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