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         newseq.reserve(seq.size());
 420         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 421                 ex restexp = i->rest.expand();
 422                 if (!restexp.is_zero())
 423                         newseq.push_back(expair(restexp, i->coeff));
 424         }
 425         return (new pseries(relational(var,point), newseq))
 426                 ->setflag(status_flags::dynallocated | status_flags::expanded);
 427 }
 428
 429
 430 /** Implementation of ex::diff() for a power series.  It treats the series as a
 431  *  polynomial.
 432  *  @see ex::diff */
 433 ex pseries::derivative(const symbol & s) const
 434 {
 435         if (s == var) {
 436                 epvector new_seq;
 437                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 438
 439                 // FIXME: coeff might depend on var
 440                 while (it != itend) {
 441                         if (is_order_function(it->rest)) {
 442                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 443                         } else {
 444                                 ex c = it->rest * it->coeff;
 445                                 if (!c.is_zero())
 446                                         new_seq.push_back(expair(c, it->coeff - 1));
 447                         }
 448                         ++it;
 449                 }
 450                 return pseries(relational(var,point), new_seq);
 451         } else {
 452                 return *this;
 453         }
 454 }
 455
 456
 457 /*
 458  *  Construct ordinary polynomial out of series
 459  */
 460
 461 /** Convert a pseries object to an ordinary polynomial.
 462  *
 463  *  @param no_order flag: discard higher order terms */
 464 ex pseries::convert_to_poly(bool no_order) const
 465 {
 466         ex e;
 467         epvector::const_iterator it = seq.begin(), itend = seq.end();
 468
 469         while (it != itend) {
 470                 if (is_order_function(it->rest)) {
 471                         if (!no_order)
 472                                 e += Order(power(var - point, it->coeff));
 473                 } else
 474                         e += it->rest * power(var - point, it->coeff);
 475                 ++it;
 476         }
 477         return e;
 478 }
 479
 480 /** Returns true if there is no order term, i.e. the series terminates and
 481  *  false otherwise. */
 482 bool pseries::is_terminating(void) const
 483 {
 484         return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
 485 }
 486
 487
 488 /*
 489  *  Implementation of series expansion
 490  */
 491
 492 /** Default implementation of ex::series(). This performs Taylor expansion.
 493  *  @see ex::series */
 494 ex basic::series(const relational & r, int order, unsigned options) const
 495 {
 496         epvector seq;
 497         numeric fac(1);
 498         ex deriv = *this;
 499         ex coeff = deriv.subs(r);
 500         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 501
 502         if (!coeff.is_zero())
 503                 seq.push_back(expair(coeff, numeric(0)));
 504
 505         int n;
 506         for (n=1; n<order; ++n) {
 507                 fac = fac.mul(numeric(n));
 508                 deriv = deriv.diff(*s).expand();
 509                 if (deriv.is_zero()) {
 510                         // Series terminates
 511                         return pseries(r, seq);
 512                 }
 513                 coeff = deriv.subs(r);
 514                 if (!coeff.is_zero())
 515                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
 516         }
 517
 518         // Higher-order terms, if present
 519         deriv = deriv.diff(*s);
 520         if (!deriv.expand().is_zero())
 521                 seq.push_back(expair(Order(_ex1()), numeric(n)));
 522         return pseries(r, seq);
 523 }
 524
 525
 526 /** Implementation of ex::series() for symbols.
 527  *  @see ex::series */
 528 ex symbol::series(const relational & r, int order, unsigned options) const
 529 {
 530         epvector seq;
 531         const ex point = r.rhs();
 532         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 533         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 534
 535         if (this->is_equal(*s)) {
 536                 if (order > 0 && !point.is_zero())
 537                         seq.push_back(expair(point, _ex0()));
 538                 if (order > 1)
 539                         seq.push_back(expair(_ex1(), _ex1()));
 540                 else
 541                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 542         } else
 543                 seq.push_back(expair(*this, _ex0()));
 544         return pseries(r, seq);
 545 }
 546
 547
 548 /** Add one series object to another, producing a pseries object that
 549  *  represents the sum.
 550  *
 551  *  @param other  pseries object to add with
 552  *  @return the sum as a pseries */
 553 ex pseries::add_series(const pseries &other) const
 554 {
 555         // Adding two series with different variables or expansion points
 556         // results in an empty (constant) series
 557         if (!is_compatible_to(other)) {
 558                 epvector nul;
 559                 nul.push_back(expair(Order(_ex1()), _ex0()));
 560                 return pseries(relational(var,point), nul);
 561         }
 562
 563         // Series addition
 564         epvector new_seq;
 565         epvector::const_iterator a = seq.begin();
 566         epvector::const_iterator b = other.seq.begin();
 567         epvector::const_iterator a_end = seq.end();
 568         epvector::const_iterator b_end = other.seq.end();
 569         int pow_a = INT_MAX, pow_b = INT_MAX;
 570         for (;;) {
 571                 // If a is empty, fill up with elements from b and stop
 572                 if (a == a_end) {
 573                         while (b != b_end) {
 574                                 new_seq.push_back(*b);
 575                                 ++b;
 576                         }
 577                         break;
 578                 } else
 579                         pow_a = ex_to_numeric((*a).coeff).to_int();
 580
 581                 // If b is empty, fill up with elements from a and stop
 582                 if (b == b_end) {
 583                         while (a != a_end) {
 584                                 new_seq.push_back(*a);
 585                                 ++a;
 586                         }
 587                         break;
 588                 } else
 589                         pow_b = ex_to_numeric((*b).coeff).to_int();
 590
 591                 // a and b are non-empty, compare powers
 592                 if (pow_a < pow_b) {
 593                         // a has lesser power, get coefficient from a
 594                         new_seq.push_back(*a);
 595                         if (is_order_function((*a).rest))
 596                                 break;
 597                         ++a;
 598                 } else if (pow_b < pow_a) {
 599                         // b has lesser power, get coefficient from b
 600                         new_seq.push_back(*b);
 601                         if (is_order_function((*b).rest))
 602                                 break;
 603                         ++b;
 604                 } else {
 605                         // Add coefficient of a and b
 606                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 607                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 608                                 break;  // Order term ends the sequence
 609                         } else {
 610                                 ex sum = (*a).rest + (*b).rest;
 611                                 if (!(sum.is_zero()))
 612                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 613                                 ++a;
 614                                 ++b;
 615                         }
 616                 }
 617         }
 618         return pseries(relational(var,point), new_seq);
 619 }
 620
 621
 622 /** Implementation of ex::series() for sums. This performs series addition when
 623  *  adding pseries objects.
 624  *  @see ex::series */
 625 ex add::series(const relational & r, int order, unsigned options) const
 626 {
 627         ex acc; // Series accumulator
 628
 629         // Get first term from overall_coeff
 630         acc = overall_coeff.series(r, order, options);
 631
 632         // Add remaining terms
 633         epvector::const_iterator it = seq.begin();
 634         epvector::const_iterator itend = seq.end();
 635         for (; it!=itend; ++it) {
 636                 ex op;
 637                 if (is_ex_exactly_of_type(it->rest, pseries))
 638                         op = it->rest;
 639                 else
 640                         op = it->rest.series(r, order, options);
 641                 if (!it->coeff.is_equal(_ex1()))
 642                         op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 643
 644                 // Series addition
 645                 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 646         }
 647         return acc;
 648 }
 649
 650
 651 /** Multiply a pseries object with a numeric constant, producing a pseries
 652  *  object that represents the product.
 653  *
 654  *  @param other  constant to multiply with
 655  *  @return the product as a pseries */
 656 ex pseries::mul_const(const numeric &other) const
 657 {
 658         epvector new_seq;
 659         new_seq.reserve(seq.size());
 660
 661         epvector::const_iterator it = seq.begin(), itend = seq.end();
 662         while (it != itend) {
 663                 if (!is_order_function(it->rest))
 664                         new_seq.push_back(expair(it->rest * other, it->coeff));
 665                 else
 666                         new_seq.push_back(*it);
 667                 ++it;
 668         }
 669         return pseries(relational(var,point), new_seq);
 670 }
 671
 672
 673 /** Multiply one pseries object to another, producing a pseries object that
 674  *  represents the product.
 675  *
 676  *  @param other  pseries object to multiply with
 677  *  @return the product as a pseries */
 678 ex pseries::mul_series(const pseries &other) const
 679 {
 680         // Multiplying two series with different variables or expansion points
 681         // results in an empty (constant) series
 682         if (!is_compatible_to(other)) {
 683                 epvector nul;
 684                 nul.push_back(expair(Order(_ex1()), _ex0()));
 685                 return pseries(relational(var,point), nul);
 686         }
 687
 688         // Series multiplication
 689         epvector new_seq;
 690
 691         const symbol *s = static_cast<symbol *>(var.bp);
 692         int a_max = degree(*s);
 693         int b_max = other.degree(*s);
 694         int a_min = ldegree(*s);
 695         int b_min = other.ldegree(*s);
 696         int cdeg_min = a_min + b_min;
 697         int cdeg_max = a_max + b_max;
 698
 699         int higher_order_a = INT_MAX;
 700         int higher_order_b = INT_MAX;
 701         if (is_order_function(coeff(*s, a_max)))
 702                 higher_order_a = a_max + b_min;
 703         if (is_order_function(other.coeff(*s, b_max)))
 704                 higher_order_b = b_max + a_min;
 705         int higher_order_c = std::min(higher_order_a, higher_order_b);
 706         if (cdeg_max >= higher_order_c)
 707                 cdeg_max = higher_order_c - 1;
 708
 709         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 710                 ex co = _ex0();
 711                 // c(i)=a(0)b(i)+...+a(i)b(0)
 712                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 713                         ex a_coeff = coeff(*s, i);
 714                         ex b_coeff = other.coeff(*s, cdeg-i);
 715                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 716                                 co += a_coeff * b_coeff;
 717                 }
 718                 if (!co.is_zero())
 719                         new_seq.push_back(expair(co, numeric(cdeg)));
 720         }
 721         if (higher_order_c < INT_MAX)
 722                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 723         return pseries(relational(var,point), new_seq);
 724 }
 725
 726
 727 /** Implementation of ex::series() for product. This performs series
 728  *  multiplication when multiplying series.
 729  *  @see ex::series */
 730 ex mul::series(const relational & r, int order, unsigned options) const
 731 {
 732         ex acc; // Series accumulator
 733
 734         // Get first term from overall_coeff
 735         acc = overall_coeff.series(r, order, options);
 736
 737         // Multiply with remaining terms
 738         epvector::const_iterator it = seq.begin();
 739         epvector::const_iterator itend = seq.end();
 740         for (; it!=itend; ++it) {
 741                 ex op = it->rest;
 742                 if (op.info(info_flags::numeric)) {
 743                         // series * const (special case, faster)
 744                         ex f = power(op, it->coeff);
 745                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 746                         continue;
 747                 } else if (!is_ex_exactly_of_type(op, pseries))
 748                         op = op.series(r, order, options);
 749                 if (!it->coeff.is_equal(_ex1()))
 750                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 751
 752                 // Series multiplication
 753                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 754         }
 755         return acc;
 756 }
 757
 758
 759 /** Compute the p-th power of a series.
 760  *
 761  *  @param p  power to compute
 762  *  @param deg  truncation order of series calculation */
 763 ex pseries::power_const(const numeric &p, int deg) const
 764 {
 765         // method:
 766         // let A(x) be this series and for the time being let it start with a
 767         // constant (later we'll generalize):
 768         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 769         // We want to compute
 770         //     C(x) = A(x)^p
 771         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 772         // Taking the derivative on both sides and multiplying with A(x) one
 773         // immediately arrives at
 774         //     C'(x)*A(x) = p*C(x)*A'(x)
 775         // Multiplying this out and comparing coefficients we get the recurrence
 776         // formula
 777         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 778         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 779         // which can easily be solved given the starting value c_0 = (a_0)^p.
 780         // For the more general case where the leading coefficient of A(x) is not
 781         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 782         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 783         // then of course x^(p*m) but the recurrence formula still holds.
 784         const symbol *s = static_cast<symbol *>(var.bp);
 785         int ldeg = ldegree(*s);
 786
 787         // Compute coefficients of the powered series
 788         exvector co;
 789         co.reserve(deg);
 790         co.push_back(power(coeff(*s, ldeg), p));
 791         bool all_sums_zero = true;
 792         for (int i=1; i<deg; ++i) {
 793                 ex sum = _ex0();
 794                 for (int j=1; j<=i; ++j) {
 795                         ex c = coeff(*s, j + ldeg);
 796                         if (is_order_function(c)) {
 797                                 co.push_back(Order(_ex1()));
 798                                 break;
 799                         } else
 800                                 sum += (p * j - (i - j)) * co[i - j] * c;
 801                 }
 802                 if (!sum.is_zero())
 803                         all_sums_zero = false;
 804                 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
 805         }
 806
 807         // Construct new series (of non-zero coefficients)
 808         epvector new_seq;
 809         bool higher_order = false;
 810         for (int i=0; i<deg; ++i) {
 811                 if (!co[i].is_zero())
 812                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 813                 if (is_order_function(co[i])) {
 814                         higher_order = true;
 815                         break;
 816                 }
 817         }
 818         if (!higher_order && !all_sums_zero)
 819                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 820         return pseries(relational(var,point), new_seq);
 821 }
 822
 823
 824 /** Return a new pseries object with the powers shifted by deg. */
 825 pseries pseries::shift_exponents(int deg) const
 826 {
 827         epvector newseq(seq);
 828         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
 829                 i->coeff = i->coeff + deg;
 830         return pseries(relational(var, point), newseq);
 831 }
 832
 833
 834 /** Implementation of ex::series() for powers. This performs Laurent expansion
 835  *  of reciprocals of series at singularities.
 836  *  @see ex::series */
 837 ex power::series(const relational & r, int order, unsigned options) const
 838 {
 839         ex e;
 840         if (!is_ex_exactly_of_type(basis, pseries)) {
 841                 // Basis is not a series, may there be a singularity?
 842                 bool must_expand_basis = false;
 843                 try {
 844                         basis.subs(r);
 845                 } catch (pole_error) {
 846                         must_expand_basis = true;
 847                 }
 848
 849                 // Is the expression of type something^(-int)?
 850                 if (!must_expand_basis && !exponent.info(info_flags::negint))
 851                         return basic::series(r, order, options);
 852
 853                 // Is the expression of type 0^something?
 854                 if (!must_expand_basis && !basis.subs(r).is_zero())
 855                         return basic::series(r, order, options);
 856
 857                 // Singularity encountered, expand basis into series
 858                 e = basis.series(r, order, options);
 859         } else {
 860                 // Basis is a series
 861                 e = basis;
 862         }
 863
 864         // Power e
 865         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 866 }
 867
 868
 869 /** Re-expansion of a pseries object. */
 870 ex pseries::series(const relational & r, int order, unsigned options) const
 871 {
 872         const ex p = r.rhs();
 873         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 874         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 875
 876         if (var.is_equal(*s) && point.is_equal(p)) {
 877                 if (order > degree(*s))
 878                         return *this;
 879                 else {
 880                         epvector new_seq;
 881                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 882                         while (it != itend) {
 883                                 int o = ex_to_numeric(it->coeff).to_int();
 884                                 if (o >= order) {
 885                                         new_seq.push_back(expair(Order(_ex1()), o));
 886                                         break;
 887                                 }
 888                                 new_seq.push_back(*it);
 889                                 ++it;
 890                         }
 891                         return pseries(r, new_seq);
 892                 }
 893         } else
 894                 return convert_to_poly().series(r, order, options);
 895 }
 896
 897
 898 /** Compute the truncated series expansion of an expression.
 899  *  This function returns an expression containing an object of class pseries
 900  *  to represent the series. If the series does not terminate within the given
 901  *  truncation order, the last term of the series will be an order term.
 902  *
 903  *  @param r  expansion relation, lhs holds variable and rhs holds point
 904  *  @param order  truncation order of series calculations
 905  *  @param options  of class series_options
 906  *  @return an expression holding a pseries object */
 907 ex ex::series(const ex & r, int order, unsigned options) const
 908 {
 909         GINAC_ASSERT(bp!=0);
 910         ex e;
 911         relational rel_;
 912
 913         if (is_ex_exactly_of_type(r,relational))
 914                 rel_ = ex_to_relational(r);
 915         else if (is_ex_exactly_of_type(r,symbol))
 916                 rel_ = relational(r,_ex0());
 917         else
 918                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 919
 920         try {
 921                 e = bp->series(rel_, order, options);
 922         } catch (std::exception &x) {
 923                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 924         }
 925         return e;
 926 }
 927
 928 //////////
 929 // static member variables
 930 //////////
 931
 932 // protected
 933
 934 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
 935
 936 #ifndef NO_NAMESPACE_GINAC
 937 } // namespace GiNaC
 938 #endif // ndef NO_NAMESPACE_GINAC