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) 1999-2016 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  22  */
  23
  24 #include "pseries.h"
  25 #include "add.h"
  26 #include "inifcns.h" // for Order function
  27 #include "lst.h"
  28 #include "mul.h"
  29 #include "power.h"
  30 #include "relational.h"
  31 #include "operators.h"
  32 #include "symbol.h"
  33 #include "integral.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36
  37 #include <limits>
  38 #include <numeric>
  39 #include <stdexcept>
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
  44   print_func<print_context>(&pseries::do_print).
  45   print_func<print_latex>(&pseries::do_print_latex).
  46   print_func<print_tree>(&pseries::do_print_tree).
  47   print_func<print_python>(&pseries::do_print_python).
  48   print_func<print_python_repr>(&pseries::do_print_python_repr))
  49
  50
  51 /*
  52  *  Default constructor
  53  */
  54
  55 pseries::pseries() { }
  56
  57
  58 /*
  59  *  Other ctors
  60  */
  61
  62 /** Construct pseries from a vector of coefficients and powers.
  63  *  expair.rest holds the coefficient, expair.coeff holds the power.
  64  *  The powers must be integers (positive or negative) and in ascending order;
  65  *  the last coefficient can be Order(_ex1) to represent a truncated,
  66  *  non-terminating series.
  67  *
  68  *  @param rel_  expansion variable and point (must hold a relational)
  69  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  70  *  @return newly constructed pseries */
  71 pseries::pseries(const ex &rel_, const epvector &ops_)
  72   : seq(ops_)
  73 {
  74 #ifdef DO_GINAC_ASSERT
  75         auto i = seq.begin();
  76         while (i != seq.end()) {
  77                 auto ip1 = i+1;
  78                 if (ip1 != seq.end())
  79                         GINAC_ASSERT(!is_order_function(i->rest));
  80                 else
  81                         break;
  82                 GINAC_ASSERT(is_a<numeric>(i->coeff));
  83                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
  84                 ++i;
  85         }
  86 #endif // def DO_GINAC_ASSERT
  87         GINAC_ASSERT(is_a<relational>(rel_));
  88         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
  89         point = rel_.rhs();
  90         var = rel_.lhs();
  91 }
  92 pseries::pseries(const ex &rel_, epvector &&ops_)
  93   : seq(std::move(ops_))
  94 {
  95 #ifdef DO_GINAC_ASSERT
  96         auto i = seq.begin();
  97         while (i != seq.end()) {
  98                 auto ip1 = i+1;
  99                 if (ip1 != seq.end())
 100                         GINAC_ASSERT(!is_order_function(i->rest));
 101                 else
 102                         break;
 103                 GINAC_ASSERT(is_a<numeric>(i->coeff));
 104                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
 105                 ++i;
 106         }
 107 #endif // def DO_GINAC_ASSERT
 108         GINAC_ASSERT(is_a<relational>(rel_));
 109         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
 110         point = rel_.rhs();
 111         var = rel_.lhs();
 112 }
 113
 114
 115 /*
 116  *  Archiving
 117  */
 118
 119 void pseries::read_archive(const archive_node &n, lst &sym_lst)
 120 {
 121         inherited::read_archive(n, sym_lst);
 122         auto first = n.find_first("coeff");
 123         auto last = n.find_last("power");
 124         ++last;
 125         seq.reserve((last-first)/2);
 126
 127         for (auto loc = first; loc < last;) {
 128                 ex rest;
 129                 ex coeff;
 130                 n.find_ex_by_loc(loc++, rest, sym_lst);
 131                 n.find_ex_by_loc(loc++, coeff, sym_lst);
 132                 seq.push_back(expair(rest, coeff));
 133         }
 134
 135         n.find_ex("var", var, sym_lst);
 136         n.find_ex("point", point, sym_lst);
 137 }
 138
 139 void pseries::archive(archive_node &n) const
 140 {
 141         inherited::archive(n);
 142         for (auto & it : seq) {
 143                 n.add_ex("coeff", it.rest);
 144                 n.add_ex("power", it.coeff);
 145         }
 146         n.add_ex("var", var);
 147         n.add_ex("point", point);
 148 }
 149
 150
 151 //////////
 152 // functions overriding virtual functions from base classes
 153 //////////
 154
 155 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
 156 {
 157         if (precedence() <= level)
 158                 c.s << '(';
 159
 160         // objects of type pseries must not have any zero entries, so the
 161         // trivial (zero) pseries needs a special treatment here:
 162         if (seq.empty())
 163                 c.s << '0';
 164
 165         auto i = seq.begin(), end = seq.end();
 166         while (i != end) {
 167
 168                 // print a sign, if needed
 169                 if (i != seq.begin())
 170                         c.s << '+';
 171
 172                 if (!is_order_function(i->rest)) {
 173
 174                         // print 'rest', i.e. the expansion coefficient
 175                         if (i->rest.info(info_flags::numeric) &&
 176                                 i->rest.info(info_flags::positive)) {
 177                                 i->rest.print(c);
 178                         } else {
 179                                 c.s << openbrace << '(';
 180                                 i->rest.print(c);
 181                                 c.s << ')' << closebrace;
 182                         }
 183
 184                         // print 'coeff', something like (x-1)^42
 185                         if (!i->coeff.is_zero()) {
 186                                 c.s << mul_sym;
 187                                 if (!point.is_zero()) {
 188                                         c.s << openbrace << '(';
 189                                         (var-point).print(c);
 190                                         c.s << ')' << closebrace;
 191                                 } else
 192                                         var.print(c);
 193                                 if (i->coeff.compare(_ex1)) {
 194                                         c.s << pow_sym;
 195                                         c.s << openbrace;
 196                                         if (i->coeff.info(info_flags::negative)) {
 197                                                 c.s << '(';
 198                                                 i->coeff.print(c);
 199                                                 c.s << ')';
 200                                         } else
 201                                                 i->coeff.print(c);
 202                                         c.s << closebrace;
 203                                 }
 204                         }
 205                 } else
 206                         Order(pow(var - point, i->coeff)).print(c);
 207                 ++i;
 208         }
 209
 210         if (precedence() <= level)
 211                 c.s << ')';
 212 }
 213
 214 void pseries::do_print(const print_context & c, unsigned level) const
 215 {
 216         print_series(c, "", "", "*", "^", level);
 217 }
 218
 219 void pseries::do_print_latex(const print_latex & c, unsigned level) const
 220 {
 221         print_series(c, "{", "}", " ", "^", level);
 222 }
 223
 224 void pseries::do_print_python(const print_python & c, unsigned level) const
 225 {
 226         print_series(c, "", "", "*", "**", level);
 227 }
 228
 229 void pseries::do_print_tree(const print_tree & c, unsigned level) const
 230 {
 231         c.s << std::string(level, ' ') << class_name() << " @" << this
 232             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 233             << std::endl;
 234         size_t num = seq.size();
 235         for (size_t i=0; i<num; ++i) {
 236                 seq[i].rest.print(c, level + c.delta_indent);
 237                 seq[i].coeff.print(c, level + c.delta_indent);
 238                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
 239         }
 240         var.print(c, level + c.delta_indent);
 241         point.print(c, level + c.delta_indent);
 242 }
 243
 244 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
 245 {
 246         c.s << class_name() << "(relational(";
 247         var.print(c);
 248         c.s << ',';
 249         point.print(c);
 250         c.s << "),[";
 251         size_t num = seq.size();
 252         for (size_t i=0; i<num; ++i) {
 253                 if (i)
 254                         c.s << ',';
 255                 c.s << '(';
 256                 seq[i].rest.print(c);
 257                 c.s << ',';
 258                 seq[i].coeff.print(c);
 259                 c.s << ')';
 260         }
 261         c.s << "])";
 262 }
 263
 264 int pseries::compare_same_type(const basic & other) const
 265 {
 266         GINAC_ASSERT(is_a<pseries>(other));
 267         const pseries &o = static_cast<const pseries &>(other);
 268
 269         // first compare the lengths of the series...
 270         if (seq.size()>o.seq.size())
 271                 return 1;
 272         if (seq.size()<o.seq.size())
 273                 return -1;
 274
 275         // ...then the expansion point...
 276         int cmpval = var.compare(o.var);
 277         if (cmpval)
 278                 return cmpval;
 279         cmpval = point.compare(o.point);
 280         if (cmpval)
 281                 return cmpval;
 282
 283         // ...and if that failed the individual elements
 284         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 285         while (it!=seq.end() && o_it!=o.seq.end()) {
 286                 cmpval = it->compare(*o_it);
 287                 if (cmpval)
 288                         return cmpval;
 289                 ++it;
 290                 ++o_it;
 291         }
 292
 293         // so they are equal.
 294         return 0;
 295 }
 296
 297 /** Return the number of operands including a possible order term. */
 298 size_t pseries::nops() const
 299 {
 300         return seq.size();
 301 }
 302
 303 /** Return the ith term in the series when represented as a sum. */
 304 ex pseries::op(size_t i) const
 305 {
 306         if (i >= seq.size())
 307                 throw (std::out_of_range("op() out of range"));
 308
 309         if (is_order_function(seq[i].rest))
 310                 return Order(pow(var-point, seq[i].coeff));
 311         return seq[i].rest * pow(var - point, seq[i].coeff);
 312 }
 313
 314 /** Return degree of highest power of the series.  This is usually the exponent
 315  *  of the Order term.  If s is not the expansion variable of the series, the
 316  *  series is examined termwise. */
 317 int pseries::degree(const ex &s) const
 318 {
 319         if (seq.empty())
 320                 return 0;
 321
 322         if (var.is_equal(s))
 323                 // Return last/greatest exponent
 324                 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 325
 326         int max_pow = std::numeric_limits<int>::min();
 327         for (auto & it : seq)
 328                 max_pow = std::max(max_pow, it.rest.degree(s));
 329         return max_pow;
 330 }
 331
 332 /** Return degree of lowest power of the series.  This is usually the exponent
 333  *  of the leading term.  If s is not the expansion variable of the series, the
 334  *  series is examined termwise.  If s is the expansion variable but the
 335  *  expansion point is not zero the series is not expanded to find the degree.
 336  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 337 int pseries::ldegree(const ex &s) const
 338 {
 339         if (seq.empty())
 340                 return 0;
 341
 342         if (var.is_equal(s))
 343                 // Return first/smallest exponent
 344                 return ex_to<numeric>((seq.begin())->coeff).to_int();
 345
 346         int min_pow = std::numeric_limits<int>::max();
 347         for (auto & it : seq)
 348                 min_pow = std::min(min_pow, it.rest.degree(s));
 349         return min_pow;
 350 }
 351
 352 /** Return coefficient of degree n in power series if s is the expansion
 353  *  variable.  If the expansion point is nonzero, by definition the n=1
 354  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 355  *  the expansion took place in the s in the first place).
 356  *  If s is not the expansion variable, an attempt is made to convert the
 357  *  series to a polynomial and return the corresponding coefficient from
 358  *  there. */
 359 ex pseries::coeff(const ex &s, int n) const
 360 {
 361         if (var.is_equal(s)) {
 362                 if (seq.empty())
 363                         return _ex0;
 364
 365                 // Binary search in sequence for given power
 366                 numeric looking_for = numeric(n);
 367                 int lo = 0, hi = seq.size() - 1;
 368                 while (lo <= hi) {
 369                         int mid = (lo + hi) / 2;
 370                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 371                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 372                         switch (cmp) {
 373                                 case -1:
 374                                         lo = mid + 1;
 375                                         break;
 376                                 case 0:
 377                                         return seq[mid].rest;
 378                                 case 1:
 379                                         hi = mid - 1;
 380                                         break;
 381                                 default:
 382                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 383                         }
 384                 }
 385                 return _ex0;
 386         } else
 387                 return convert_to_poly().coeff(s, n);
 388 }
 389
 390 /** Does nothing. */
 391 ex pseries::collect(const ex &s, bool distributed) const
 392 {
 393         return *this;
 394 }
 395
 396 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 397 ex pseries::eval() const
 398 {
 399         if (flags & status_flags::evaluated) {
 400                 return *this;
 401         }
 402
 403         // Construct a new series with evaluated coefficients
 404         epvector new_seq;
 405         new_seq.reserve(seq.size());
 406         for (auto & it : seq)
 407                 new_seq.push_back(expair(it.rest, it.coeff));
 408
 409         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 410 }
 411
 412 /** Evaluate coefficients numerically. */
 413 ex pseries::evalf(int level) const
 414 {
 415         if (level == 1)
 416                 return *this;
 417
 418         if (level == -max_recursion_level)
 419                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 420
 421         // Construct a new series with evaluated coefficients
 422         epvector new_seq;
 423         new_seq.reserve(seq.size());
 424         for (auto & it : seq)
 425                 new_seq.push_back(expair(it.rest, it.coeff));
 426
 427         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
 428 }
 429
 430 ex pseries::conjugate() const
 431 {
 432         if(!var.info(info_flags::real))
 433                 return conjugate_function(*this).hold();
 434
 435         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
 436         ex newpoint = point.conjugate();
 437
 438         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
 439                 return *this;
 440         }
 441
 442         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
 443 }
 444
 445 ex pseries::real_part() const
 446 {
 447         if(!var.info(info_flags::real))
 448                 return real_part_function(*this).hold();
 449         ex newpoint = point.real_part();
 450         if(newpoint != point)
 451                 return real_part_function(*this).hold();
 452
 453         epvector v;
 454         v.reserve(seq.size());
 455         for (auto & it : seq)
 456                 v.push_back(expair((it.rest).real_part(), it.coeff));
 457         return dynallocate<pseries>(var==point, std::move(v));
 458 }
 459
 460 ex pseries::imag_part() const
 461 {
 462         if(!var.info(info_flags::real))
 463                 return imag_part_function(*this).hold();
 464         ex newpoint = point.real_part();
 465         if(newpoint != point)
 466                 return imag_part_function(*this).hold();
 467
 468         epvector v;
 469         v.reserve(seq.size());
 470         for (auto & it : seq)
 471                 v.push_back(expair((it.rest).imag_part(), it.coeff));
 472         return dynallocate<pseries>(var==point, std::move(v));
 473 }
 474
 475 ex pseries::eval_integ() const
 476 {
 477         std::unique_ptr<epvector> newseq(nullptr);
 478         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 479                 if (newseq) {
 480                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
 481                         continue;
 482                 }
 483                 ex newterm = i->rest.eval_integ();
 484                 if (!are_ex_trivially_equal(newterm, i->rest)) {
 485                         newseq.reset(new epvector);
 486                         newseq->reserve(seq.size());
 487                         for (auto j=seq.begin(); j!=i; ++j)
 488                                 newseq->push_back(*j);
 489                         newseq->push_back(expair(newterm, i->coeff));
 490                 }
 491         }
 492
 493         ex newpoint = point.eval_integ();
 494         if (newseq || !are_ex_trivially_equal(newpoint, point))
 495                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
 496         return *this;
 497 }
 498
 499 ex pseries::evalm() const
 500 {
 501         // evalm each coefficient
 502         epvector newseq;
 503         bool something_changed = false;
 504         for (auto i=seq.begin(); i!=seq.end(); ++i) {
 505                 if (something_changed) {
 506                         ex newcoeff = i->rest.evalm();
 507                         if (!newcoeff.is_zero())
 508                                 newseq.push_back(expair(newcoeff, i->coeff));
 509                 } else {
 510                         ex newcoeff = i->rest.evalm();
 511                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
 512                                 something_changed = true;
 513                                 newseq.reserve(seq.size());
 514                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
 515                                 if (!newcoeff.is_zero())
 516                                         newseq.push_back(expair(newcoeff, i->coeff));
 517                         }
 518                 }
 519         }
 520         if (something_changed)
 521                 return dynallocate<pseries>(var==point, std::move(newseq));
 522         else
 523                 return *this;
 524 }
 525
 526 ex pseries::subs(const exmap & m, unsigned options) const
 527 {
 528         // If expansion variable is being substituted, convert the series to a
 529         // polynomial and do the substitution there because the result might
 530         // no longer be a power series
 531         if (m.find(var) != m.end())
 532                 return convert_to_poly(true).subs(m, options);
 533
 534         // Otherwise construct a new series with substituted coefficients and
 535         // expansion point
 536         epvector newseq;
 537         newseq.reserve(seq.size());
 538         for (auto & it : seq)
 539                 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
 540         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
 541 }
 542
 543 /** Implementation of ex::expand() for a power series.  It expands all the
 544  *  terms individually and returns the resulting series as a new pseries. */
 545 ex pseries::expand(unsigned options) const
 546 {
 547         epvector newseq;
 548         for (auto & it : seq) {
 549                 ex restexp = it.rest.expand();
 550                 if (!restexp.is_zero())
 551                         newseq.push_back(expair(restexp, it.coeff));
 552         }
 553         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
 554 }
 555
 556 /** Implementation of ex::diff() for a power series.
 557  *  @see ex::diff */
 558 ex pseries::derivative(const symbol & s) const
 559 {
 560         epvector new_seq;
 561
 562         if (s == var) {
 563
 564                 // FIXME: coeff might depend on var
 565                 for (auto & it : seq) {
 566                         if (is_order_function(it.rest)) {
 567                                 new_seq.push_back(expair(it.rest, it.coeff - 1));
 568                         } else {
 569                                 ex c = it.rest * it.coeff;
 570                                 if (!c.is_zero())
 571                                         new_seq.push_back(expair(c, it.coeff - 1));
 572                         }
 573                 }
 574
 575         } else {
 576
 577                 for (auto & it : seq) {
 578                         if (is_order_function(it.rest)) {
 579                                 new_seq.push_back(it);
 580                         } else {
 581                                 ex c = it.rest.diff(s);
 582                                 if (!c.is_zero())
 583                                         new_seq.push_back(expair(c, it.coeff));
 584                         }
 585                 }
 586         }
 587
 588         return pseries(relational(var,point), std::move(new_seq));
 589 }
 590
 591 ex pseries::convert_to_poly(bool no_order) const
 592 {
 593         ex e;
 594         for (auto & it : seq) {
 595                 if (is_order_function(it.rest)) {
 596                         if (!no_order)
 597                                 e += Order(pow(var - point, it.coeff));
 598                 } else
 599                         e += it.rest * pow(var - point, it.coeff);
 600         }
 601         return e;
 602 }
 603
 604 bool pseries::is_terminating() const
 605 {
 606         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 607 }
 608
 609 ex pseries::coeffop(size_t i) const
 610 {
 611         if (i >= nops())
 612                 throw (std::out_of_range("coeffop() out of range"));
 613         return seq[i].rest;
 614 }
 615
 616 ex pseries::exponop(size_t i) const
 617 {
 618         if (i >= nops())
 619                 throw (std::out_of_range("exponop() out of range"));
 620         return seq[i].coeff;
 621 }
 622
 623
 624 /*
 625  *  Implementations of series expansion
 626  */
 627
 628 /** Default implementation of ex::series(). This performs Taylor expansion.
 629  *  @see ex::series */
 630 ex basic::series(const relational & r, int order, unsigned options) const
 631 {
 632         epvector seq;
 633         const symbol &s = ex_to<symbol>(r.lhs());
 634
 635         // default for order-values that make no sense for Taylor expansion
 636         if ((order <= 0) && this->has(s)) {
 637                 seq.push_back(expair(Order(_ex1), order));
 638                 return pseries(r, std::move(seq));
 639         }
 640
 641         // do Taylor expansion
 642         numeric fac = 1;
 643         ex deriv = *this;
 644         ex coeff = deriv.subs(r, subs_options::no_pattern);
 645
 646         if (!coeff.is_zero()) {
 647                 seq.push_back(expair(coeff, _ex0));
 648         }
 649
 650         int n;
 651         for (n=1; n<order; ++n) {
 652                 fac = fac.div(n);
 653                 // We need to test for zero in order to see if the series terminates.
 654                 // The problem is that there is no such thing as a perfect test for
 655                 // zero.  Expanding the term occasionally helps a little...
 656                 deriv = deriv.diff(s).expand();
 657                 if (deriv.is_zero())  // Series terminates
 658                         return pseries(r, std::move(seq));
 659
 660                 coeff = deriv.subs(r, subs_options::no_pattern);
 661                 if (!coeff.is_zero())
 662                         seq.push_back(expair(fac * coeff, n));
 663         }
 664
 665         // Higher-order terms, if present
 666         deriv = deriv.diff(s);
 667         if (!deriv.expand().is_zero())
 668                 seq.push_back(expair(Order(_ex1), n));
 669         return pseries(r, std::move(seq));
 670 }
 671
 672
 673 /** Implementation of ex::series() for symbols.
 674  *  @see ex::series */
 675 ex symbol::series(const relational & r, int order, unsigned options) const
 676 {
 677         epvector seq;
 678         const ex point = r.rhs();
 679         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 680
 681         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 682                 if (order > 0 && !point.is_zero())
 683                         seq.push_back(expair(point, _ex0));
 684                 if (order > 1)
 685                         seq.push_back(expair(_ex1, _ex1));
 686                 else
 687                         seq.push_back(expair(Order(_ex1), numeric(order)));
 688         } else
 689                 seq.push_back(expair(*this, _ex0));
 690         return pseries(r, std::move(seq));
 691 }
 692
 693
 694 /** Add one series object to another, producing a pseries object that
 695  *  represents the sum.
 696  *
 697  *  @param other  pseries object to add with
 698  *  @return the sum as a pseries */
 699 ex pseries::add_series(const pseries &other) const
 700 {
 701         // Adding two series with different variables or expansion points
 702         // results in an empty (constant) series
 703         if (!is_compatible_to(other)) {
 704                 epvector nul { expair(Order(_ex1), _ex0) };
 705                 return pseries(relational(var,point), std::move(nul));
 706         }
 707
 708         // Series addition
 709         epvector new_seq;
 710         auto a = seq.begin(), a_end = seq.end();
 711         auto b = other.seq.begin(), b_end = other.seq.end();
 712         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
 713         for (;;) {
 714                 // If a is empty, fill up with elements from b and stop
 715                 if (a == a_end) {
 716                         while (b != b_end) {
 717                                 new_seq.push_back(*b);
 718                                 ++b;
 719                         }
 720                         break;
 721                 } else
 722                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 723
 724                 // If b is empty, fill up with elements from a and stop
 725                 if (b == b_end) {
 726                         while (a != a_end) {
 727                                 new_seq.push_back(*a);
 728                                 ++a;
 729                         }
 730                         break;
 731                 } else
 732                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 733
 734                 // a and b are non-empty, compare powers
 735                 if (pow_a < pow_b) {
 736                         // a has lesser power, get coefficient from a
 737                         new_seq.push_back(*a);
 738                         if (is_order_function((*a).rest))
 739                                 break;
 740                         ++a;
 741                 } else if (pow_b < pow_a) {
 742                         // b has lesser power, get coefficient from b
 743                         new_seq.push_back(*b);
 744                         if (is_order_function((*b).rest))
 745                                 break;
 746                         ++b;
 747                 } else {
 748                         // Add coefficient of a and b
 749                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 750                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 751                                 break;  // Order term ends the sequence
 752                         } else {
 753                                 ex sum = (*a).rest + (*b).rest;
 754                                 if (!(sum.is_zero()))
 755                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 756                                 ++a;
 757                                 ++b;
 758                         }
 759                 }
 760         }
 761         return pseries(relational(var,point), std::move(new_seq));
 762 }
 763
 764
 765 /** Implementation of ex::series() for sums. This performs series addition when
 766  *  adding pseries objects.
 767  *  @see ex::series */
 768 ex add::series(const relational & r, int order, unsigned options) const
 769 {
 770         ex acc; // Series accumulator
 771
 772         // Get first term from overall_coeff
 773         acc = overall_coeff.series(r, order, options);
 774
 775         // Add remaining terms
 776         for (auto & it : seq) {
 777                 ex op;
 778                 if (is_exactly_a<pseries>(it.rest))
 779                         op = it.rest;
 780                 else
 781                         op = it.rest.series(r, order, options);
 782                 if (!it.coeff.is_equal(_ex1))
 783                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
 784
 785                 // Series addition
 786                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 787         }
 788         return acc;
 789 }
 790
 791
 792 /** Multiply a pseries object with a numeric constant, producing a pseries
 793  *  object that represents the product.
 794  *
 795  *  @param other  constant to multiply with
 796  *  @return the product as a pseries */
 797 ex pseries::mul_const(const numeric &other) const
 798 {
 799         epvector new_seq;
 800         new_seq.reserve(seq.size());
 801
 802         for (auto & it : seq) {
 803                 if (!is_order_function(it.rest))
 804                         new_seq.push_back(expair(it.rest * other, it.coeff));
 805                 else
 806                         new_seq.push_back(it);
 807         }
 808         return pseries(relational(var,point), std::move(new_seq));
 809 }
 810
 811
 812 /** Multiply one pseries object to another, producing a pseries object that
 813  *  represents the product.
 814  *
 815  *  @param other  pseries object to multiply with
 816  *  @return the product as a pseries */
 817 ex pseries::mul_series(const pseries &other) const
 818 {
 819         // Multiplying two series with different variables or expansion points
 820         // results in an empty (constant) series
 821         if (!is_compatible_to(other)) {
 822                 epvector nul { expair(Order(_ex1), _ex0) };
 823                 return pseries(relational(var,point), std::move(nul));
 824         }
 825
 826         if (seq.empty() || other.seq.empty()) {
 827                 return dynallocate<pseries>(var==point, epvector());
 828         }
 829
 830         // Series multiplication
 831         epvector new_seq;
 832         const int a_max = degree(var);
 833         const int b_max = other.degree(var);
 834         const int a_min = ldegree(var);
 835         const int b_min = other.ldegree(var);
 836         const int cdeg_min = a_min + b_min;
 837         int cdeg_max = a_max + b_max;
 838
 839         int higher_order_a = std::numeric_limits<int>::max();
 840         int higher_order_b = std::numeric_limits<int>::max();
 841         if (is_order_function(coeff(var, a_max)))
 842                 higher_order_a = a_max + b_min;
 843         if (is_order_function(other.coeff(var, b_max)))
 844                 higher_order_b = b_max + a_min;
 845         const int higher_order_c = std::min(higher_order_a, higher_order_b);
 846         if (cdeg_max >= higher_order_c)
 847                 cdeg_max = higher_order_c - 1;
 848
 849         std::map<int, ex> rest_map_a, rest_map_b;
 850         for (const auto& it : seq)
 851                 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 852
 853         if (other.var.is_equal(var))
 854                 for (const auto& it : other.seq)
 855                         rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
 856
 857         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 858                 ex co = _ex0;
 859                 // c(i)=a(0)b(i)+...+a(i)b(0)
 860                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 861                         const auto& ita = rest_map_a.find(i);
 862                         if (ita == rest_map_a.end())
 863                                 continue;
 864                         const auto& itb = rest_map_b.find(cdeg-i);
 865                         if (itb == rest_map_b.end())
 866                                 continue;
 867                         if (!is_order_function(ita->second) && !is_order_function(itb->second))
 868                                 co += ita->second * itb->second;
 869                 }
 870                 if (!co.is_zero())
 871                         new_seq.push_back(expair(co, numeric(cdeg)));
 872         }
 873         if (higher_order_c < std::numeric_limits<int>::max())
 874                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 875         return pseries(relational(var, point), std::move(new_seq));
 876 }
 877
 878
 879 /** Implementation of ex::series() for product. This performs series
 880  *  multiplication when multiplying series.
 881  *  @see ex::series */
 882 ex mul::series(const relational & r, int order, unsigned options) const
 883 {
 884         pseries acc; // Series accumulator
 885
 886         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 887         const ex& sym = r.lhs();
 888
 889         // holds ldegrees of the series of individual factors
 890         std::vector<int> ldegrees;
 891         std::vector<bool> ldegree_redo;
 892
 893         // find minimal degrees
 894         // first round: obtain a bound up to which minimal degrees have to be
 895         // considered
 896         for (auto & it : seq) {
 897
 898                 ex expon = it.coeff;
 899                 int factor = 1;
 900                 ex buf;
 901                 if (expon.info(info_flags::integer)) {
 902                         buf = it.rest;
 903                         factor = ex_to<numeric>(expon).to_int();
 904                 } else {
 905                         buf = recombine_pair_to_ex(it);
 906                 }
 907
 908                 int real_ldegree = 0;
 909                 bool flag_redo = false;
 910                 try {
 911                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
 912                 } catch (std::runtime_error) {}
 913
 914                 if (real_ldegree == 0) {
 915                         if ( factor < 0 ) {
 916                                 // This case must terminate, otherwise we would have division by
 917                                 // zero.
 918                                 int orderloop = 0;
 919                                 do {
 920                                         orderloop++;
 921                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 922                                 } while (real_ldegree == orderloop);
 923                         } else {
 924                                 // Here it is possible that buf does not have a ldegree, therefore
 925                                 // check only if ldegree is negative, otherwise reconsider the case
 926                                 // in the second round.
 927                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
 928                                 if (real_ldegree == 0)
 929                                         flag_redo = true;
 930                         }
 931                 }
 932
 933                 ldegrees.push_back(factor * real_ldegree);
 934                 ldegree_redo.push_back(flag_redo);
 935         }
 936
 937         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 938         // Second round: determine the remaining positive ldegrees by the series
 939         // method.
 940         // here we can ignore ldegrees larger than degbound
 941         size_t j = 0;
 942         for (auto & it : seq) {
 943                 if ( ldegree_redo[j] ) {
 944                         ex expon = it.coeff;
 945                         int factor = 1;
 946                         ex buf;
 947                         if (expon.info(info_flags::integer)) {
 948                                 buf = it.rest;
 949                                 factor = ex_to<numeric>(expon).to_int();
 950                         } else {
 951                                 buf = recombine_pair_to_ex(it);
 952                         }
 953                         int real_ldegree = 0;
 954                         int orderloop = 0;
 955                         do {
 956                                 orderloop++;
 957                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
 958                         } while ((real_ldegree == orderloop)
 959                               && (factor*real_ldegree < degbound));
 960                         ldegrees[j] = factor * real_ldegree;
 961                         degbound -= factor * real_ldegree;
 962                 }
 963                 j++;
 964         }
 965
 966         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
 967
 968         if (degsum >= order) {
 969                 epvector epv { expair(Order(_ex1), order) };
 970                 return dynallocate<pseries>(r, std::move(epv));
 971         }
 972
 973         // Multiply with remaining terms
 974         auto itd = ldegrees.begin();
 975         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
 976
 977                 // do series expansion with adjusted order
 978                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
 979
 980                 // Series multiplication
 981                 if (it == seq.begin())
 982                         acc = ex_to<pseries>(op);
 983                 else
 984                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 985         }
 986
 987         return acc.mul_const(ex_to<numeric>(overall_coeff));
 988 }
 989
 990
 991 /** Compute the p-th power of a series.
 992  *
 993  *  @param p  power to compute
 994  *  @param deg  truncation order of series calculation */
 995 ex pseries::power_const(const numeric &p, int deg) const
 996 {
 997         // method:
 998         // (due to Leonhard Euler)
 999         // let A(x) be this series and for the time being let it start with a
1000         // constant (later we'll generalize):
1001         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
1002         // We want to compute
1003         //     C(x) = A(x)^p
1004         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
1005         // Taking the derivative on both sides and multiplying with A(x) one
1006         // immediately arrives at
1007         //     C'(x)*A(x) = p*C(x)*A'(x)
1008         // Multiplying this out and comparing coefficients we get the recurrence
1009         // formula
1010         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1011         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1012         // which can easily be solved given the starting value c_0 = (a_0)^p.
1013         // For the more general case where the leading coefficient of A(x) is not
1014         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1015         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
1016         // then of course x^(p*m) but the recurrence formula still holds.
1017
1018         if (seq.empty()) {
1019                 // as a special case, handle the empty (zero) series honoring the
1020                 // usual power laws such as implemented in power::eval()
1021                 if (p.real().is_zero())
1022                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1023                 else if (p.real().is_negative())
1024                         throw pole_error("pseries::power_const(): division by zero",1);
1025                 else
1026                         return *this;
1027         }
1028
1029         const int ldeg = ldegree(var);
1030         if (!(p*ldeg).is_integer())
1031                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1032
1033         // adjust number of coefficients
1034         int numcoeff = deg - (p*ldeg).to_int();
1035         if (numcoeff <= 0) {
1036                 epvector epv { expair(Order(_ex1), deg) };
1037                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1038         }
1039
1040         // O(x^n)^(-m) is undefined
1041         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1042                 throw pole_error("pseries::power_const(): division by zero",1);
1043
1044         // Compute coefficients of the powered series
1045         exvector co;
1046         co.reserve(numcoeff);
1047         co.push_back(pow(coeff(var, ldeg), p));
1048         for (int i=1; i<numcoeff; ++i) {
1049                 ex sum = _ex0;
1050                 for (int j=1; j<=i; ++j) {
1051                         ex c = coeff(var, j + ldeg);
1052                         if (is_order_function(c)) {
1053                                 co.push_back(Order(_ex1));
1054                                 break;
1055                         } else
1056                                 sum += (p * j - (i - j)) * co[i - j] * c;
1057                 }
1058                 co.push_back(sum / coeff(var, ldeg) / i);
1059         }
1060
1061         // Construct new series (of non-zero coefficients)
1062         epvector new_seq;
1063         bool higher_order = false;
1064         for (int i=0; i<numcoeff; ++i) {
1065                 if (!co[i].is_zero())
1066                         new_seq.push_back(expair(co[i], p * ldeg + i));
1067                 if (is_order_function(co[i])) {
1068                         higher_order = true;
1069                         break;
1070                 }
1071         }
1072         if (!higher_order)
1073                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1074
1075         return pseries(relational(var,point), std::move(new_seq));
1076 }
1077
1078
1079 /** Return a new pseries object with the powers shifted by deg. */
1080 pseries pseries::shift_exponents(int deg) const
1081 {
1082         epvector newseq = seq;
1083         for (auto & it : newseq)
1084                 it.coeff += deg;
1085         return pseries(relational(var, point), std::move(newseq));
1086 }
1087
1088
1089 /** Implementation of ex::series() for powers. This performs Laurent expansion
1090  *  of reciprocals of series at singularities.
1091  *  @see ex::series */
1092 ex power::series(const relational & r, int order, unsigned options) const
1093 {
1094         // If basis is already a series, just power it
1095         if (is_exactly_a<pseries>(basis))
1096                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1097
1098         // Basis is not a series, may there be a singularity?
1099         bool must_expand_basis = false;
1100         try {
1101                 basis.subs(r, subs_options::no_pattern);
1102         } catch (pole_error) {
1103                 must_expand_basis = true;
1104         }
1105
1106         bool exponent_is_regular = true;
1107         try {
1108                 exponent.subs(r, subs_options::no_pattern);
1109         } catch (pole_error) {
1110                 exponent_is_regular = false;
1111         }
1112
1113         if (!exponent_is_regular) {
1114                 ex l = exponent*log(basis);
1115                 // this == exp(l);
1116                 ex le = l.series(r, order, options);
1117                 // Note: expanding exp(l) won't help, since that will attempt
1118                 // Taylor expansion, and fail (because exponent is "singular")
1119                 // Still l itself might be expanded in Taylor series.
1120                 // Examples:
1121                 // sin(x)/x*log(cos(x))
1122                 // 1/x*log(1 + x)
1123                 return exp(le).series(r, order, options);
1124                 // Note: if l happens to have a Laurent expansion (with
1125                 // negative powers of (var - point)), expanding exp(le)
1126                 // will barf (which is The Right Thing).
1127         }
1128
1129         // Is the expression of type something^(-int)?
1130         if (!must_expand_basis && !exponent.info(info_flags::negint)
1131          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1132                 return basic::series(r, order, options);
1133
1134         // Is the expression of type 0^something?
1135         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1136          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1137                 return basic::series(r, order, options);
1138
1139         // Singularity encountered, is the basis equal to (var - point)?
1140         if (basis.is_equal(r.lhs() - r.rhs())) {
1141                 epvector new_seq;
1142                 if (ex_to<numeric>(exponent).to_int() < order)
1143                         new_seq.push_back(expair(_ex1, exponent));
1144                 else
1145                         new_seq.push_back(expair(Order(_ex1), exponent));
1146                 return pseries(r, std::move(new_seq));
1147         }
1148
1149         // No, expand basis into series
1150
1151         numeric numexp;
1152         if (is_a<numeric>(exponent)) {
1153                 numexp = ex_to<numeric>(exponent);
1154         } else {
1155                 numexp = 0;
1156         }
1157         const ex& sym = r.lhs();
1158         // find existing minimal degree
1159         ex eb = basis.expand();
1160         int real_ldegree = 0;
1161         if (eb.info(info_flags::rational_function))
1162                 real_ldegree = eb.ldegree(sym-r.rhs());
1163         if (real_ldegree == 0) {
1164                 int orderloop = 0;
1165                 do {
1166                         orderloop++;
1167                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1168                 } while (real_ldegree == orderloop);
1169         }
1170
1171         if (!(real_ldegree*numexp).is_integer())
1172                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1173         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1174
1175         ex result;
1176         try {
1177                 result = ex_to<pseries>(e).power_const(numexp, order);
1178         } catch (pole_error) {
1179                 epvector ser { expair(Order(_ex1), order) };
1180                 result = pseries(r, std::move(ser));
1181         }
1182
1183         return result;
1184 }
1185
1186
1187 /** Re-expansion of a pseries object. */
1188 ex pseries::series(const relational & r, int order, unsigned options) const
1189 {
1190         const ex p = r.rhs();
1191         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1192         const symbol &s = ex_to<symbol>(r.lhs());
1193
1194         if (var.is_equal(s) && point.is_equal(p)) {
1195                 if (order > degree(s))
1196                         return *this;
1197                 else {
1198                         epvector new_seq;
1199                         for (auto & it : seq) {
1200                                 int o = ex_to<numeric>(it.coeff).to_int();
1201                                 if (o >= order) {
1202                                         new_seq.push_back(expair(Order(_ex1), o));
1203                                         break;
1204                                 }
1205                                 new_seq.push_back(it);
1206                         }
1207                         return pseries(r, std::move(new_seq));
1208                 }
1209         } else
1210                 return convert_to_poly().series(r, order, options);
1211 }
1212
1213 ex integral::series(const relational & r, int order, unsigned options) const
1214 {
1215         if (x.subs(r) != x)
1216                 throw std::logic_error("Cannot series expand wrt dummy variable");
1217
1218         // Expanding integrand with r substituted taken in boundaries.
1219         ex fseries = f.series(r, order, options);
1220         epvector fexpansion;
1221         fexpansion.reserve(fseries.nops());
1222         for (size_t i=0; i<fseries.nops(); ++i) {
1223                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1224                 currcoeff = (currcoeff == Order(_ex1))
1225                         ? currcoeff
1226                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1227                 if (currcoeff != 0)
1228                         fexpansion.push_back(
1229                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1230         }
1231
1232         // Expanding lower boundary
1233         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1234         ex aseries = (a-a.subs(r)).series(r, order, options);
1235         fseries = f.series(x == (a.subs(r)), order, options);
1236         for (size_t i=0; i<fseries.nops(); ++i) {
1237                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1238                 if (is_order_function(currcoeff))
1239                         break;
1240                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1241                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1242                 currcoeff = currcoeff.series(r, orderforf);
1243                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1244                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1245                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1246                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1247         }
1248
1249         // Expanding upper boundary
1250         ex bseries = (b-b.subs(r)).series(r, order, options);
1251         fseries = f.series(x == (b.subs(r)), order, options);
1252         for (size_t i=0; i<fseries.nops(); ++i) {
1253                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1254                 if (is_order_function(currcoeff))
1255                         break;
1256                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1257                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1258                 currcoeff = currcoeff.series(r, orderforf);
1259                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1260                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1261                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1262                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1263         }
1264
1265         return result;
1266 }
1267
1268
1269 /** Compute the truncated series expansion of an expression.
1270  *  This function returns an expression containing an object of class pseries
1271  *  to represent the series. If the series does not terminate within the given
1272  *  truncation order, the last term of the series will be an order term.
1273  *
1274  *  @param r  expansion relation, lhs holds variable and rhs holds point
1275  *  @param order  truncation order of series calculations
1276  *  @param options  of class series_options
1277  *  @return an expression holding a pseries object */
1278 ex ex::series(const ex & r, int order, unsigned options) const
1279 {
1280         ex e;
1281         relational rel_;
1282
1283         if (is_a<relational>(r))
1284                 rel_ = ex_to<relational>(r);
1285         else if (is_a<symbol>(r))
1286                 rel_ = relational(r,_ex0);
1287         else
1288                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1289
1290         e = bp->series(rel_, order, options);
1291         return e;
1292 }
1293
1294 GINAC_BIND_UNARCHIVER(pseries);
1295
1296 } // namespace GiNaC