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) 1999-2001 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  22  */
  23
  24 #include <stdexcept>
  25
  26 #include "pseries.h"
  27 #include "add.h"
  28 #include "inifcns.h"
  29 #include "lst.h"
  30 #include "mul.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "symbol.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36 #include "debugmsg.h"
  37
  38 #ifndef NO_NAMESPACE_GINAC
  39 namespace GiNaC {
  40 #endif // ndef NO_NAMESPACE_GINAC
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  43
  44 /*
  45  *  Default constructor, destructor, copy constructor, assignment operator and helpers
  46  */
  47
  48 pseries::pseries() : basic(TINFO_pseries)
  49 {
  50         debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
  51 }
  52
  53 void pseries::copy(const pseries &other)
  54 {
  55         inherited::copy(other);
  56         seq = other.seq;
  57         var = other.var;
  58         point = other.point;
  59 }
  60
  61 void pseries::destroy(bool call_parent)
  62 {
  63         if (call_parent)
  64                 inherited::destroy(call_parent);
  65 }
  66
  67
  68 /*
  69  *  Other constructors
  70  */
  71
  72 /** Construct pseries from a vector of coefficients and powers.
  73  *  expair.rest holds the coefficient, expair.coeff holds the power.
  74  *  The powers must be integers (positive or negative) and in ascending order;
  75  *  the last coefficient can be Order(_ex1()) to represent a truncated,
  76  *  non-terminating series.
  77  *
  78  *  @param rel_  expansion variable and point (must hold a relational)
  79  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  80  *  @return newly constructed pseries */
  81 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  82 {
  83         debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
  84         GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
  85         GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
  86         point = rel_.rhs();
  87         var = *static_cast<symbol *>(rel_.lhs().bp);
  88 }
  89
  90
  91 /*
  92  *  Archiving
  93  */
  94
  95 /** Construct object from archive_node. */
  96 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  97 {
  98         debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
  99         for (unsigned int i=0; true; ++i) {
 100                 ex rest;
 101                 ex coeff;
 102                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
 103                         seq.push_back(expair(rest, coeff));
 104                 else
 105                         break;
 106         }
 107         n.find_ex("var", var, sym_lst);
 108         n.find_ex("point", point, sym_lst);
 109 }
 110
 111 /** Unarchive the object. */
 112 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
 113 {
 114         return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
 115 }
 116
 117 /** Archive the object. */
 118 void pseries::archive(archive_node &n) const
 119 {
 120         inherited::archive(n);
 121         epvector::const_iterator i = seq.begin(), iend = seq.end();
 122         while (i != iend) {
 123                 n.add_ex("coeff", i->rest);
 124                 n.add_ex("power", i->coeff);
 125                 ++i;
 126         }
 127         n.add_ex("var", var);
 128         n.add_ex("point", point);
 129 }
 130
 131 //////////
 132 // functions overriding virtual functions from bases classes
 133 //////////
 134
 135 void pseries::print(std::ostream &os, unsigned upper_precedence) const
 136 {
 137         debugmsg("pseries print", LOGLEVEL_PRINT);
 138         if (precedence<=upper_precedence) os << "(";
 139         // objects of type pseries must not have any zero entries, so the
 140         // trivial (zero) pseries needs a special treatment here:
 141         if (seq.size()==0)
 142                 os << '0';
 143         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 144                 // print a sign, if needed
 145                 if (i!=seq.begin())
 146                         os << '+';
 147                 if (!is_order_function(i->rest)) {
 148                         // print 'rest', i.e. the expansion coefficient
 149                         if (i->rest.info(info_flags::numeric) &&
 150                                 i->rest.info(info_flags::positive)) {
 151                                 os << i->rest;
 152                         } else
 153                                 os << "(" << i->rest << ')';
 154                         // print 'coeff', something like (x-1)^42
 155                         if (!i->coeff.is_zero()) {
 156                                 os << '*';
 157                                 if (!point.is_zero())
 158                                         os << '(' << var-point << ')';
 159                                 else
 160                                         os << var;
 161                                 if (i->coeff.compare(_ex1())) {
 162                                         os << '^';
 163                                         if (i->coeff.info(info_flags::negative))
 164                                                 os << '(' << i->coeff << ')';
 165                                         else
 166                                                 os << i->coeff;
 167                                 }
 168                         }
 169                 } else {
 170                         os << Order(power(var-point,i->coeff));
 171                 }
 172         }
 173         if (precedence<=upper_precedence) os << ")";
 174 }
 175
 176
 177 void pseries::printraw(std::ostream &os) const
 178 {
 179         debugmsg("pseries printraw", LOGLEVEL_PRINT);
 180         os << "pseries(" << var << ";" << point << ";";
 181         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 182                 os << "(" << (*i).rest << "," << (*i).coeff << "),";
 183         os << ")";
 184 }
 185
 186
 187 void pseries::printtree(std::ostream & os, unsigned indent) const
 188 {
 189         debugmsg("pseries printtree",LOGLEVEL_PRINT);
 190         os << std::string(indent,' ') << "pseries "
 191            << ", hash=" << hashvalue
 192            << " (0x" << std::hex << hashvalue << std::dec << ")"
 193            << ", flags=" << flags << std::endl;
 194         for (unsigned i=0; i<seq.size(); ++i) {
 195                 seq[i].rest.printtree(os,indent+delta_indent);
 196                 seq[i].coeff.printtree(os,indent+delta_indent);
 197                 if (i!=seq.size()-1)
 198                         os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
 199         }
 200         var.printtree(os, indent+delta_indent);
 201         point.printtree(os, indent+delta_indent);
 202 }
 203
 204 int pseries::compare_same_type(const basic & other) const
 205 {
 206         GINAC_ASSERT(is_of_type(other, pseries));
 207         const pseries &o = static_cast<const pseries &>(other);
 208
 209         int cmpval = var.compare(o.var);
 210         if (cmpval)
 211                 return cmpval;
 212         cmpval = point.compare(o.point);
 213         if (cmpval)
 214                 return cmpval;
 215
 216         epvector::const_iterator it1 = seq.begin(), it2 = o.seq.begin(), it1end = seq.end(), it2end = o.seq.end();
 217         while ((it1 != it1end) && (it2 != it2end)) {
 218                 cmpval = it1->compare(*it2);
 219                 if (cmpval)
 220                         return cmpval;
 221                 it1++; it2++;
 222         }
 223         if (it1 == it1end)
 224                 return it2 == it2end ? 0 : -1;
 225
 226         return 0;
 227 }
 228
 229 /** Return the number of operands including a possible order term. */
 230 unsigned pseries::nops(void) const
 231 {
 232         return seq.size();
 233 }
 234
 235
 236 /** Return the ith term in the series when represented as a sum. */
 237 ex pseries::op(int i) const
 238 {
 239         if (i < 0 || unsigned(i) >= seq.size())
 240                 throw (std::out_of_range("op() out of range"));
 241         return seq[i].rest * power(var - point, seq[i].coeff);
 242 }
 243
 244
 245 ex &pseries::let_op(int i)
 246 {
 247         throw (std::logic_error("let_op not defined for pseries"));
 248 }
 249
 250
 251 /** Return degree of highest power of the series.  This is usually the exponent
 252  *  of the Order term.  If s is not the expansion variable of the series, the
 253  *  series is examined termwise. */
 254 int pseries::degree(const symbol &s) const
 255 {
 256         if (var.is_equal(s)) {
 257                 // Return last exponent
 258                 if (seq.size())
 259                         return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 260                 else
 261                         return 0;
 262         } else {
 263                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 264                 if (it == itend)
 265                         return 0;
 266                 int max_pow = INT_MIN;
 267                 while (it != itend) {
 268                         int pow = it->rest.degree(s);
 269                         if (pow > max_pow)
 270                                 max_pow = pow;
 271                         ++it;
 272                 }
 273                 return max_pow;
 274         }
 275 }
 276
 277 /** Return degree of lowest power of the series.  This is usually the exponent
 278  *  of the leading term.  If s is not the expansion variable of the series, the
 279  *  series is examined termwise.  If s is the expansion variable but the
 280  *  expansion point is not zero the series is not expanded to find the degree.
 281  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 282 int pseries::ldegree(const symbol &s) const
 283 {
 284         if (var.is_equal(s)) {
 285                 // Return first exponent
 286                 if (seq.size())
 287                         return ex_to_numeric((*(seq.begin())).coeff).to_int();
 288                 else
 289                         return 0;
 290         } else {
 291                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 292                 if (it == itend)
 293                         return 0;
 294                 int min_pow = INT_MAX;
 295                 while (it != itend) {
 296                         int pow = it->rest.ldegree(s);
 297                         if (pow < min_pow)
 298                                 min_pow = pow;
 299                         ++it;
 300                 }
 301                 return min_pow;
 302         }
 303 }
 304
 305 /** Return coefficient of degree n in power series if s is the expansion
 306  *  variable.  If the expansion point is nonzero, by definition the n=1
 307  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 308  *  the expansion took place in the s in the first place).
 309  *  If s is not the expansion variable, an attempt is made to convert the
 310  *  series to a polynomial and return the corresponding coefficient from
 311  *  there. */
 312 ex pseries::coeff(const symbol &s, int n) const
 313 {
 314         if (var.is_equal(s)) {
 315                 if (seq.size() == 0)
 316                         return _ex0();
 317
 318                 // Binary search in sequence for given power
 319                 numeric looking_for = numeric(n);
 320                 int lo = 0, hi = seq.size() - 1;
 321                 while (lo <= hi) {
 322                         int mid = (lo + hi) / 2;
 323                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 324                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 325                         switch (cmp) {
 326                                 case -1:
 327                                         lo = mid + 1;
 328                                         break;
 329                                 case 0:
 330                                         return seq[mid].rest;
 331                                 case 1:
 332                                         hi = mid - 1;
 333                                         break;
 334                                 default:
 335                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 336                         }
 337                 }
 338                 return _ex0();
 339         } else
 340                 return convert_to_poly().coeff(s, n);
 341 }
 342
 343 /** Does nothing. */
 344 ex pseries::collect(const symbol &s) const
 345 {
 346         return *this;
 347 }
 348
 349
 350 /** Evaluate coefficients. */
 351 ex pseries::eval(int level) const
 352 {
 353         if (level == 1)
 354                 return this->hold();
 355
 356         if (level == -max_recursion_level)
 357                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 358
 359         // Construct a new series with evaluated coefficients
 360         epvector new_seq;
 361         new_seq.reserve(seq.size());
 362         epvector::const_iterator it = seq.begin(), itend = seq.end();
 363         while (it != itend) {
 364                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 365                 ++it;
 366         }
 367         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 368 }
 369
 370
 371 /** Evaluate coefficients numerically. */
 372 ex pseries::evalf(int level) const
 373 {
 374         if (level == 1)
 375                 return *this;
 376
 377         if (level == -max_recursion_level)
 378                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 379
 380         // Construct a new series with evaluated coefficients
 381         epvector new_seq;
 382         new_seq.reserve(seq.size());
 383         epvector::const_iterator it = seq.begin(), itend = seq.end();
 384         while (it != itend) {
 385                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 386                 ++it;
 387         }
 388         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 389 }
 390
 391
 392 ex pseries::subs(const lst & ls, const lst & lr) const
 393 {
 394         // If expansion variable is being substituted, convert the series to a
 395         // polynomial and do the substitution there because the result might
 396         // no longer be a power series
 397         if (ls.has(var))
 398                 return convert_to_poly(true).subs(ls, lr);
 399
 400         // Otherwise construct a new series with substituted coefficients and
 401         // expansion point
 402         epvector newseq;
 403         newseq.reserve(seq.size());
 404         epvector::const_iterator it = seq.begin(), itend = seq.end();
 405         while (it != itend) {
 406                 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 407                 ++it;
 408         }
 409         return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
 410 }
 411
 412
 413 /** Implementation of ex::expand() for a power series.  It expands all the
 414  *  terms individually and returns the resulting series as a new pseries.
 415  *  @see ex::diff */
 416 ex pseries::expand(unsigned options) const
 417 {
 418         epvector newseq;
 419         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 420                 ex restexp = i->rest.expand();
 421                 if (!restexp.is_zero())
 422                         newseq.push_back(expair(restexp, i->coeff));
 423         }
 424         return (new pseries(relational(var,point), newseq))
 425                 ->setflag(status_flags::dynallocated | status_flags::expanded);
 426 }
 427
 428
 429 /** Implementation of ex::diff() for a power series.  It treats the series as a
 430  *  polynomial.
 431  *  @see ex::diff */
 432 ex pseries::derivative(const symbol & s) const
 433 {
 434         if (s == var) {
 435                 epvector new_seq;
 436                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 437
 438                 // FIXME: coeff might depend on var
 439                 while (it != itend) {
 440                         if (is_order_function(it->rest)) {
 441                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 442                         } else {
 443                                 ex c = it->rest * it->coeff;
 444                                 if (!c.is_zero())
 445                                         new_seq.push_back(expair(c, it->coeff - 1));
 446                         }
 447                         ++it;
 448                 }
 449                 return pseries(relational(var,point), new_seq);
 450         } else {
 451                 return *this;
 452         }
 453 }
 454
 455
 456 /*
 457  *  Construct ordinary polynomial out of series
 458  */
 459
 460 /** Convert a pseries object to an ordinary polynomial.
 461  *
 462  *  @param no_order flag: discard higher order terms */
 463 ex pseries::convert_to_poly(bool no_order) const
 464 {
 465         ex e;
 466         epvector::const_iterator it = seq.begin(), itend = seq.end();
 467
 468         while (it != itend) {
 469                 if (is_order_function(it->rest)) {
 470                         if (!no_order)
 471                                 e += Order(power(var - point, it->coeff));
 472                 } else
 473                         e += it->rest * power(var - point, it->coeff);
 474                 ++it;
 475         }
 476         return e;
 477 }
 478
 479 /** Returns true if there is no order term, i.e. the series terminates and
 480  *  false otherwise. */
 481 bool pseries::is_terminating(void) const
 482 {
 483         return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
 484 }
 485
 486
 487 /*
 488  *  Implementation of series expansion
 489  */
 490
 491 /** Default implementation of ex::series(). This performs Taylor expansion.
 492  *  @see ex::series */
 493 ex basic::series(const relational & r, int order, unsigned options) const
 494 {
 495         epvector seq;
 496         numeric fac(1);
 497         ex deriv = *this;
 498         ex coeff = deriv.subs(r);
 499         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 500
 501         if (!coeff.is_zero())
 502                 seq.push_back(expair(coeff, numeric(0)));
 503
 504         int n;
 505         for (n=1; n<order; ++n) {
 506                 fac = fac.mul(numeric(n));
 507                 deriv = deriv.diff(*s).expand();
 508                 if (deriv.is_zero()) {
 509                         // Series terminates
 510                         return pseries(r, seq);
 511                 }
 512                 coeff = deriv.subs(r);
 513                 if (!coeff.is_zero())
 514                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
 515         }
 516
 517         // Higher-order terms, if present
 518         deriv = deriv.diff(*s);
 519         if (!deriv.expand().is_zero())
 520                 seq.push_back(expair(Order(_ex1()), numeric(n)));
 521         return pseries(r, seq);
 522 }
 523
 524
 525 /** Implementation of ex::series() for symbols.
 526  *  @see ex::series */
 527 ex symbol::series(const relational & r, int order, unsigned options) const
 528 {
 529         epvector seq;
 530         const ex point = r.rhs();
 531         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 532         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 533
 534         if (this->is_equal(*s)) {
 535                 if (order > 0 && !point.is_zero())
 536                         seq.push_back(expair(point, _ex0()));
 537                 if (order > 1)
 538                         seq.push_back(expair(_ex1(), _ex1()));
 539                 else
 540                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 541         } else
 542                 seq.push_back(expair(*this, _ex0()));
 543         return pseries(r, seq);
 544 }
 545
 546
 547 /** Add one series object to another, producing a pseries object that
 548  *  represents the sum.
 549  *
 550  *  @param other  pseries object to add with
 551  *  @return the sum as a pseries */
 552 ex pseries::add_series(const pseries &other) const
 553 {
 554         // Adding two series with different variables or expansion points
 555         // results in an empty (constant) series
 556         if (!is_compatible_to(other)) {
 557                 epvector nul;
 558                 nul.push_back(expair(Order(_ex1()), _ex0()));
 559                 return pseries(relational(var,point), nul);
 560         }
 561
 562         // Series addition
 563         epvector new_seq;
 564         epvector::const_iterator a = seq.begin();
 565         epvector::const_iterator b = other.seq.begin();
 566         epvector::const_iterator a_end = seq.end();
 567         epvector::const_iterator b_end = other.seq.end();
 568         int pow_a = INT_MAX, pow_b = INT_MAX;
 569         for (;;) {
 570                 // If a is empty, fill up with elements from b and stop
 571                 if (a == a_end) {
 572                         while (b != b_end) {
 573                                 new_seq.push_back(*b);
 574                                 ++b;
 575                         }
 576                         break;
 577                 } else
 578                         pow_a = ex_to_numeric((*a).coeff).to_int();
 579
 580                 // If b is empty, fill up with elements from a and stop
 581                 if (b == b_end) {
 582                         while (a != a_end) {
 583                                 new_seq.push_back(*a);
 584                                 ++a;
 585                         }
 586                         break;
 587                 } else
 588                         pow_b = ex_to_numeric((*b).coeff).to_int();
 589
 590                 // a and b are non-empty, compare powers
 591                 if (pow_a < pow_b) {
 592                         // a has lesser power, get coefficient from a
 593                         new_seq.push_back(*a);
 594                         if (is_order_function((*a).rest))
 595                                 break;
 596                         ++a;
 597                 } else if (pow_b < pow_a) {
 598                         // b has lesser power, get coefficient from b
 599                         new_seq.push_back(*b);
 600                         if (is_order_function((*b).rest))
 601                                 break;
 602                         ++b;
 603                 } else {
 604                         // Add coefficient of a and b
 605                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 606                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 607                                 break;  // Order term ends the sequence
 608                         } else {
 609                                 ex sum = (*a).rest + (*b).rest;
 610                                 if (!(sum.is_zero()))
 611                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 612                                 ++a;
 613                                 ++b;
 614                         }
 615                 }
 616         }
 617         return pseries(relational(var,point), new_seq);
 618 }
 619
 620
 621 /** Implementation of ex::series() for sums. This performs series addition when
 622  *  adding pseries objects.
 623  *  @see ex::series */
 624 ex add::series(const relational & r, int order, unsigned options) const
 625 {
 626         ex acc; // Series accumulator
 627
 628         // Get first term from overall_coeff
 629         acc = overall_coeff.series(r, order, options);
 630
 631         // Add remaining terms
 632         epvector::const_iterator it = seq.begin();
 633         epvector::const_iterator itend = seq.end();
 634         for (; it!=itend; ++it) {
 635                 ex op;
 636                 if (is_ex_exactly_of_type(it->rest, pseries))
 637                         op = it->rest;
 638                 else
 639                         op = it->rest.series(r, order, options);
 640                 if (!it->coeff.is_equal(_ex1()))
 641                         op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 642
 643                 // Series addition
 644                 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 645         }
 646         return acc;
 647 }
 648
 649
 650 /** Multiply a pseries object with a numeric constant, producing a pseries
 651  *  object that represents the product.
 652  *
 653  *  @param other  constant to multiply with
 654  *  @return the product as a pseries */
 655 ex pseries::mul_const(const numeric &other) const
 656 {
 657         epvector new_seq;
 658         new_seq.reserve(seq.size());
 659
 660         epvector::const_iterator it = seq.begin(), itend = seq.end();
 661         while (it != itend) {
 662                 if (!is_order_function(it->rest))
 663                         new_seq.push_back(expair(it->rest * other, it->coeff));
 664                 else
 665                         new_seq.push_back(*it);
 666                 ++it;
 667         }
 668         return pseries(relational(var,point), new_seq);
 669 }
 670
 671
 672 /** Multiply one pseries object to another, producing a pseries object that
 673  *  represents the product.
 674  *
 675  *  @param other  pseries object to multiply with
 676  *  @return the product as a pseries */
 677 ex pseries::mul_series(const pseries &other) const
 678 {
 679         // Multiplying two series with different variables or expansion points
 680         // results in an empty (constant) series
 681         if (!is_compatible_to(other)) {
 682                 epvector nul;
 683                 nul.push_back(expair(Order(_ex1()), _ex0()));
 684                 return pseries(relational(var,point), nul);
 685         }
 686
 687         // Series multiplication
 688         epvector new_seq;
 689
 690         const symbol *s = static_cast<symbol *>(var.bp);
 691         int a_max = degree(*s);
 692         int b_max = other.degree(*s);
 693         int a_min = ldegree(*s);
 694         int b_min = other.ldegree(*s);
 695         int cdeg_min = a_min + b_min;
 696         int cdeg_max = a_max + b_max;
 697
 698         int higher_order_a = INT_MAX;
 699         int higher_order_b = INT_MAX;
 700         if (is_order_function(coeff(*s, a_max)))
 701                 higher_order_a = a_max + b_min;
 702         if (is_order_function(other.coeff(*s, b_max)))
 703                 higher_order_b = b_max + a_min;
 704         int higher_order_c = std::min(higher_order_a, higher_order_b);
 705         if (cdeg_max >= higher_order_c)
 706                 cdeg_max = higher_order_c - 1;
 707
 708         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 709                 ex co = _ex0();
 710                 // c(i)=a(0)b(i)+...+a(i)b(0)
 711                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 712                         ex a_coeff = coeff(*s, i);
 713                         ex b_coeff = other.coeff(*s, cdeg-i);
 714                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 715                                 co += a_coeff * b_coeff;
 716                 }
 717                 if (!co.is_zero())
 718                         new_seq.push_back(expair(co, numeric(cdeg)));
 719         }
 720         if (higher_order_c < INT_MAX)
 721                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 722         return pseries(relational(var,point), new_seq);
 723 }
 724
 725
 726 /** Implementation of ex::series() for product. This performs series
 727  *  multiplication when multiplying series.
 728  *  @see ex::series */
 729 ex mul::series(const relational & r, int order, unsigned options) const
 730 {
 731         ex acc; // Series accumulator
 732
 733         // Get first term from overall_coeff
 734         acc = overall_coeff.series(r, order, options);
 735
 736         // Multiply with remaining terms
 737         epvector::const_iterator it = seq.begin();
 738         epvector::const_iterator itend = seq.end();
 739         for (; it!=itend; ++it) {
 740                 ex op = it->rest;
 741                 if (op.info(info_flags::numeric)) {
 742                         // series * const (special case, faster)
 743                         ex f = power(op, it->coeff);
 744                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 745                         continue;
 746                 } else if (!is_ex_exactly_of_type(op, pseries))
 747                         op = op.series(r, order, options);
 748                 if (!it->coeff.is_equal(_ex1()))
 749                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 750
 751                 // Series multiplication
 752                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 753         }
 754         return acc;
 755 }
 756
 757
 758 /** Compute the p-th power of a series.
 759  *
 760  *  @param p  power to compute
 761  *  @param deg  truncation order of series calculation */
 762 ex pseries::power_const(const numeric &p, int deg) const
 763 {
 764         // method:
 765         // let A(x) be this series and for the time being let it start with a
 766         // constant (later we'll generalize):
 767         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 768         // We want to compute
 769         //     C(x) = A(x)^p
 770         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 771         // Taking the derivative on both sides and multiplying with A(x) one
 772         // immediately arrives at
 773         //     C'(x)*A(x) = p*C(x)*A'(x)
 774         // Multiplying this out and comparing coefficients we get the recurrence
 775         // formula
 776         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 777         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 778         // which can easily be solved given the starting value c_0 = (a_0)^p.
 779         // For the more general case where the leading coefficient of A(x) is not
 780         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 781         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 782         // then of course x^(p*m) but the recurrence formula still holds.
 783         const symbol *s = static_cast<symbol *>(var.bp);
 784         int ldeg = ldegree(*s);
 785
 786         // Compute coefficients of the powered series
 787         exvector co;
 788         co.reserve(deg);
 789         co.push_back(power(coeff(*s, ldeg), p));
 790         bool all_sums_zero = true;
 791         for (int i=1; i<deg; ++i) {
 792                 ex sum = _ex0();
 793                 for (int j=1; j<=i; ++j) {
 794                         ex c = coeff(*s, j + ldeg);
 795                         if (is_order_function(c)) {
 796                                 co.push_back(Order(_ex1()));
 797                                 break;
 798                         } else
 799                                 sum += (p * j - (i - j)) * co[i - j] * c;
 800                 }
 801                 if (!sum.is_zero())
 802                         all_sums_zero = false;
 803                 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
 804         }
 805
 806         // Construct new series (of non-zero coefficients)
 807         epvector new_seq;
 808         bool higher_order = false;
 809         for (int i=0; i<deg; ++i) {
 810                 if (!co[i].is_zero())
 811                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 812                 if (is_order_function(co[i])) {
 813                         higher_order = true;
 814                         break;
 815                 }
 816         }
 817         if (!higher_order && !all_sums_zero)
 818                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 819         return pseries(relational(var,point), new_seq);
 820 }
 821
 822
 823 /** Return a new pseries object with the powers shifted by deg. */
 824 pseries pseries::shift_exponents(int deg) const
 825 {
 826         epvector newseq(seq);
 827         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
 828                 i->coeff = i->coeff + deg;
 829         return pseries(relational(var, point), newseq);
 830 }
 831
 832
 833 /** Implementation of ex::series() for powers. This performs Laurent expansion
 834  *  of reciprocals of series at singularities.
 835  *  @see ex::series */
 836 ex power::series(const relational & r, int order, unsigned options) const
 837 {
 838         ex e;
 839         if (!is_ex_exactly_of_type(basis, pseries)) {
 840                 // Basis is not a series, may there be a singularity?
 841                 bool must_expand_basis = false;
 842                 try {
 843                         basis.subs(r);
 844                 } catch (pole_error) {
 845                         must_expand_basis = true;
 846                 }
 847
 848                 // Is the expression of type something^(-int)?
 849                 if (!must_expand_basis && !exponent.info(info_flags::negint))
 850                         return basic::series(r, order, options);
 851
 852                 // Is the expression of type 0^something?
 853                 if (!must_expand_basis && !basis.subs(r).is_zero())
 854                         return basic::series(r, order, options);
 855
 856                 // Singularity encountered, expand basis into series
 857                 e = basis.series(r, order, options);
 858         } else {
 859                 // Basis is a series
 860                 e = basis;
 861         }
 862
 863         // Power e
 864         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 865 }
 866
 867
 868 /** Re-expansion of a pseries object. */
 869 ex pseries::series(const relational & r, int order, unsigned options) const
 870 {
 871         const ex p = r.rhs();
 872         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 873         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 874
 875         if (var.is_equal(*s) && point.is_equal(p)) {
 876                 if (order > degree(*s))
 877                         return *this;
 878                 else {
 879                         epvector new_seq;
 880                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 881                         while (it != itend) {
 882                                 int o = ex_to_numeric(it->coeff).to_int();
 883                                 if (o >= order) {
 884                                         new_seq.push_back(expair(Order(_ex1()), o));
 885                                         break;
 886                                 }
 887                                 new_seq.push_back(*it);
 888                                 ++it;
 889                         }
 890                         return pseries(r, new_seq);
 891                 }
 892         } else
 893                 return convert_to_poly().series(r, order, options);
 894 }
 895
 896
 897 /** Compute the truncated series expansion of an expression.
 898  *  This function returns an expression containing an object of class pseries
 899  *  to represent the series. If the series does not terminate within the given
 900  *  truncation order, the last term of the series will be an order term.
 901  *
 902  *  @param r  expansion relation, lhs holds variable and rhs holds point
 903  *  @param order  truncation order of series calculations
 904  *  @param options  of class series_options
 905  *  @return an expression holding a pseries object */
 906 ex ex::series(const ex & r, int order, unsigned options) const
 907 {
 908         GINAC_ASSERT(bp!=0);
 909         ex e;
 910         relational rel_;
 911
 912         if (is_ex_exactly_of_type(r,relational))
 913                 rel_ = ex_to_relational(r);
 914         else if (is_ex_exactly_of_type(r,symbol))
 915                 rel_ = relational(r,_ex0());
 916         else
 917                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 918
 919         try {
 920                 e = bp->series(rel_, order, options);
 921         } catch (std::exception &x) {
 922                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 923         }
 924         return e;
 925 }
 926
 927 //////////
 928 // static member variables
 929 //////////
 930
 931 // protected
 932
 933 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
 934
 935 #ifndef NO_NAMESPACE_GINAC
 936 } // namespace GiNaC
 937 #endif // ndef NO_NAMESPACE_GINAC