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