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 "print.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37 #include "debugmsg.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  42
  43
  44 /*
  45  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  46  */
  47
  48 pseries::pseries() : basic(TINFO_pseries)
  49 {
  50         debugmsg("pseries default ctor", 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 DEFAULT_DESTROY(pseries)
  62
  63
  64 /*
  65  *  Other ctors
  66  */
  67
  68 /** Construct pseries from a vector of coefficients and powers.
  69  *  expair.rest holds the coefficient, expair.coeff holds the power.
  70  *  The powers must be integers (positive or negative) and in ascending order;
  71  *  the last coefficient can be Order(_ex1()) to represent a truncated,
  72  *  non-terminating series.
  73  *
  74  *  @param rel_  expansion variable and point (must hold a relational)
  75  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  76  *  @return newly constructed pseries */
  77 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  78 {
  79         debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
  80         GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
  81         GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
  82         point = rel_.rhs();
  83         var = *static_cast<symbol *>(rel_.lhs().bp);
  84 }
  85
  86
  87 /*
  88  *  Archiving
  89  */
  90
  91 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  92 {
  93         debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
  94         for (unsigned int i=0; true; ++i) {
  95                 ex rest;
  96                 ex coeff;
  97                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  98                         seq.push_back(expair(rest, coeff));
  99                 else
 100                         break;
 101         }
 102         n.find_ex("var", var, sym_lst);
 103         n.find_ex("point", point, sym_lst);
 104 }
 105
 106 void pseries::archive(archive_node &n) const
 107 {
 108         inherited::archive(n);
 109         epvector::const_iterator i = seq.begin(), iend = seq.end();
 110         while (i != iend) {
 111                 n.add_ex("coeff", i->rest);
 112                 n.add_ex("power", i->coeff);
 113                 ++i;
 114         }
 115         n.add_ex("var", var);
 116         n.add_ex("point", point);
 117 }
 118
 119 DEFAULT_UNARCHIVE(pseries)
 120
 121 //////////
 122 // functions overriding virtual functions from bases classes
 123 //////////
 124
 125 void pseries::print(const print_context & c, unsigned level) const
 126 {
 127         debugmsg("pseries print", LOGLEVEL_PRINT);
 128
 129         if (is_of_type(c, print_tree)) {
 130
 131                 c.s << std::string(level, ' ') << class_name()
 132                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 133                     << std::endl;
 134                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 135                 for (unsigned i=0; i<seq.size(); ++i) {
 136                         seq[i].rest.print(c, level + delta_indent);
 137                         seq[i].coeff.print(c, level + delta_indent);
 138                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 139                 }
 140                 var.print(c, level + delta_indent);
 141                 point.print(c, level + delta_indent);
 142
 143         } else {
 144
 145                 if (precedence <= level)
 146                         c.s << "(";
 147
 148                 std::string par_open = is_of_type(c, print_latex) ? "{(" : "(";
 149                 std::string par_close = is_of_type(c, print_latex) ? ")}" : ")";
 150
 151                 // objects of type pseries must not have any zero entries, so the
 152                 // trivial (zero) pseries needs a special treatment here:
 153                 if (seq.size() == 0)
 154                         c.s << '0';
 155                 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 156                         // print a sign, if needed
 157                         if (i != seq.begin())
 158                                 c.s << '+';
 159                         if (!is_order_function(i->rest)) {
 160                                 // print 'rest', i.e. the expansion coefficient
 161                                 if (i->rest.info(info_flags::numeric) &&
 162                                         i->rest.info(info_flags::positive)) {
 163                                         i->rest.print(c);
 164                                 } else {
 165                                         c.s << par_open;
 166                                         i->rest.print(c);
 167                                         c.s << par_close;
 168                                 }
 169                                 // print 'coeff', something like (x-1)^42
 170                                 if (!i->coeff.is_zero()) {
 171                                         if (is_of_type(c, print_latex))
 172                                                 c.s << ' ';
 173                                         else
 174                                                 c.s << '*';
 175                                         if (!point.is_zero()) {
 176                                                 c.s << par_open;
 177                                                 (var-point).print(c);
 178                                                 c.s << par_close;
 179                                         } else
 180                                                 var.print(c);
 181                                         if (i->coeff.compare(_ex1())) {
 182                                                 c.s << '^';
 183                                                 if (i->coeff.info(info_flags::negative)) {
 184                                                         c.s << par_open;
 185                                                         i->coeff.print(c);
 186                                                         c.s << par_close;
 187                                                 } else {
 188                                                         if (is_of_type(c, print_latex)) {
 189                                                                 c.s << '{';
 190                                                                 i->coeff.print(c);
 191                                                                 c.s << '}';
 192                                                         } else
 193                                                                 i->coeff.print(c);
 194                                                 }
 195                                         }
 196                                 }
 197                         } else
 198                                 Order(power(var-point,i->coeff)).print(c);
 199                 }
 200
 201                 if (precedence <= level)
 202                         c.s << ")";
 203         }
 204 }
 205
 206 int pseries::compare_same_type(const basic & other) const
 207 {
 208         GINAC_ASSERT(is_of_type(other, pseries));
 209         const pseries &o = static_cast<const pseries &>(other);
 210
 211         // first compare the lengths of the series...
 212         if (seq.size()>o.seq.size())
 213                 return 1;
 214         if (seq.size()<o.seq.size())
 215                 return -1;
 216
 217         // ...then the expansion point...
 218         int cmpval = var.compare(o.var);
 219         if (cmpval)
 220                 return cmpval;
 221         cmpval = point.compare(o.point);
 222         if (cmpval)
 223                 return cmpval;
 224
 225         // ...and if that failed the individual elements
 226         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 227         while (it!=seq.end() && o_it!=o.seq.end()) {
 228                 cmpval = it->compare(*o_it);
 229                 if (cmpval)
 230                         return cmpval;
 231                 ++it;
 232                 ++o_it;
 233         }
 234
 235         // so they are equal.
 236         return 0;
 237 }
 238
 239 /** Return the number of operands including a possible order term. */
 240 unsigned pseries::nops(void) const
 241 {
 242         return seq.size();
 243 }
 244
 245 /** Return the ith term in the series when represented as a sum. */
 246 ex pseries::op(int i) const
 247 {
 248         if (i < 0 || unsigned(i) >= seq.size())
 249                 throw (std::out_of_range("op() out of range"));
 250         return seq[i].rest * power(var - point, seq[i].coeff);
 251 }
 252
 253 ex &pseries::let_op(int i)
 254 {
 255         throw (std::logic_error("let_op not defined for pseries"));
 256 }
 257
 258 /** Return degree of highest power of the series.  This is usually the exponent
 259  *  of the Order term.  If s is not the expansion variable of the series, the
 260  *  series is examined termwise. */
 261 int pseries::degree(const ex &s) const
 262 {
 263         if (var.is_equal(s)) {
 264                 // Return last exponent
 265                 if (seq.size())
 266                         return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 267                 else
 268                         return 0;
 269         } else {
 270                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 271                 if (it == itend)
 272                         return 0;
 273                 int max_pow = INT_MIN;
 274                 while (it != itend) {
 275                         int pow = it->rest.degree(s);
 276                         if (pow > max_pow)
 277                                 max_pow = pow;
 278                         ++it;
 279                 }
 280                 return max_pow;
 281         }
 282 }
 283
 284 /** Return degree of lowest power of the series.  This is usually the exponent
 285  *  of the leading term.  If s is not the expansion variable of the series, the
 286  *  series is examined termwise.  If s is the expansion variable but the
 287  *  expansion point is not zero the series is not expanded to find the degree.
 288  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 289 int pseries::ldegree(const ex &s) const
 290 {
 291         if (var.is_equal(s)) {
 292                 // Return first exponent
 293                 if (seq.size())
 294                         return ex_to_numeric((*(seq.begin())).coeff).to_int();
 295                 else
 296                         return 0;
 297         } else {
 298                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 299                 if (it == itend)
 300                         return 0;
 301                 int min_pow = INT_MAX;
 302                 while (it != itend) {
 303                         int pow = it->rest.ldegree(s);
 304                         if (pow < min_pow)
 305                                 min_pow = pow;
 306                         ++it;
 307                 }
 308                 return min_pow;
 309         }
 310 }
 311
 312 /** Return coefficient of degree n in power series if s is the expansion
 313  *  variable.  If the expansion point is nonzero, by definition the n=1
 314  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 315  *  the expansion took place in the s in the first place).
 316  *  If s is not the expansion variable, an attempt is made to convert the
 317  *  series to a polynomial and return the corresponding coefficient from
 318  *  there. */
 319 ex pseries::coeff(const ex &s, int n) const
 320 {
 321         if (var.is_equal(s)) {
 322                 if (seq.size() == 0)
 323                         return _ex0();
 324
 325                 // Binary search in sequence for given power
 326                 numeric looking_for = numeric(n);
 327                 int lo = 0, hi = seq.size() - 1;
 328                 while (lo <= hi) {
 329                         int mid = (lo + hi) / 2;
 330                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 331                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 332                         switch (cmp) {
 333                                 case -1:
 334                                         lo = mid + 1;
 335                                         break;
 336                                 case 0:
 337                                         return seq[mid].rest;
 338                                 case 1:
 339                                         hi = mid - 1;
 340                                         break;
 341                                 default:
 342                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 343                         }
 344                 }
 345                 return _ex0();
 346         } else
 347                 return convert_to_poly().coeff(s, n);
 348 }
 349
 350 /** Does nothing. */
 351 ex pseries::collect(const ex &s, bool distributed) const
 352 {
 353         return *this;
 354 }
 355
 356 /** Evaluate coefficients. */
 357 ex pseries::eval(int level) const
 358 {
 359         if (level == 1)
 360                 return this->hold();
 361
 362         if (level == -max_recursion_level)
 363                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 364
 365         // Construct a new series with evaluated coefficients
 366         epvector new_seq;
 367         new_seq.reserve(seq.size());
 368         epvector::const_iterator it = seq.begin(), itend = seq.end();
 369         while (it != itend) {
 370                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 371                 ++it;
 372         }
 373         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 374 }
 375
 376 /** Evaluate coefficients numerically. */
 377 ex pseries::evalf(int level) const
 378 {
 379         if (level == 1)
 380                 return *this;
 381
 382         if (level == -max_recursion_level)
 383                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 384
 385         // Construct a new series with evaluated coefficients
 386         epvector new_seq;
 387         new_seq.reserve(seq.size());
 388         epvector::const_iterator it = seq.begin(), itend = seq.end();
 389         while (it != itend) {
 390                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 391                 ++it;
 392         }
 393         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 394 }
 395
 396 ex pseries::subs(const lst & ls, const lst & lr) const
 397 {
 398         // If expansion variable is being substituted, convert the series to a
 399         // polynomial and do the substitution there because the result might
 400         // no longer be a power series
 401         if (ls.has(var))
 402                 return convert_to_poly(true).subs(ls, lr);
 403
 404         // Otherwise construct a new series with substituted coefficients and
 405         // expansion point
 406         epvector newseq;
 407         newseq.reserve(seq.size());
 408         epvector::const_iterator it = seq.begin(), itend = seq.end();
 409         while (it != itend) {
 410                 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 411                 ++it;
 412         }
 413         return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
 414 }
 415
 416 /** Implementation of ex::expand() for a power series.  It expands all the
 417  *  terms individually and returns the resulting series as a new pseries. */
 418 ex pseries::expand(unsigned options) const
 419 {
 420         epvector newseq;
 421         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 422                 ex restexp = i->rest.expand();
 423                 if (!restexp.is_zero())
 424                         newseq.push_back(expair(restexp, i->coeff));
 425         }
 426         return (new pseries(relational(var,point), newseq))
 427                 ->setflag(status_flags::dynallocated | status_flags::expanded);
 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 ex pseries::convert_to_poly(bool no_order) const
 457 {
 458         ex e;
 459         epvector::const_iterator it = seq.begin(), itend = seq.end();
 460
 461         while (it != itend) {
 462                 if (is_order_function(it->rest)) {
 463                         if (!no_order)
 464                                 e += Order(power(var - point, it->coeff));
 465                 } else
 466                         e += it->rest * power(var - point, it->coeff);
 467                 ++it;
 468         }
 469         return e;
 470 }
 471
 472 bool pseries::is_terminating(void) const
 473 {
 474         return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
 475 }
 476
 477
 478 /*
 479  *  Implementations of series expansion
 480  */
 481
 482 /** Default implementation of ex::series(). This performs Taylor expansion.
 483  *  @see ex::series */
 484 ex basic::series(const relational & r, int order, unsigned options) const
 485 {
 486         epvector seq;
 487         numeric fac(1);
 488         ex deriv = *this;
 489         ex coeff = deriv.subs(r);
 490         const symbol &s = static_cast<symbol &>(*r.lhs().bp);
 491
 492         if (!coeff.is_zero())
 493                 seq.push_back(expair(coeff, _ex0()));
 494
 495         int n;
 496         for (n=1; n<order; ++n) {
 497                 fac = fac.mul(numeric(n));
 498                 deriv = deriv.diff(s).expand();
 499                 if (deriv.is_zero()) {
 500                         // Series terminates
 501                         return pseries(r, seq);
 502                 }
 503                 coeff = deriv.subs(r);
 504                 if (!coeff.is_zero())
 505                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
 506         }
 507
 508         // Higher-order terms, if present
 509         deriv = deriv.diff(s);
 510         if (!deriv.expand().is_zero())
 511                 seq.push_back(expair(Order(_ex1()), numeric(n)));
 512         return pseries(r, seq);
 513 }
 514
 515
 516 /** Implementation of ex::series() for symbols.
 517  *  @see ex::series */
 518 ex symbol::series(const relational & r, int order, unsigned options) const
 519 {
 520         epvector seq;
 521         const ex point = r.rhs();
 522         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 523         ex s = r.lhs();
 524
 525         if (this->is_equal(*s.bp)) {
 526                 if (order > 0 && !point.is_zero())
 527                         seq.push_back(expair(point, _ex0()));
 528                 if (order > 1)
 529                         seq.push_back(expair(_ex1(), _ex1()));
 530                 else
 531                         seq.push_back(expair(Order(_ex1()), numeric(order)));
 532         } else
 533                 seq.push_back(expair(*this, _ex0()));
 534         return pseries(r, seq);
 535 }
 536
 537
 538 /** Add one series object to another, producing a pseries object that
 539  *  represents the sum.
 540  *
 541  *  @param other  pseries object to add with
 542  *  @return the sum as a pseries */
 543 ex pseries::add_series(const pseries &other) const
 544 {
 545         // Adding two series with different variables or expansion points
 546         // results in an empty (constant) series
 547         if (!is_compatible_to(other)) {
 548                 epvector nul;
 549                 nul.push_back(expair(Order(_ex1()), _ex0()));
 550                 return pseries(relational(var,point), nul);
 551         }
 552
 553         // Series addition
 554         epvector new_seq;
 555         epvector::const_iterator a = seq.begin();
 556         epvector::const_iterator b = other.seq.begin();
 557         epvector::const_iterator a_end = seq.end();
 558         epvector::const_iterator b_end = other.seq.end();
 559         int pow_a = INT_MAX, pow_b = INT_MAX;
 560         for (;;) {
 561                 // If a is empty, fill up with elements from b and stop
 562                 if (a == a_end) {
 563                         while (b != b_end) {
 564                                 new_seq.push_back(*b);
 565                                 ++b;
 566                         }
 567                         break;
 568                 } else
 569                         pow_a = ex_to_numeric((*a).coeff).to_int();
 570
 571                 // If b is empty, fill up with elements from a and stop
 572                 if (b == b_end) {
 573                         while (a != a_end) {
 574                                 new_seq.push_back(*a);
 575                                 ++a;
 576                         }
 577                         break;
 578                 } else
 579                         pow_b = ex_to_numeric((*b).coeff).to_int();
 580
 581                 // a and b are non-empty, compare powers
 582                 if (pow_a < pow_b) {
 583                         // a has lesser power, get coefficient from a
 584                         new_seq.push_back(*a);
 585                         if (is_order_function((*a).rest))
 586                                 break;
 587                         ++a;
 588                 } else if (pow_b < pow_a) {
 589                         // b has lesser power, get coefficient from b
 590                         new_seq.push_back(*b);
 591                         if (is_order_function((*b).rest))
 592                                 break;
 593                         ++b;
 594                 } else {
 595                         // Add coefficient of a and b
 596                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 597                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
 598                                 break;  // Order term ends the sequence
 599                         } else {
 600                                 ex sum = (*a).rest + (*b).rest;
 601                                 if (!(sum.is_zero()))
 602                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 603                                 ++a;
 604                                 ++b;
 605                         }
 606                 }
 607         }
 608         return pseries(relational(var,point), new_seq);
 609 }
 610
 611
 612 /** Implementation of ex::series() for sums. This performs series addition when
 613  *  adding pseries objects.
 614  *  @see ex::series */
 615 ex add::series(const relational & r, int order, unsigned options) const
 616 {
 617         ex acc; // Series accumulator
 618
 619         // Get first term from overall_coeff
 620         acc = overall_coeff.series(r, order, options);
 621
 622         // Add remaining terms
 623         epvector::const_iterator it = seq.begin();
 624         epvector::const_iterator itend = seq.end();
 625         for (; it!=itend; ++it) {
 626                 ex op;
 627                 if (is_ex_exactly_of_type(it->rest, pseries))
 628                         op = it->rest;
 629                 else
 630                         op = it->rest.series(r, order, options);
 631                 if (!it->coeff.is_equal(_ex1()))
 632                         op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
 633
 634                 // Series addition
 635                 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
 636         }
 637         return acc;
 638 }
 639
 640
 641 /** Multiply a pseries object with a numeric constant, producing a pseries
 642  *  object that represents the product.
 643  *
 644  *  @param other  constant to multiply with
 645  *  @return the product as a pseries */
 646 ex pseries::mul_const(const numeric &other) const
 647 {
 648         epvector new_seq;
 649         new_seq.reserve(seq.size());
 650
 651         epvector::const_iterator it = seq.begin(), itend = seq.end();
 652         while (it != itend) {
 653                 if (!is_order_function(it->rest))
 654                         new_seq.push_back(expair(it->rest * other, it->coeff));
 655                 else
 656                         new_seq.push_back(*it);
 657                 ++it;
 658         }
 659         return pseries(relational(var,point), new_seq);
 660 }
 661
 662
 663 /** Multiply one pseries object to another, producing a pseries object that
 664  *  represents the product.
 665  *
 666  *  @param other  pseries object to multiply with
 667  *  @return the product as a pseries */
 668 ex pseries::mul_series(const pseries &other) const
 669 {
 670         // Multiplying two series with different variables or expansion points
 671         // results in an empty (constant) series
 672         if (!is_compatible_to(other)) {
 673                 epvector nul;
 674                 nul.push_back(expair(Order(_ex1()), _ex0()));
 675                 return pseries(relational(var,point), nul);
 676         }
 677
 678         // Series multiplication
 679         epvector new_seq;
 680
 681         int a_max = degree(var);
 682         int b_max = other.degree(var);
 683         int a_min = ldegree(var);
 684         int b_min = other.ldegree(var);
 685         int cdeg_min = a_min + b_min;
 686         int cdeg_max = a_max + b_max;
 687
 688         int higher_order_a = INT_MAX;
 689         int higher_order_b = INT_MAX;
 690         if (is_order_function(coeff(var, a_max)))
 691                 higher_order_a = a_max + b_min;
 692         if (is_order_function(other.coeff(var, b_max)))
 693                 higher_order_b = b_max + a_min;
 694         int higher_order_c = std::min(higher_order_a, higher_order_b);
 695         if (cdeg_max >= higher_order_c)
 696                 cdeg_max = higher_order_c - 1;
 697
 698         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 699                 ex co = _ex0();
 700                 // c(i)=a(0)b(i)+...+a(i)b(0)
 701                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 702                         ex a_coeff = coeff(var, i);
 703                         ex b_coeff = other.coeff(var, cdeg-i);
 704                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 705                                 co += a_coeff * b_coeff;
 706                 }
 707                 if (!co.is_zero())
 708                         new_seq.push_back(expair(co, numeric(cdeg)));
 709         }
 710         if (higher_order_c < INT_MAX)
 711                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 712         return pseries(relational(var, point), new_seq);
 713 }
 714
 715
 716 /** Implementation of ex::series() for product. This performs series
 717  *  multiplication when multiplying series.
 718  *  @see ex::series */
 719 ex mul::series(const relational & r, int order, unsigned options) const
 720 {
 721         ex acc; // Series accumulator
 722
 723         // Get first term from overall_coeff
 724         acc = overall_coeff.series(r, order, options);
 725
 726         // Multiply with remaining terms
 727         epvector::const_iterator it = seq.begin();
 728         epvector::const_iterator itend = seq.end();
 729         for (; it!=itend; ++it) {
 730                 ex op = it->rest;
 731                 if (op.info(info_flags::numeric)) {
 732                         // series * const (special case, faster)
 733                         ex f = power(op, it->coeff);
 734                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 735                         continue;
 736                 } else if (!is_ex_exactly_of_type(op, pseries))
 737                         op = op.series(r, order, options);
 738                 if (!it->coeff.is_equal(_ex1()))
 739                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 740
 741                 // Series multiplication
 742                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 743         }
 744         return acc;
 745 }
 746
 747
 748 /** Compute the p-th power of a series.
 749  *
 750  *  @param p  power to compute
 751  *  @param deg  truncation order of series calculation */
 752 ex pseries::power_const(const numeric &p, int deg) const
 753 {
 754         // method:
 755         // let A(x) be this series and for the time being let it start with a
 756         // constant (later we'll generalize):
 757         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 758         // We want to compute
 759         //     C(x) = A(x)^p
 760         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 761         // Taking the derivative on both sides and multiplying with A(x) one
 762         // immediately arrives at
 763         //     C'(x)*A(x) = p*C(x)*A'(x)
 764         // Multiplying this out and comparing coefficients we get the recurrence
 765         // formula
 766         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 767         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 768         // which can easily be solved given the starting value c_0 = (a_0)^p.
 769         // For the more general case where the leading coefficient of A(x) is not
 770         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 771         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 772         // then of course x^(p*m) but the recurrence formula still holds.
 773
 774         if (seq.size()==0) {
 775                 // as a spacial case, handle the empty (zero) series honoring the
 776                 // usual power laws such as implemented in power::eval()
 777                 if (p.real().is_zero())
 778                         throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
 779                 else if (p.real().is_negative())
 780                         throw (pole_error("pseries::power_const(): division by zero",1));
 781                 else
 782                         return *this;
 783         }
 784
 785         int ldeg = ldegree(var);
 786
 787         // Compute coefficients of the powered series
 788         exvector co;
 789         co.reserve(deg);
 790         co.push_back(power(coeff(var, 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(var, 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(var, 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 } // namespace GiNaC