return first/smallest exponent
 342                 return ex_to<numeric>((seq.begin())->coeff).to_int();
 343
 344         int min_pow = std::numeric_limits<int>::max();
 345         for (auto & it : seq)
 346                 min_pow = std::min(min_pow, it.rest.degree(s));
 347         return min_pow;
 348 }
 349
 350 /** Return coefficient of degree n in power series if s is the expansion
 351  *  variable.  If the expansion point is nonzero, by definition the n=1
 352  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 353  *  the expansion took place in the s in the first place).
 354  *  If s is not the expansion variable, an attempt is made to convert the
 355  *  series to a polynomial and return the corresponding coefficient from
 356  *  there. */
 357 ex pseries::coeff(const ex &s, int n) const
 358 {
 359         if (var.is_equal(s)) {
 360                 if (seq.empty())
 361                         return _ex0;
 362
 363                 // Binary search in sequence for given power
 364                 numeric looking_for = numeric(n);
 365                 int lo = 0, hi = seq.size() - 1;
 366                 while (lo <= hi) {
 367                         int mid = (lo + hi) / 2;
 368                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 369                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 370                         switch (cmp) {
 371                                 case -1:
 372                                         lo = mid + 1;
 373                                         break;
 374                                 case 0:
 375                                         return seq[mid].rest;
 376                                 case 1:
 377                                         hi = mid - 1;
 378                                         break;
 379                                 default:
 380                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 381                         }
 382                 }
 383                 return _ex0;
 384         } else
 385                 return convert_to_poly().coeff(s, n);
 386 }
 387
 388 /** Does nothing. */
 389 ex pseries::collect(const ex &s, bool distributed) const
 390 {
 391         return *this;
 392 }
 393
 394 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 395 ex pseries::eval() const
 396 {
 397         return hold();
 398 }
 399
 400 /** Evaluate coefficients numerically. */
 401 ex pseries::evalf() const
 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.emplace_back(expair(it.rest.evalf(), it.coeff));
 408
 409         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 410 }
 411
 412 ex pseries::conjugate() const
 413 {
 414         if(!var.info(info_flags::real))
 415                 return conjugate_function(*this).hold();
 416
 417         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
 418         ex newpoint = point.conjugate();
 419
 420         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 421                 return *this;
 422         }
 423
 424         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
 425 }
 426
 427 ex pseries::real_part() const
 428 {
 429         if(!var.info(info_flags::real))
 430                 return real_part_function(*this).hold();
 431         ex newpoint = point.real_part();
 432         if(newpoint != point)
 433                 return real_part_function(*this).hold();
 434
 435         epvector v;
 436         v.reserve(seq.size());
 437         for (auto & it : seq)
 438                 v.emplace_back(expair(it.rest.real_part(), it.coeff));
 439         return dynallocate<pseries>(var==point, std::move(v));
 440 }
 441
 442 ex pseries::imag_part() const
 443 {
 444         if(!var.info(info_flags::real))
 445                 return imag_part_function(*this).hold();
 446         ex newpoint = point.real_part();
 447         if(newpoint != point)
 448                 return imag_part_function(*this).hold();
 449
 450         epvector v;
 451         v.reserve(seq.size());
 452         for (auto & it : seq)
 453                 v.emplace_back(expair(it.rest.imag_part(), it.coeff));
 454         return dynallocate<pseries>(var==point, std::move(v));
 455 }
 456
 457 ex pseries::eval_integ() const
 458 {
 459         std::unique_ptr<epvector> newseq(nullptr);
 460         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 461                 if (newseq) {
 462                         newseq->emplace_back(expair(i->rest.eval_integ(), i->coeff));
 463                         continue;
 464                 }
 465                 ex newterm = i->rest.eval_integ();
 466                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 467                         newseq.reset(new epvector);
 468                         newseq->reserve(seq.size());
 469                         for (auto j=seq.begin(); j!=i; ++j)
 470                                 newseq->push_back(*j);
 471                         newseq->emplace_back(expair(newterm, i->coeff));
 472                 }
 473         }
 474
 475         ex newpoint = point.eval_integ();
 476         if (newseq || !are_ex_trivially_equal(newpoint, point))
 477                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
 478         return *this;
 479 }
 480
 481 ex pseries::evalm() const
 482 {
 483         // evalm each coefficient
 484         epvector newseq;
 485         bool something_changed = false;
 486         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 487                 if (something_changed) {
 488                         ex newcoeff = i->rest.evalm();
 489                         if (!newcoeff.is_zero())
 490                                 newseq.emplace_back(expair(newcoeff, i->coeff));
 491                 } else {
 492                         ex newcoeff = i->rest.evalm();
 493                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 494                                 something_changed = true;
 495                                 newseq.reserve(seq.size());
 496                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 497                                 if (!newcoeff.is_zero())
 498                                         newseq.emplace_back(expair(newcoeff, i->coeff));
 499                         }
 500                 }
 501         }
 502         if (something_changed)
 503                 return dynallocate<pseries>(var==point, std::move(newseq));
 504         else
 505                 return *this;
 506 }
 507
 508 ex pseries::subs(const exmap & m, unsigned options) const
 509 {
 510         // If expansion variable is being substituted, convert the series to a
 511         // polynomial and do the substitution there because the result might
 512         // no longer be a power series
 513         if (m.find(var) != m.end())
 514                 return convert_to_poly(true).subs(m, options);
 515
 516         // Otherwise construct a new series with substituted coefficients and
 517         // expansion point
 518         epvector newseq;
 519         newseq.reserve(seq.size());
 520         for (auto & it : seq)
 521                 newseq.emplace_back(expair(it.rest.subs(m, options), it.coeff));
 522         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
 523 }
 524
 525 /** Implementation of ex::expand() for a power series.  It expands all the
 526  *  terms individually and returns the resulting series as a new pseries. */
 527 ex pseries::expand(unsigned options) const
 528 {
 529         epvector newseq;
 530         for (auto & it : seq) {
 531                 ex restexp = it.rest.expand();
 532                 if (!restexp.is_zero())
 533                         newseq.emplace_back(expair(restexp, it.coeff));
 534         }
 535         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
 536 }
 537
 538 /** Implementation of ex::diff() for a power series.
 539  *  @see ex::diff */
 540 ex pseries::derivative(const symbol & s) const
 541 {
 542         epvector new_seq;
 543
 544         if (s == var) {
 545
 546                 // FIXME: coeff might depend on var
 547                 for (auto & it : seq) {
 548                         if (is_order_function(it.rest)) {
 549                                 new_seq.emplace_back(expair(it.rest, it.coeff - 1));
 550                         } else {
 551                                 ex c = it.rest * it.coeff;
 552                                 if (!c.is_zero())
 553                                         new_seq.emplace_back(expair(c, it.coeff - 1));
 554                         }
 555                 }
 556
 557         } else {
 558
 559                 for (auto & it : seq) {
 560                         if (is_order_function(it.rest)) {
 561                                 new_seq.push_back(it);
 562                         } else {
 563                                 ex c = it.rest.diff(s);
 564                                 if (!c.is_zero())
 565                                         new_seq.emplace_back(expair(c, it.coeff));
 566                         }
 567                 }
 568         }
 569
 570         return pseries(relational(var,point), std::move(new_seq));
 571 }
 572
 573 ex pseries::convert_to_poly(bool no_order) const
 574 {
 575         ex e;
 576         for (auto & it : seq) {
 577                 if (is_order_function(it.rest)) {
 578                         if (!no_order)
 579                                 e += Order(pow(var - point, it.coeff));
 580                 } else
 581                         e += it.rest * pow(var - point, it.coeff);
 582         }
 583         return e;
 584 }
 585
 586 bool pseries::is_terminating() const
 587 {
 588         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 589 }
 590
 591 ex pseries::coeffop(size_t i) const
 592 {
 593         if (i >= nops())
 594                 throw (std::out_of_range("coeffop() out of range"));
 595         return seq[i].rest;
 596 }
 597
 598 ex pseries::exponop(size_t i) const
 599 {
 600         if (i >= nops())
 601                 throw (std::out_of_range("exponop() out of range"));
 602         return seq[i].coeff;
 603 }
 604
 605
 606 /*
 607  *  Implementations of series expansion
 608  */
 609
 610 /** Default implementation of ex::series(). This performs Taylor expansion.
 611  *  @see ex::series */
 612 ex basic::series(const relational & r, int order, unsigned options) const
 613 {
 614         epvector seq;
 615         const symbol &s = ex_to<symbol>(r.lhs());
 616
 617         // default for order-values that make no sense for Taylor expansion
 618         if ((order <= 0) && this->has(s)) {
 619                 seq.emplace_back(expair(Order(_ex1), order));
 620                 return pseries(r, std::move(seq));
 621         }
 622
 623         // do Taylor expansion
 624         numeric fac = 1;
 625         ex deriv = *this;
 626         ex coeff = deriv.subs(r, subs_options::no_pattern);
 627
 628         if (!coeff.is_zero()) {
 629                 seq.emplace_back(expair(coeff, _ex0));
 630         }
 631
 632         int n;
 633         for (n=1; n<order; ++n) {
 634                 fac = fac.div(n);
 635                 // We need to test for zero in order to see if the series terminates.
 636                 // The problem is that there is no such thing as a perfect test for
 637                 // zero.  Expanding the term occasionally helps a little...
 638                 deriv = deriv.diff(s).expand();
 639                 if (deriv.is_zero())  // Series terminates
 640                         return pseries(r, std::move(seq));
 641
 642                 coeff = deriv.subs(r, subs_options::no_pattern);
 643                 if (!coeff.is_zero())
 644                         seq.emplace_back(expair(fac * coeff, n));
 645         }
 646
 647         // Higher-order terms, if present
 648         deriv = deriv.diff(s);
 649         if (!deriv.expand().is_zero())
 650                 seq.emplace_back(expair(Order(_ex1), n));
 651         return pseries(r, std::move(seq));
 652 }
 653
 654
 655 /** Implementation of ex::series() for symbols.
 656  *  @see ex::series */
 657 ex symbol::series(const relational & r, int order, unsigned options) const
 658 {
 659         epvector seq;
 660         const ex point = r.rhs();
 661         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 662
 663         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 664                 if (order > 0 && !point.is_zero())
 665                         seq.emplace_back(expair(point, _ex0));
 666                 if (order > 1)
 667                         seq.emplace_back(expair(_ex1, _ex1));
 668                 else
 669                         seq.emplace_back(expair(Order(_ex1), numeric(order)));
 670         } else
 671                 seq.emplace_back(expair(*this, _ex0));
 672         return pseries(r, std::move(seq));
 673 }
 674
 675
 676 /** Add one series object to another, producing a pseries object that
 677  *  represents the sum.
 678  *
 679  *  @param other  pseries object to add with
 680  *  @return the sum as a pseries */
 681 ex pseries::add_series(const pseries &other) const
 682 {
 683         // Adding two series with different variables or expansion points
 684         // results in an empty (constant) series
 685         if (!is_compatible_to(other)) {
 686                 epvector nul { expair(Order(_ex1), _ex0) };
 687                 return pseries(relational(var,point), std::move(nul));
 688         }
 689
 690         // Series addition
 691         epvector new_seq;
 692         auto a = seq.begin(), a_end = seq.end();
 693         auto b = other.seq.begin(), b_end = other.seq.end();
 694         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 695         for (;;) {
 696                 // If a is empty, fill up with elements from b and stop
 697                 if (a == a_end) {
 698                         while (b != b_end) {
 699                                 new_seq.push_back(*b);
 700                                 ++b;
 701                         }
 702                         break;
 703                 } else
 704                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 705
 706                 // If b is empty, fill up with elements from a and stop
 707                 if (b == b_end) {
 708                         while (a != a_end) {
 709                                 new_seq.push_back(*a);
 710                                 ++a;
 711                         }
 712                         break;
 713                 } else
 714                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 715
 716                 // a and b are non-empty, compare powers
 717                 if (pow_a < pow_b) {
 718                         // a has lesser power, get coefficient from a
 719                         new_seq.push_back(*a);
 720                         if (is_order_function((*a).rest))
 721                                 break;
 722                         ++a;
 723                 } else if (pow_b < pow_a) {
 724                         // b has lesser power, get coefficient from b
 725                         new_seq.push_back(*b);
 726                         if (is_order_function((*b).rest))
 727                                 break;
 728                         ++b;
 729                 } else {
 730                         // Add coefficient of a and b
 731                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 732                                 new_seq.emplace_back(expair(Order(_ex1), (*a).coeff));
 733                                 break;  // Order term ends the sequence
 734                         } else {
 735                                 ex sum = (*a).rest + (*b).rest;
 736                                 if (!(sum.is_zero()))
 737                                         new_seq.emplace_back(expair(sum, numeric(pow_a)));
 738                                 ++a;
 739                                 ++b;
 740                         }
 741                 }
 742         }
 743         return pseries(relational(var,point), std::move(new_seq));
 744 }
 745
 746
 747 /** Implementation of ex::series() for sums. This performs series addition when
 748  *  adding pseries objects.
 749  *  @see ex::series */
 750 ex add::series(const relational & r, int order, unsigned options) const
 751 {
 752         ex acc; // Series accumulator
 753
 754         // Get first term from overall_coeff
 755         acc = overall_coeff.series(r, order, options);
 756
 757         // Add remaining terms
 758         for (auto & it : seq) {
 759                 ex op;
 760                 if (is_exactly_a<pseries>(it.rest))
 761                         op = it.rest;
 762                 else
 763                         op = it.rest.series(r, order, options);
 764                 if (!it.coeff.is_equal(_ex1))
 765                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
 766
 767                 // Series addition
 768                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 769         }
 770         return acc;
 771 }
 772
 773
 774 /** Multiply a pseries object with a numeric constant, producing a pseries
 775  *  object that represents the product.
 776  *
 777  *  @param other  constant to multiply with
 778  *  @return the product as a pseries */
 779 ex pseries::mul_const(const numeric &other) const
 780 {
 781         epvector new_seq;
 782         new_seq.reserve(seq.size());
 783
 784         for (auto & it : seq) {
 785                 if (!is_order_function(it.rest))
 786                         new_seq.emplace_back(expair(it.rest * other, it.coeff));
 787                 else
 788                         new_seq.push_back(it);
 789         }
 790         return pseries(relational(var,point), std::move(new_seq));
 791 }
 792
 793
 794 /** Multiply one pseries object to another, producing a pseries object that
 795  *  represents the product.
 796  *
 797  *  @param other  pseries object to multiply with
 798  *  @return the product as a pseries */
 799 ex pseries::mul_series(const pseries &other) const
 800 {
 801         // Multiplying two series with different variables or expansion points
 802         // results in an empty (constant) series
 803         if (!is_compatible_to(other)) {
 804                 epvector nul { expair(Order(_ex1), _ex0) };
 805                 return pseries(relational(var,point), std::move(nul));
 806         }
 807
 808         if (seq.empty() || other.seq.empty()) {
 809                 return dynallocate<pseries>(var==point, epvector());
 810         }
 811
 812         // Series multiplication
 813         epvector new_seq;
 814         const int a_max = degree(var);
 815         const int b_max = other.degree(var);
 816         const int a_min = ldegree(var);
 817         const int b_min = other.ldegree(var);
 818         const int cdeg_min = a_min + b_min;
 819         int cdeg_max = a_max + b_max;
 820
 821         int higher_order_a = std::numeric_limits<int>::max();
 822         int higher_order_b = std::numeric_limits<int>::max();
 823         if (is_order_function(coeff(var, a_max)))
 824                 higher_order_a = a_max + b_min;
 825         if (is_order_function(other.coeff(var, b_max)))
 826                 higher_order_b = b_max + a_min;
 827         const int higher_order_c = std::min(higher_order_a, higher_order_b);
 828         if (cdeg_max >= higher_order_c)
 829                 cdeg_max = higher_order_c - 1;
 830
 831         std::map<int, ex> rest_map_a, rest_map_b;
 832         for (const auto& it : seq)
 833                 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 834
 835         if (other.var.is_equal(var))
 836                 for (const auto& it : other.seq)
 837                         rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 838
 839         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 840                 ex co = _ex0;
 841                 // c(i)=a(0)b(i)+...+a(i)b(0)
 842                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 843                         const auto& ita = rest_map_a.find(i);
 844                         if (ita == rest_map_a.end())
 845                                 continue;
 846                         const auto& itb = rest_map_b.find(cdeg-i);
 847                         if (itb == rest_map_b.end())
 848                                 continue;
 849                         if (!is_order_function(ita->second) && !is_order_function(itb->second))
 850                                 co += ita->second * itb->second;
 851                 }
 852                 if (!co.is_zero())
 853                         new_seq.emplace_back(expair(co, numeric(cdeg)));
 854         }
 855         if (higher_order_c < std::numeric_limits<int>::max())
 856                 new_seq.emplace_back(expair(Order(_ex1), numeric(higher_order_c)));
 857         return pseries(relational(var, point), std::move(new_seq));
 858 }
 859
 860
 861 /** Implementation of ex::series() for product. This performs series
 862  *  multiplication when multiplying series.
 863  *  @see ex::series */
 864 ex mul::series(const relational & r, int order, unsigned options) const
 865 {
 866         pseries acc; // Series accumulator
 867
 868         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 869         const ex& sym = r.lhs();
 870
 871         // holds ldegrees of the series of individual factors
 872         std::vector<int> ldegrees;
 873         std::vector<bool> ldegree_redo;
 874
 875         // find minimal degrees
 876         // first round: obtain a bound up to which minimal degrees have to be
 877         // considered
 878         for (auto & it : seq) {
 879
 880                 ex expon = it.coeff;
 881                 int factor = 1;
 882                 ex buf;
 883                 if (expon.info(info_flags::integer)) {
 884                         buf = it.rest;
 885                         factor = ex_to<numeric>(expon).to_int();
 886                 } else {
 887                         buf = recombine_pair_to_ex(it);
 888                 }
 889
 890                 int real_ldegree = 0;
 891                 bool flag_redo = false;
 892                 try {
 893                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 894                 } catch (std::runtime_error &) {}
 895
 896                 if (real_ldegree == 0) {
 897                         if ( factor < 0 ) {
 898                                 // This case must terminate, otherwise we would have division by
 899                                 // zero.
 900                                 int orderloop = 0;
 901                                 do {
 902                                         orderloop++;
 903                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 904                                 } while (real_ldegree == orderloop);
 905                         } else {
 906                                 // Here it is possible that buf does not have a ldegree, therefore
 907                                 // check only if ldegree is negative, otherwise reconsider the case
 908                                 // in the second round.
 909                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 910                                 if (real_ldegree == 0)
 911                                         flag_redo = true;
 912                         }
 913                 }
 914
 915                 ldegrees.push_back(factor * real_ldegree);
 916                 ldegree_redo.push_back(flag_redo);
 917         }
 918
 919         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 920         // Second round: determine the remaining positive ldegrees by the series
 921         // method.
 922         // here we can ignore ldegrees larger than degbound
 923         size_t j = 0;
 924         for (auto & it : seq) {
 925                 if ( ldegree_redo[j] ) {
 926                         ex expon = it.coeff;
 927                         int factor = 1;
 928                         ex buf;
 929                         if (expon.info(info_flags::integer)) {
 930                                 buf = it.rest;
 931                                 factor = ex_to<numeric>(expon).to_int();
 932                         } else {
 933                                 buf = recombine_pair_to_ex(it);
 934                         }
 935                         int real_ldegree = 0;
 936                         int orderloop = 0;
 937                         do {
 938                                 orderloop++;
 939                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 940                         } while ((real_ldegree == orderloop)
 941                               && (factor*real_ldegree < degbound));
 942                         ldegrees[j] = factor * real_ldegree;
 943                         degbound -= factor * real_ldegree;
 944                 }
 945                 j++;
 946         }
 947
 948         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 949
 950         if (degsum > order) {
 951                 return dynallocate<pseries>(r, epvector{{Order(_ex1), order}});
 952         }
 953
 954         // Multiply with remaining terms
 955         auto itd = ldegrees.begin();
 956         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 957
 958                 // do series expansion with adjusted order
 959                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 960
 961                 // Series multiplication
 962                 if (it == seq.begin())
 963                         acc = ex_to<pseries>(op);
 964                 else
 965                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 966         }
 967
 968         return acc.mul_const(ex_to<numeric>(overall_coeff));
 969 }
 970
 971
 972 /** Compute the p-th power of a series.
 973  *
 974  *  @param p  power to compute
 975  *  @param deg  truncation order of series calculation */
 976 ex pseries::power_const(const numeric &p, int deg) const
 977 {
 978         // method:
 979         // (due to Leonhard Euler)
 980         // let A(x) be this series and for the time being let it start with a
 981         // constant (later we'll generalize):
 982         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 983         // We want to compute
 984         //     C(x) = A(x)^p
 985         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 986         // Taking the derivative on both sides and multiplying with A(x) one
 987         // immediately arrives at
 988         //     C'(x)*A(x) = p*C(x)*A'(x)
 989         // Multiplying this out and comparing coefficients we get the recurrence
 990         // formula
 991         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 992         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 993         // which can easily be solved given the starting value c_0 = (a_0)^p.
 994         // For the more general case where the leading coefficient of A(x) is not
 995         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 996         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 997         // then of course x^(p*m) but the recurrence formula still holds.
 998
 999         if (seq.empty()) {
1000                 // as a special case, handle the empty (zero) series honoring the
1001                 // usual power laws such as implemented in power::eval()
1002                 if (p.real().is_zero())
1003                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1004                 else if (p.real().is_negative())
1005                         throw pole_error("pseries::power_const(): division by zero",1);
1006                 else
1007                         return *this;
1008         }
1009
1010         const int ldeg = ldegree(var);
1011         if (!(p*ldeg).is_integer())
1012                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1013
1014         // adjust number of coefficients
1015         int numcoeff = deg - (p*ldeg).to_int();
1016         if (numcoeff <= 0) {
1017                 epvector epv { expair(Order(_ex1), deg) };
1018                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1019         }
1020
1021         // O(x^n)^(-m) is undefined
1022         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1023                 throw pole_error("pseries::power_const(): division by zero",1);
1024
1025         // Compute coefficients of the powered series
1026         exvector co;
1027         co.reserve(numcoeff);
1028         co.push_back(pow(coeff(var, ldeg), p));
1029         for (int i=1; i<numcoeff; ++i) {
1030                 ex sum = _ex0;
1031                 for (int j=1; j<=i; ++j) {
1032                         ex c = coeff(var, j + ldeg);
1033                         if (is_order_function(c)) {
1034                                 co.push_back(Order(_ex1));
1035                                 break;
1036                         } else
1037                                 sum += (p * j - (i - j)) * co[i - j] * c;
1038                 }
1039                 co.push_back(sum / coeff(var, ldeg) / i);
1040         }
1041
1042         // Construct new series (of non-zero coefficients)
1043         epvector new_seq;
1044         bool higher_order = false;
1045         for (int i=0; i<numcoeff; ++i) {
1046                 if (!co[i].is_zero())
1047                         new_seq.emplace_back(expair(co[i], p * ldeg + i));
1048                 if (is_order_function(co[i])) {
1049                         higher_order = true;
1050                         break;
1051                 }
1052         }
1053         if (!higher_order)
1054                 new_seq.emplace_back(expair(Order(_ex1), p * ldeg + numcoeff));
1055
1056         return pseries(relational(var,point), std::move(new_seq));
1057 }
1058
1059
1060 /** Return a new pseries object with the powers shifted by deg. */
1061 pseries pseries::shift_exponents(int deg) const
1062 {
1063         epvector newseq = seq;
1064         for (auto & it : newseq)
1065                 it.coeff += deg;
1066         return pseries(relational(var, point), std::move(newseq));
1067 }
1068
1069
1070 /** Implementation of ex::series() for powers. This performs Laurent expansion
1071  *  of reciprocals of series at singularities.
1072  *  @see ex::series */
1073 ex power::series(const relational & r, int order, unsigned options) const
1074 {
1075         // If basis is already a series, just power it
1076         if (is_exactly_a<pseries>(basis))
1077                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1078
1079         // Basis is not a series, may there be a singularity?
1080         bool must_expand_basis = false;
1081         try {
1082                 basis.subs(r, subs_options::no_pattern);
1083         } catch (pole_error &) {
1084                 must_expand_basis = true;
1085         }
1086
1087         bool exponent_is_regular = true;
1088         try {
1089                 exponent.subs(r, subs_options::no_pattern);
1090         } catch (pole_error &) {
1091                 exponent_is_regular = false;
1092         }
1093
1094         if (!exponent_is_regular) {
1095                 ex l = exponent*log(basis);
1096                 // this == exp(l);
1097                 ex le = l.series(r, order, options);
1098                 // Note: expanding exp(l) won't help, since that will attempt
1099                 // Taylor expansion, and fail (because exponent is "singular")
1100                 // Still l itself might be expanded in Taylor series.
1101                 // Examples:
1102                 // sin(x)/x*log(cos(x))
1103                 // 1/x*log(1 + x)
1104                 return exp(le).series(r, order, options);
1105                 // Note: if l happens to have a Laurent expansion (with
1106                 // negative powers of (var - point)), expanding exp(le)
1107                 // will barf (which is The Right Thing).
1108         }
1109
1110         // Is the expression of type something^(-int)?
1111         if (!must_expand_basis && !exponent.info(info_flags::negint)
1112          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1113                 return basic::series(r, order, options);
1114
1115         // Is the expression of type 0^something?
1116         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1117          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1118                 return basic::series(r, order, options);
1119
1120         // Singularity encountered, is the basis equal to (var - point)?
1121         if (basis.is_equal(r.lhs() - r.rhs())) {
1122                 epvector new_seq;
1123                 if (ex_to<numeric>(exponent).to_int() < order)
1124                         new_seq.emplace_back(expair(_ex1, exponent));
1125                 else
1126                         new_seq.emplace_back(expair(Order(_ex1), exponent));
1127                 return pseries(r, std::move(new_seq));
1128         }
1129
1130         // No, expand basis into series
1131
1132         numeric numexp;
1133         if (is_a<numeric>(exponent)) {
1134                 numexp = ex_to<numeric>(exponent);
1135         } else {
1136                 numexp = 0;
1137         }
1138         const ex& sym = r.lhs();
1139         // find existing minimal degree
1140         ex eb = basis.expand();
1141         int real_ldegree = 0;
1142         if (eb.info(info_flags::rational_function))
1143                 real_ldegree = eb.ldegree(sym-r.rhs());
1144         if (real_ldegree == 0) {
1145                 int orderloop = 0;
1146                 do {
1147                         orderloop++;
1148                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1149                 } while (real_ldegree == orderloop);
1150         }
1151
1152         if (!(real_ldegree*numexp).is_integer())
1153                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1154         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1155
1156         ex result;
1157         try {
1158                 result = ex_to<pseries>(e).power_const(numexp, order);
1159         } catch (pole_error &) {
1160                 epvector ser { expair(Order(_ex1), order) };
1161                 result = pseries(r, std::move(ser));
1162         }
1163
1164         return result;
1165 }
1166
1167
1168 /** Re-expansion of a pseries object. */
1169 ex pseries::series(const relational & r, int order, unsigned options) const
1170 {
1171         const ex p = r.rhs();
1172         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1173         const symbol &s = ex_to<symbol>(r.lhs());
1174
1175         if (var.is_equal(s) && point.is_equal(p)) {
1176                 if (order > degree(s))
1177                         return *this;
1178                 else {
1179                         epvector new_seq;
1180                         for (auto & it : seq) {
1181                                 int o = ex_to<numeric>(it.coeff).to_int();
1182                                 if (o >= order) {
1183                                         new_seq.emplace_back(expair(Order(_ex1), o));
1184                                         break;
1185                                 }
1186                                 new_seq.push_back(it);
1187                         }
1188                         return pseries(r, std::move(new_seq));
1189                 }
1190         } else
1191                 return convert_to_poly().series(r, order, options);
1192 }
1193
1194 ex integral::series(const relational & r, int order, unsigned options) const
1195 {
1196         if (x.subs(r) != x)
1197                 throw std::logic_error("Cannot series expand wrt dummy variable");
1198
1199         // Expanding integrand with r substituted taken in boundaries.
1200         ex fseries = f.series(r, order, options);
1201         epvector fexpansion;
1202         fexpansion.reserve(fseries.nops());
1203         for (size_t i=0; i<fseries.nops(); ++i) {
1204                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1205                 currcoeff = (currcoeff == Order(_ex1))
1206                         ? currcoeff
1207                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1208                 if (currcoeff != 0)
1209                         fexpansion.emplace_back(
1210                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1211         }
1212
1213         // Expanding lower boundary
1214         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1215         ex aseries = (a-a.subs(r)).series(r, order, options);
1216         fseries = f.series(x == (a.subs(r)), order, options);
1217         for (size_t i=0; i<fseries.nops(); ++i) {
1218                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1219                 if (is_order_function(currcoeff))
1220                         break;
1221                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1222                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1223                 currcoeff = currcoeff.series(r, orderforf);
1224                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1225                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1226                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1227                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1228         }
1229
1230         // Expanding upper boundary
1231         ex bseries = (b-b.subs(r)).series(r, order, options);
1232         fseries = f.series(x == (b.subs(r)), order, options);
1233         for (size_t i=0; i<fseries.nops(); ++i) {
1234                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1235                 if (is_order_function(currcoeff))
1236                         break;
1237                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1238                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1239                 currcoeff = currcoeff.series(r, orderforf);
1240                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1241                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1242                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1243                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1244         }
1245
1246         return result;
1247 }
1248
1249
1250 /** Compute the truncated series expansion of an expression.
1251  *  This function returns an expression containing an object of class pseries
1252  *  to represent the series. If the series does not terminate within the given
1253  *  truncation order, the last term of the series will be an order term.
1254  *
1255  *  @param r  expansion relation, lhs holds variable and rhs holds point
1256  *  @param order  truncation order of series calculations
1257  *  @param options  of class series_options
1258  *  @return an expression holding a pseries object */
1259 ex ex::series(const ex & r, int order, unsigned options) const
1260 {
1261         ex e;
1262         relational rel_;
1263
1264         if (is_a<relational>(r))
1265                 rel_ = ex_to<relational>(r);
1266         else if (is_a<symbol>(r))
1267                 rel_ = relational(r,_ex0);
1268         else
1269                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1270
1271         e = bp->series(rel_, order, options);
1272         return e;
1273 }
1274
1275 GINAC_BIND_UNARCHIVER(pseries);
1276
1277 } // namespace GiNaC