ginac/pseries.cpp

   1 /** @file pseries.cpp
   2  *
   3  *  Implementation of class for extended truncated power series and
   4  *  methods for series expansion. */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include "pseries.h"
  25 #include "add.h"
  26 #include "inifcns.h" // for Order function
  27 #include "lst.h"
  28 #include "mul.h"
  29 #include "power.h"
  30 #include "relational.h"
  31 #include "operators.h"
  32 #include "symbol.h"
  33 #include "integral.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36
  37 #include <limits>
  38 #include <numeric>
  39 #include <stdexcept>
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
  44   print_func<print_context>(&pseries::do_print).
  45   print_func<print_latex>(&pseries::do_print_latex).
  46   print_func<print_tree>(&pseries::do_print_tree).
  47   print_func<print_python>(&pseries::do_print_python).
  48   print_func<print_python_repr>(&pseries::do_print_python_repr))
  49
  50
  51 /*
  52  *  Default constructor
  53  */
  54
  55 pseries::pseries() { }
  56
  57
  58 /*
  59  *  Other ctors
  60  */
  61
  62 /** Construct pseries from a vector of coefficients and powers.
  63  *  expair.rest holds the coefficient, expair.coeff holds the power.
  64  *  The powers must be integers (positive or negative) and in ascending order;
  65  *  the last coefficient can be Order(_ex1) to represent a truncated,
  66  *  non-terminating series.
  67  *
  68  *  @param rel_  expansion variable and point (must hold a relational)
  69  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  70  *  @return newly constructed pseries */
  71 pseries::pseries(const ex &rel_, const epvector &ops_)
  72   : seq(ops_)
  73 {
  74         GINAC_ASSERT(is_a<relational>(rel_));
  75         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  76         point = rel_.rhs();
  77         var = rel_.lhs();
  78 }
  79 pseries::pseries(const ex &rel_, epvector &&ops_)
  80   : seq(std::move(ops_))
  81 {
  82         GINAC_ASSERT(is_a<relational>(rel_));
  83         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  84         point = rel_.rhs();
  85         var = rel_.lhs();
  86 }
  87
  88
  89 /*
  90  *  Archiving
  91  */
  92
  93 void pseries::read_archive(const archive_node &n, lst &sym_lst)
  94 {
  95         inherited::read_archive(n, sym_lst);
  96         auto first = n.find_first("coeff");
  97         auto last = n.find_last("power");
  98         ++last;
  99         seq.reserve((last-first)/2);
 100
 101         for (auto loc = first; loc < last;) {
 102                 ex rest;
 103                 ex coeff;
 104                 n.find_ex_by_loc(loc++, rest, sym_lst);
 105                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 106                 seq.push_back(expair(rest, coeff));
 107         }
 108
 109         n.find_ex("var", var, sym_lst);
 110         n.find_ex("point", point, sym_lst);
 111 }
 112
 113 void pseries::archive(archive_node &n) const
 114 {
 115         inherited::archive(n);
 116         for (auto & it : seq) {
 117                 n.add_ex("coeff", it.rest);
 118                 n.add_ex("power", it.coeff);
 119         }
 120         n.add_ex("var", var);
 121         n.add_ex("point", point);
 122 }
 123
 124
 125 //////////
 126 // functions overriding virtual functions from base classes
 127 //////////
 128
 129 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
 130 {
 131         if (precedence() <= level)
 132                 c.s << '(';
 133
 134         // objects of type pseries must not have any zero entries, so the
 135         // trivial (zero) pseries needs a special treatment here:
 136         if (seq.empty())
 137                 c.s << '0';
 138
 139         auto i = seq.begin(), end = seq.end();
 140         while (i != end) {
 141
 142                 // print a sign, if needed
 143                 if (i != seq.begin())
 144                         c.s << '+';
 145
 146                 if (!is_order_function(i->rest)) {
 147
 148                         // print 'rest', i.e. the expansion coefficient
 149                         if (i->rest.info(info_flags::numeric) &&
 150                                 i->rest.info(info_flags::positive)) {
 151                                 i->rest.print(c);
 152                         } else {
 153                                 c.s << openbrace << '(';
 154                                 i->rest.print(c);
 155                                 c.s << ')' << closebrace;
 156                         }
 157
 158                         // print 'coeff', something like (x-1)^42
 159                         if (!i->coeff.is_zero()) {
 160                                 c.s << mul_sym;
 161                                 if (!point.is_zero()) {
 162                                         c.s << openbrace << '(';
 163                                         (var-point).print(c);
 164                                         c.s << ')' << closebrace;
 165                                 } else
 166                                         var.print(c);
 167                                 if (i->coeff.compare(_ex1)) {
 168                                         c.s << pow_sym;
 169                                         c.s << openbrace;
 170                                         if (i->coeff.info(info_flags::negative)) {
 171                                                 c.s << '(';
 172                                                 i->coeff.print(c);
 173                                                 c.s << ')';
 174                                         } else
 175                                                 i->coeff.print(c);
 176                                         c.s << closebrace;
 177                                 }
 178                         }
 179                 } else
 180                         Order(power(var-point,i->coeff)).print(c);
 181                 ++i;
 182         }
 183
 184         if (precedence() <= level)
 185                 c.s << ')';
 186 }
 187
 188 void pseries::do_print(const print_context & c, unsigned level) const
 189 {
 190         print_series(c, "", "", "*", "^", level);
 191 }
 192
 193 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 194 {
 195         print_series(c, "{", "}", " ", "^", level);
 196 }
 197
 198 void pseries::do_print_python(const print_python & c, unsigned level) const
 199 {
 200         print_series(c, "", "", "*", "**", level);
 201 }
 202
 203 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 204 {
 205         c.s << std::string(level, ' ') << class_name() << " @" << this
 206             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 207             << std::endl;
 208         size_t num = seq.size();
 209         for (size_t i=0; i<num; ++i) {
 210                 seq[i].rest.print(c, level + c.delta_indent);
 211                 seq[i].coeff.print(c, level + c.delta_indent);
 212                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 213         }
 214         var.print(c, level + c.delta_indent);
 215         point.print(c, level + c.delta_indent);
 216 }
 217
 218 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 219 {
 220         c.s << class_name() << "(relational(";
 221         var.print(c);
 222         c.s << ',';
 223         point.print(c);
 224         c.s << "),[";
 225         size_t num = seq.size();
 226         for (size_t i=0; i<num; ++i) {
 227                 if (i)
 228                         c.s << ',';
 229                 c.s << '(';
 230                 seq[i].rest.print(c);
 231                 c.s << ',';
 232                 seq[i].coeff.print(c);
 233                 c.s << ')';
 234         }
 235         c.s << "])";
 236 }
 237
 238 int pseries::compare_same_type(const basic & other) const
 239 {
 240         GINAC_ASSERT(is_a<pseries>(other));
 241         const pseries &o = static_cast<const pseries &>(other);
 242
 243         // first compare the lengths of the series...
 244         if (seq.size()>o.seq.size())
 245                 return 1;
 246         if (seq.size()<o.seq.size())
 247                 return -1;
 248
 249         // ...then the expansion point...
 250         int cmpval = var.compare(o.var);
 251         if (cmpval)
 252                 return cmpval;
 253         cmpval = point.compare(o.point);
 254         if (cmpval)
 255                 return cmpval;
 256
 257         // ...and if that failed the individual elements
 258         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 259         while (it!=seq.end() && o_it!=o.seq.end()) {
 260                 cmpval = it->compare(*o_it);
 261                 if (cmpval)
 262                         return cmpval;
 263                 ++it;
 264                 ++o_it;
 265         }
 266
 267         // so they are equal.
 268         return 0;
 269 }
 270
 271 /** Return the number of operands including a possible order term. */
 272 size_t pseries::nops() const
 273 {
 274         return seq.size();
 275 }
 276
 277 /** Return the ith term in the series when represented as a sum. */
 278 ex pseries::op(size_t i) const
 279 {
 280         if (i >= seq.size())
 281                 throw (std::out_of_range("op() out of range"));
 282
 283         if (is_order_function(seq[i].rest))
 284                 return Order(power(var-point, seq[i].coeff));
 285         return seq[i].rest * power(var - point, seq[i].coeff);
 286 }
 287
 288 /** Return degree of highest power of the series.  This is usually the exponent
 289  *  of the Order term.  If s is not the expansion variable of the series, the
 290  *  series is examined termwise. */
 291 int pseries::degree(const ex &s) const
 292 {
 293         if (var.is_equal(s)) {
 294                 // Return last exponent
 295                 if (seq.size())
 296                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 297                 else
 298                         return 0;
 299         } else {
 300                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 301                 if (it == itend)
 302                         return 0;
 303                 int max_pow = std::numeric_limits<int>::min();
 304                 while (it != itend) {
 305                         int pow = it->rest.degree(s);
 306                         if (pow > max_pow)
 307                                 max_pow = pow;
 308                         ++it;
 309                 }
 310                 return max_pow;
 311         }
 312 }
 313
 314 /** Return degree of lowest power of the series.  This is usually the exponent
 315  *  of the leading term.  If s is not the expansion variable of the series, the
 316  *  series is examined termwise.  If s is the expansion variable but the
 317  *  expansion point is not zero the series is not expanded to find the degree.
 318  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 319 int pseries::ldegree(const ex &s) const
 320 {
 321         if (var.is_equal(s)) {
 322                 // Return first exponent
 323                 if (seq.size())
 324                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 325                 else
 326                         return 0;
 327         } else {
 328                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 329                 if (it == itend)
 330                         return 0;
 331                 int min_pow = std::numeric_limits<int>::max();
 332                 while (it != itend) {
 333                         int pow = it->rest.ldegree(s);
 334                         if (pow < min_pow)
 335                                 min_pow = pow;
 336                         ++it;
 337                 }
 338                 return min_pow;
 339         }
 340 }
 341
 342 /** Return coefficient of degree n in power series if s is the expansion
 343  *  variable.  If the expansion point is nonzero, by definition the n=1
 344  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 345  *  the expansion took place in the s in the first place).
 346  *  If s is not the expansion variable, an attempt is made to convert the
 347  *  series to a polynomial and return the corresponding coefficient from
 348  *  there. */
 349 ex pseries::coeff(const ex &s, int n) const
 350 {
 351         if (var.is_equal(s)) {
 352                 if (seq.empty())
 353                         return _ex0;
 354
 355                 // Binary search in sequence for given power
 356                 numeric looking_for = numeric(n);
 357                 int lo = 0, hi = seq.size() - 1;
 358                 while (lo <= hi) {
 359                         int mid = (lo + hi) / 2;
 360                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 361                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 362                         switch (cmp) {
 363                                 case -1:
 364                                         lo = mid + 1;
 365                                         break;
 366                                 case 0:
 367                                         return seq[mid].rest;
 368                                 case 1:
 369                                         hi = mid - 1;
 370                                         break;
 371                                 default:
 372                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 373                         }
 374                 }
 375                 return _ex0;
 376         } else
 377                 return convert_to_poly().coeff(s, n);
 378 }
 379
 380 /** Does nothing. */
 381 ex pseries::collect(const ex &s, bool distributed) const
 382 {
 383         return *this;
 384 }
 385
 386 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 387 ex pseries::eval() const
 388 {
 389         if (flags & status_flags::evaluated) {
 390                 return *this;
 391         }
 392
 393         // Construct a new series with evaluated coefficients
 394         epvector new_seq;
 395         new_seq.reserve(seq.size());
 396         epvector::const_iterator it = seq.begin(), itend = seq.end();
 397         while (it != itend) {
 398                 new_seq.push_back(expair(it->rest, it->coeff));
 399                 ++it;
 400         }
 401         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 402 }
 403
 404 /** Evaluate coefficients numerically. */
 405 ex pseries::evalf(int level) const
 406 {
 407         if (level == 1)
 408                 return *this;
 409
 410         if (level == -max_recursion_level)
 411                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 412
 413         // Construct a new series with evaluated coefficients
 414         epvector new_seq;
 415         new_seq.reserve(seq.size());
 416         epvector::const_iterator it = seq.begin(), itend = seq.end();
 417         while (it != itend) {
 418                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 419                 ++it;
 420         }
 421         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 422 }
 423
 424 ex pseries::conjugate() const
 425 {
 426         if(!var.info(info_flags::real))
 427                 return conjugate_function(*this).hold();
 428
 429         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
 430         ex newpoint = point.conjugate();
 431
 432         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 433                 return *this;
 434         }
 435
 436         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
 437 }
 438
 439 ex pseries::real_part() const
 440 {
 441         if(!var.info(info_flags::real))
 442                 return real_part_function(*this).hold();
 443         ex newpoint = point.real_part();
 444         if(newpoint != point)
 445                 return real_part_function(*this).hold();
 446
 447         epvector v;
 448         v.reserve(seq.size());
 449         for (auto & it : seq)
 450                 v.push_back(expair((it.rest).real_part(), it.coeff));
 451         return dynallocate<pseries>(var==point, std::move(v));
 452 }
 453
 454 ex pseries::imag_part() const
 455 {
 456         if(!var.info(info_flags::real))
 457                 return imag_part_function(*this).hold();
 458         ex newpoint = point.real_part();
 459         if(newpoint != point)
 460                 return imag_part_function(*this).hold();
 461
 462         epvector v;
 463         v.reserve(seq.size());
 464         for (auto & it : seq)
 465                 v.push_back(expair((it.rest).imag_part(), it.coeff));
 466         return dynallocate<pseries>(var==point, std::move(v));
 467 }
 468
 469 ex pseries::eval_integ() const
 470 {
 471         std::unique_ptr<epvector> newseq(nullptr);
 472         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 473                 if (newseq) {
 474                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 475                         continue;
 476                 }
 477                 ex newterm = i->rest.eval_integ();
 478                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 479                         newseq.reset(new epvector);
 480                         newseq->reserve(seq.size());
 481                         for (auto j=seq.begin(); j!=i; ++j)
 482                                 newseq->push_back(*j);
 483                         newseq->push_back(expair(newterm, i->coeff));
 484                 }
 485         }
 486
 487         ex newpoint = point.eval_integ();
 488         if (newseq || !are_ex_trivially_equal(newpoint, point))
 489                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
 490         return *this;
 491 }
 492
 493 ex pseries::evalm() const
 494 {
 495         // evalm each coefficient
 496         epvector newseq;
 497         bool something_changed = false;
 498         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 499                 if (something_changed) {
 500                         ex newcoeff = i->rest.evalm();
 501                         if (!newcoeff.is_zero())
 502                                 newseq.push_back(expair(newcoeff, i->coeff));
 503                 }
 504                 else {
 505                         ex newcoeff = i->rest.evalm();
 506                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 507                                 something_changed = true;
 508                                 newseq.reserve(seq.size());
 509                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 510                                 if (!newcoeff.is_zero())
 511                                         newseq.push_back(expair(newcoeff, i->coeff));
 512                         }
 513                 }
 514         }
 515         if (something_changed)
 516                 return dynallocate<pseries>(var==point, std::move(newseq));
 517         else
 518                 return *this;
 519 }
 520
 521 ex pseries::subs(const exmap & m, unsigned options) const
 522 {
 523         // If expansion variable is being substituted, convert the series to a
 524         // polynomial and do the substitution there because the result might
 525         // no longer be a power series
 526         if (m.find(var) != m.end())
 527                 return convert_to_poly(true).subs(m, options);
 528
 529         // Otherwise construct a new series with substituted coefficients and
 530         // expansion point
 531         epvector newseq;
 532         newseq.reserve(seq.size());
 533         for (auto & it : seq)
 534                 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
 535         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
 536 }
 537
 538 /** Implementation of ex::expand() for a power series.  It expands all the
 539  *  terms individually and returns the resulting series as a new pseries. */
 540 ex pseries::expand(unsigned options) const
 541 {
 542         epvector newseq;
 543         for (auto & it : seq) {
 544                 ex restexp = it.rest.expand();
 545                 if (!restexp.is_zero())
 546                         newseq.push_back(expair(restexp, it.coeff));
 547         }
 548         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
 549 }
 550
 551 /** Implementation of ex::diff() for a power series.
 552  *  @see ex::diff */
 553 ex pseries::derivative(const symbol & s) const
 554 {
 555         epvector new_seq;
 556
 557         if (s == var) {
 558
 559                 // FIXME: coeff might depend on var
 560                 for (auto & it : seq) {
 561                         if (is_order_function(it.rest)) {
 562                                 new_seq.push_back(expair(it.rest, it.coeff - 1));
 563                         } else {
 564                                 ex c = it.rest * it.coeff;
 565                                 if (!c.is_zero())
 566                                         new_seq.push_back(expair(c, it.coeff - 1));
 567                         }
 568                 }
 569
 570         } else {
 571
 572                 for (auto & it : seq) {
 573                         if (is_order_function(it.rest)) {
 574                                 new_seq.push_back(it);
 575                         } else {
 576                                 ex c = it.rest.diff(s);
 577                                 if (!c.is_zero())
 578                                         new_seq.push_back(expair(c, it.coeff));
 579                         }
 580                 }
 581         }
 582
 583         return pseries(relational(var,point), std::move(new_seq));
 584 }
 585
 586 ex pseries::convert_to_poly(bool no_order) const
 587 {
 588         ex e;
 589         for (auto & it : seq) {
 590                 if (is_order_function(it.rest)) {
 591                         if (!no_order)
 592                                 e += Order(power(var - point, it.coeff));
 593                 } else
 594                         e += it.rest * power(var - point, it.coeff);
 595         }
 596         return e;
 597 }
 598
 599 bool pseries::is_terminating() const
 600 {
 601         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 602 }
 603
 604 ex pseries::coeffop(size_t i) const
 605 {
 606         if (i >= nops())
 607                 throw (std::out_of_range("coeffop() out of range"));
 608         return seq[i].rest;
 609 }
 610
 611 ex pseries::exponop(size_t i) const
 612 {
 613         if (i >= nops())
 614                 throw (std::out_of_range("exponop() out of range"));
 615         return seq[i].coeff;
 616 }
 617
 618
 619 /*
 620  *  Implementations of series expansion
 621  */
 622
 623 /** Default implementation of ex::series(). This performs Taylor expansion.
 624  *  @see ex::series */
 625 ex basic::series(const relational & r, int order, unsigned options) const
 626 {
 627         epvector seq;
 628         const symbol &s = ex_to<symbol>(r.lhs());
 629
 630         // default for order-values that make no sense for Taylor expansion
 631         if ((order <= 0) && this->has(s)) {
 632                 seq.push_back(expair(Order(_ex1), order));
 633                 return pseries(r, std::move(seq));
 634         }
 635
 636         // do Taylor expansion
 637         numeric fac = 1;
 638         ex deriv = *this;
 639         ex coeff = deriv.subs(r, subs_options::no_pattern);
 640
 641         if (!coeff.is_zero()) {
 642                 seq.push_back(expair(coeff, _ex0));
 643         }
 644
 645         int n;
 646         for (n=1; n<order; ++n) {
 647                 fac = fac.mul(n);
 648                 // We need to test for zero in order to see if the series terminates.
 649                 // The problem is that there is no such thing as a perfect test for
 650                 // zero.  Expanding the term occasionally helps a little...
 651                 deriv = deriv.diff(s).expand();
 652                 if (deriv.is_zero())  // Series terminates
 653                         return pseries(r, std::move(seq));
 654
 655                 coeff = deriv.subs(r, subs_options::no_pattern);
 656                 if (!coeff.is_zero())
 657                         seq.push_back(expair(fac.inverse() * coeff, n));
 658         }
 659
 660         // Higher-order terms, if present
 661         deriv = deriv.diff(s);
 662         if (!deriv.expand().is_zero())
 663                 seq.push_back(expair(Order(_ex1), n));
 664         return pseries(r, std::move(seq));
 665 }
 666
 667
 668 /** Implementation of ex::series() for symbols.
 669  *  @see ex::series */
 670 ex symbol::series(const relational & r, int order, unsigned options) const
 671 {
 672         epvector seq;
 673         const ex point = r.rhs();
 674         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 675
 676         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 677                 if (order > 0 && !point.is_zero())
 678                         seq.push_back(expair(point, _ex0));
 679                 if (order > 1)
 680                         seq.push_back(expair(_ex1, _ex1));
 681                 else
 682                         seq.push_back(expair(Order(_ex1), numeric(order)));
 683         } else
 684                 seq.push_back(expair(*this, _ex0));
 685         return pseries(r, std::move(seq));
 686 }
 687
 688
 689 /** Add one series object to another, producing a pseries object that
 690  *  represents the sum.
 691  *
 692  *  @param other  pseries object to add with
 693  *  @return the sum as a pseries */
 694 ex pseries::add_series(const pseries &other) const
 695 {
 696         // Adding two series with different variables or expansion points
 697         // results in an empty (constant) series
 698         if (!is_compatible_to(other)) {
 699                 epvector nul { expair(Order(_ex1), _ex0) };
 700                 return pseries(relational(var,point), std::move(nul));
 701         }
 702
 703         // Series addition
 704         epvector new_seq;
 705         auto a = seq.begin(), a_end = seq.end();
 706         auto b = other.seq.begin(), b_end = other.seq.end();
 707         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 708         for (;;) {
 709                 // If a is empty, fill up with elements from b and stop
 710                 if (a == a_end) {
 711                         while (b != b_end) {
 712                                 new_seq.push_back(*b);
 713                                 ++b;
 714                         }
 715                         break;
 716                 } else
 717                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 718
 719                 // If b is empty, fill up with elements from a and stop
 720                 if (b == b_end) {
 721                         while (a != a_end) {
 722                                 new_seq.push_back(*a);
 723                                 ++a;
 724                         }
 725                         break;
 726                 } else
 727                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 728
 729                 // a and b are non-empty, compare powers
 730                 if (pow_a < pow_b) {
 731                         // a has lesser power, get coefficient from a
 732                         new_seq.push_back(*a);
 733                         if (is_order_function((*a).rest))
 734                                 break;
 735                         ++a;
 736                 } else if (pow_b < pow_a) {
 737                         // b has lesser power, get coefficient from b
 738                         new_seq.push_back(*b);
 739                         if (is_order_function((*b).rest))
 740                                 break;
 741                         ++b;
 742                 } else {
 743                         // Add coefficient of a and b
 744                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 745                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 746                                 break;  // Order term ends the sequence
 747                         } else {
 748                                 ex sum = (*a).rest + (*b).rest;
 749                                 if (!(sum.is_zero()))
 750                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 751                                 ++a;
 752                                 ++b;
 753                         }
 754                 }
 755         }
 756         return pseries(relational(var,point), std::move(new_seq));
 757 }
 758
 759
 760 /** Implementation of ex::series() for sums. This performs series addition when
 761  *  adding pseries objects.
 762  *  @see ex::series */
 763 ex add::series(const relational & r, int order, unsigned options) const
 764 {
 765         ex acc; // Series accumulator
 766
 767         // Get first term from overall_coeff
 768         acc = overall_coeff.series(r, order, options);
 769
 770         // Add remaining terms
 771         for (auto & it : seq) {
 772                 ex op;
 773                 if (is_exactly_a<pseries>(it.rest))
 774                         op = it.rest;
 775                 else
 776                         op = it.rest.series(r, order, options);
 777                 if (!it.coeff.is_equal(_ex1))
 778                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
 779
 780                 // Series addition
 781                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 782         }
 783         return acc;
 784 }
 785
 786
 787 /** Multiply a pseries object with a numeric constant, producing a pseries
 788  *  object that represents the product.
 789  *
 790  *  @param other  constant to multiply with
 791  *  @return the product as a pseries */
 792 ex pseries::mul_const(const numeric &other) const
 793 {
 794         epvector new_seq;
 795         new_seq.reserve(seq.size());
 796
 797         for (auto & it : seq) {
 798                 if (!is_order_function(it.rest))
 799                         new_seq.push_back(expair(it.rest * other, it.coeff));
 800                 else
 801                         new_seq.push_back(it);
 802         }
 803         return pseries(relational(var,point), std::move(new_seq));
 804 }
 805
 806
 807 /** Multiply one pseries object to another, producing a pseries object that
 808  *  represents the product.
 809  *
 810  *  @param other  pseries object to multiply with
 811  *  @return the product as a pseries */
 812 ex pseries::mul_series(const pseries &other) const
 813 {
 814         // Multiplying two series with different variables or expansion points
 815         // results in an empty (constant) series
 816         if (!is_compatible_to(other)) {
 817                 epvector nul { expair(Order(_ex1), _ex0) };
 818                 return pseries(relational(var,point), std::move(nul));
 819         }
 820
 821         if (seq.empty() || other.seq.empty()) {
 822                 return dynallocate<pseries>(var==point, epvector());
 823         }
 824
 825         // Series multiplication
 826         epvector new_seq;
 827         int a_max = degree(var);
 828         int b_max = other.degree(var);
 829         int a_min = ldegree(var);
 830         int b_min = other.ldegree(var);
 831         int cdeg_min = a_min + b_min;
 832         int cdeg_max = a_max + b_max;
 833
 834         int higher_order_a = std::numeric_limits<int>::max();
 835         int higher_order_b = std::numeric_limits<int>::max();
 836         if (is_order_function(coeff(var, a_max)))
 837                 higher_order_a = a_max + b_min;
 838         if (is_order_function(other.coeff(var, b_max)))
 839                 higher_order_b = b_max + a_min;
 840         int higher_order_c = std::min(higher_order_a, higher_order_b);
 841         if (cdeg_max >= higher_order_c)
 842                 cdeg_max = higher_order_c - 1;
 843
 844         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 845                 ex co = _ex0;
 846                 // c(i)=a(0)b(i)+...+a(i)b(0)
 847                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 848                         ex a_coeff = coeff(var, i);
 849                         ex b_coeff = other.coeff(var, cdeg-i);
 850                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 851                                 co += a_coeff * b_coeff;
 852                 }
 853                 if (!co.is_zero())
 854                         new_seq.push_back(expair(co, numeric(cdeg)));
 855         }
 856         if (higher_order_c < std::numeric_limits<int>::max())
 857                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 858         return pseries(relational(var, point), std::move(new_seq));
 859 }
 860
 861
 862 /** Implementation of ex::series() for product. This performs series
 863  *  multiplication when multiplying series.
 864  *  @see ex::series */
 865 ex mul::series(const relational & r, int order, unsigned options) const
 866 {
 867         pseries acc; // Series accumulator
 868
 869         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 870         const ex& sym = r.lhs();
 871
 872         // holds ldegrees of the series of individual factors
 873         std::vector<int> ldegrees;
 874         std::vector<bool> ldegree_redo;
 875
 876         // find minimal degrees
 877         // first round: obtain a bound up to which minimal degrees have to be
 878         // considered
 879         for (auto & it : seq) {
 880
 881                 ex expon = it.coeff;
 882                 int factor = 1;
 883                 ex buf;
 884                 if (expon.info(info_flags::integer)) {
 885                         buf = it.rest;
 886                         factor = ex_to<numeric>(expon).to_int();
 887                 } else {
 888                         buf = recombine_pair_to_ex(it);
 889                 }
 890
 891                 int real_ldegree = 0;
 892                 bool flag_redo = false;
 893                 try {
 894                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 895                 } catch (std::runtime_error) {}
 896
 897                 if (real_ldegree == 0) {
 898                         if ( factor < 0 ) {
 899                                 // This case must terminate, otherwise we would have division by
 900                                 // zero.
 901                                 int orderloop = 0;
 902                                 do {
 903                                         orderloop++;
 904                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 905                                 } while (real_ldegree == orderloop);
 906                         } else {
 907                                 // Here it is possible that buf does not have a ldegree, therefore
 908                                 // check only if ldegree is negative, otherwise reconsider the case
 909                                 // in the second round.
 910                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 911                                 if (real_ldegree == 0)
 912                                         flag_redo = true;
 913                         }
 914                 }
 915
 916                 ldegrees.push_back(factor * real_ldegree);
 917                 ldegree_redo.push_back(flag_redo);
 918         }
 919
 920         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 921         // Second round: determine the remaining positive ldegrees by the series
 922         // method.
 923         // here we can ignore ldegrees larger than degbound
 924         size_t j = 0;
 925         for (auto & it : seq) {
 926                 if ( ldegree_redo[j] ) {
 927                         ex expon = it.coeff;
 928                         int factor = 1;
 929                         ex buf;
 930                         if (expon.info(info_flags::integer)) {
 931                                 buf = it.rest;
 932                                 factor = ex_to<numeric>(expon).to_int();
 933                         } else {
 934                                 buf = recombine_pair_to_ex(it);
 935                         }
 936                         int real_ldegree = 0;
 937                         int orderloop = 0;
 938                         do {
 939                                 orderloop++;
 940                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 941                         } while ((real_ldegree == orderloop)
 942                               && (factor*real_ldegree < degbound));
 943                         ldegrees[j] = factor * real_ldegree;
 944                         degbound -= factor * real_ldegree;
 945                 }
 946                 j++;
 947         }
 948
 949         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 950
 951         if (degsum >= order) {
 952                 epvector epv { expair(Order(_ex1), order) };
 953                 return dynallocate<pseries>(r, std::move(epv));
 954         }
 955
 956         // Multiply with remaining terms
 957         auto itd = ldegrees.begin();
 958         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 959
 960                 // do series expansion with adjusted order
 961                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 962
 963                 // Series multiplication
 964                 if (it == seq.begin())
 965                         acc = ex_to<pseries>(op);
 966                 else
 967                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 968         }
 969
 970         return acc.mul_const(ex_to<numeric>(overall_coeff));
 971 }
 972
 973
 974 /** Compute the p-th power of a series.
 975  *
 976  *  @param p  power to compute
 977  *  @param deg  truncation order of series calculation */
 978 ex pseries::power_const(const numeric &p, int deg) const
 979 {
 980         // method:
 981         // (due to Leonhard Euler)
 982         // let A(x) be this series and for the time being let it start with a
 983         // constant (later we'll generalize):
 984         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 985         // We want to compute
 986         //     C(x) = A(x)^p
 987         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 988         // Taking the derivative on both sides and multiplying with A(x) one
 989         // immediately arrives at
 990         //     C'(x)*A(x) = p*C(x)*A'(x)
 991         // Multiplying this out and comparing coefficients we get the recurrence
 992         // formula
 993         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 994         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 995         // which can easily be solved given the starting value c_0 = (a_0)^p.
 996         // For the more general case where the leading coefficient of A(x) is not
 997         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 998         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 999         // then of course x^(p*m) but the recurrence formula still holds.
1000
1001         if (seq.empty()) {
1002                 // as a special case, handle the empty (zero) series honoring the
1003                 // usual power laws such as implemented in power::eval()
1004                 if (p.real().is_zero())
1005                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1006                 else if (p.real().is_negative())
1007                         throw pole_error("pseries::power_const(): division by zero",1);
1008                 else
1009                         return *this;
1010         }
1011
1012         const int ldeg = ldegree(var);
1013         if (!(p*ldeg).is_integer())
1014                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1015
1016         // adjust number of coefficients
1017         int numcoeff = deg - (p*ldeg).to_int();
1018         if (numcoeff <= 0) {
1019                 epvector epv { expair(Order(_ex1), deg) };
1020                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1021         }
1022
1023         // O(x^n)^(-m) is undefined
1024         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1025                 throw pole_error("pseries::power_const(): division by zero",1);
1026
1027         // Compute coefficients of the powered series
1028         exvector co;
1029         co.reserve(numcoeff);
1030         co.push_back(power(coeff(var, ldeg), p));
1031         for (int i=1; i<numcoeff; ++i) {
1032                 ex sum = _ex0;
1033                 for (int j=1; j<=i; ++j) {
1034                         ex c = coeff(var, j + ldeg);
1035                         if (is_order_function(c)) {
1036                                 co.push_back(Order(_ex1));
1037                                 break;
1038                         } else
1039                                 sum += (p * j - (i - j)) * co[i - j] * c;
1040                 }
1041                 co.push_back(sum / coeff(var, ldeg) / i);
1042         }
1043
1044         // Construct new series (of non-zero coefficients)
1045         epvector new_seq;
1046         bool higher_order = false;
1047         for (int i=0; i<numcoeff; ++i) {
1048                 if (!co[i].is_zero())
1049                         new_seq.push_back(expair(co[i], p * ldeg + i));
1050                 if (is_order_function(co[i])) {
1051                         higher_order = true;
1052                         break;
1053                 }
1054         }
1055         if (!higher_order)
1056                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1057
1058         return pseries(relational(var,point), std::move(new_seq));
1059 }
1060
1061
1062 /** Return a new pseries object with the powers shifted by deg. */
1063 pseries pseries::shift_exponents(int deg) const
1064 {
1065         epvector newseq = seq;
1066         for (auto & it : newseq)
1067                 it.coeff += deg;
1068         return pseries(relational(var, point), std::move(newseq));
1069 }
1070
1071
1072 /** Implementation of ex::series() for powers. This performs Laurent expansion
1073  *  of reciprocals of series at singularities.
1074  *  @see ex::series */
1075 ex power::series(const relational & r, int order, unsigned options) const
1076 {
1077         // If basis is already a series, just power it
1078         if (is_exactly_a<pseries>(basis))
1079                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1080
1081         // Basis is not a series, may there be a singularity?
1082         bool must_expand_basis = false;
1083         try {
1084                 basis.subs(r, subs_options::no_pattern);
1085         } catch (pole_error) {
1086                 must_expand_basis = true;
1087         }
1088
1089         bool exponent_is_regular = true;
1090         try {
1091                 exponent.subs(r, subs_options::no_pattern);
1092         } catch (pole_error) {
1093                 exponent_is_regular = false;
1094         }
1095
1096         if (!exponent_is_regular) {
1097                 ex l = exponent*log(basis);
1098                 // this == exp(l);
1099                 ex le = l.series(r, order, options);
1100                 // Note: expanding exp(l) won't help, since that will attempt
1101                 // Taylor expansion, and fail (because exponent is "singular")
1102                 // Still l itself might be expanded in Taylor series.
1103                 // Examples:
1104                 // sin(x)/x*log(cos(x))
1105                 // 1/x*log(1 + x)
1106                 return exp(le).series(r, order, options);
1107                 // Note: if l happens to have a Laurent expansion (with
1108                 // negative powers of (var - point)), expanding exp(le)
1109                 // will barf (which is The Right Thing).
1110         }
1111
1112         // Is the expression of type something^(-int)?
1113         if (!must_expand_basis && !exponent.info(info_flags::negint)
1114          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1115                 return basic::series(r, order, options);
1116
1117         // Is the expression of type 0^something?
1118         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1119          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1120                 return basic::series(r, order, options);
1121
1122         // Singularity encountered, is the basis equal to (var - point)?
1123         if (basis.is_equal(r.lhs() - r.rhs())) {
1124                 epvector new_seq;
1125                 if (ex_to<numeric>(exponent).to_int() < order)
1126                         new_seq.push_back(expair(_ex1, exponent));
1127                 else
1128                         new_seq.push_back(expair(Order(_ex1), exponent));
1129                 return pseries(r, std::move(new_seq));
1130         }
1131
1132         // No, expand basis into series
1133
1134         numeric numexp;
1135         if (is_a<numeric>(exponent)) {
1136                 numexp = ex_to<numeric>(exponent);
1137         } else {
1138                 numexp = 0;
1139         }
1140         const ex& sym = r.lhs();
1141         // find existing minimal degree
1142         ex eb = basis.expand();
1143         int real_ldegree = 0;
1144         if (eb.info(info_flags::rational_function))
1145                 real_ldegree = eb.ldegree(sym-r.rhs());
1146         if (real_ldegree == 0) {
1147                 int orderloop = 0;
1148                 do {
1149                         orderloop++;
1150                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1151                 } while (real_ldegree == orderloop);
1152         }
1153
1154         if (!(real_ldegree*numexp).is_integer())
1155                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1156         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1157
1158         ex result;
1159         try {
1160                 result = ex_to<pseries>(e).power_const(numexp, order);
1161         } catch (pole_error) {
1162                 epvector ser { expair(Order(_ex1), order) };
1163                 result = pseries(r, std::move(ser));
1164         }
1165
1166         return result;
1167 }
1168
1169
1170 /** Re-expansion of a pseries object. */
1171 ex pseries::series(const relational & r, int order, unsigned options) const
1172 {
1173         const ex p = r.rhs();
1174         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1175         const symbol &s = ex_to<symbol>(r.lhs());
1176
1177         if (var.is_equal(s) && point.is_equal(p)) {
1178                 if (order > degree(s))
1179                         return *this;
1180                 else {
1181                         epvector new_seq;
1182                         for (auto & it : seq) {
1183                                 int o = ex_to<numeric>(it.coeff).to_int();
1184                                 if (o >= order) {
1185                                         new_seq.push_back(expair(Order(_ex1), o));
1186                                         break;
1187                                 }
1188                                 new_seq.push_back(it);
1189                         }
1190                         return pseries(r, std::move(new_seq));
1191                 }
1192         } else
1193                 return convert_to_poly().series(r, order, options);
1194 }
1195
1196 ex integral::series(const relational & r, int order, unsigned options) const
1197 {
1198         if (x.subs(r) != x)
1199                 throw std::logic_error("Cannot series expand wrt dummy variable");
1200
1201         // Expanding integrand with r substituted taken in boundaries.
1202         ex fseries = f.series(r, order, options);
1203         epvector fexpansion;
1204         fexpansion.reserve(fseries.nops());
1205         for (size_t i=0; i<fseries.nops(); ++i) {
1206                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1207                 currcoeff = (currcoeff == Order(_ex1))
1208                         ? currcoeff
1209                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1210                 if (currcoeff != 0)
1211                         fexpansion.push_back(
1212                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1213         }
1214
1215         // Expanding lower boundary
1216         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1217         ex aseries = (a-a.subs(r)).series(r, order, options);
1218         fseries = f.series(x == (a.subs(r)), order, options);
1219         for (size_t i=0; i<fseries.nops(); ++i) {
1220                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1221                 if (is_order_function(currcoeff))
1222                         break;
1223                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1224                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1225                 currcoeff = currcoeff.series(r, orderforf);
1226                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1227                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1228                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1229                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1230         }
1231
1232         // Expanding upper boundary
1233         ex bseries = (b-b.subs(r)).series(r, order, options);
1234         fseries = f.series(x == (b.subs(r)), order, options);
1235         for (size_t i=0; i<fseries.nops(); ++i) {
1236                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1237                 if (is_order_function(currcoeff))
1238                         break;
1239                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1240                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1241                 currcoeff = currcoeff.series(r, orderforf);
1242                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1243                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1244                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1245                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1246         }
1247
1248         return result;
1249 }
1250
1251
1252 /** Compute the truncated series expansion of an expression.
1253  *  This function returns an expression containing an object of class pseries
1254  *  to represent the series. If the series does not terminate within the given
1255  *  truncation order, the last term of the series will be an order term.
1256  *
1257  *  @param r  expansion relation, lhs holds variable and rhs holds point
1258  *  @param order  truncation order of series calculations
1259  *  @param options  of class series_options
1260  *  @return an expression holding a pseries object */
1261 ex ex::series(const ex & r, int order, unsigned options) const
1262 {
1263         ex e;
1264         relational rel_;
1265
1266         if (is_a<relational>(r))
1267                 rel_ = ex_to<relational>(r);
1268         else if (is_a<symbol>(r))
1269                 rel_ = relational(r,_ex0);
1270         else
1271                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1272
1273         e = bp->series(rel_, order, options);
1274         return e;
1275 }
1276
1277 GINAC_BIND_UNARCHIVER(pseries);
1278
1279 } // namespace GiNaC