ginac/pseries.cpp

   1 /** @file pseries.cpp
   2  *
   3  *  Implementation of class for extended truncated power series and
   4  *  methods for series expansion. */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  22  */
  23
  24 #include <stdexcept>
  25
  26 #include "pseries.h"
  27 #include "add.h"
  28 #include "inifcns.h"
  29 #include "lst.h"
  30 #include "mul.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "symbol.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36 #include "debugmsg.h"
  37
  38 #ifndef NO_NAMESPACE_GINAC
  39 namespace GiNaC {
  40 #endif // ndef NO_NAMESPACE_GINAC
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  43
  44 /*
  45  *  Default constructor, destructor, copy constructor, assignment operator and helpers
  46  */
  47
  48 pseries::pseries() : basic(TINFO_pseries)
  49 {
  50         debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
  51 }
  52
  53 void pseries::copy(const pseries &other)
  54 {
  55         inherited::copy(other);
  56         seq = other.seq;
  57         var = other.var;
  58         point = other.point;
  59 }
  60
  61 void pseries::destroy(bool call_parent)
  62 {
  63         if (call_parent)
  64                 inherited::destroy(call_parent);
  65 }
  66
  67
  68 /*
  69  *  Other constructors
  70  */
  71
  72 /** Construct pseries from a vector of coefficients and powers.
  73  *  expair.rest holds the coefficient, expair.coeff holds the power.
  74  *  The powers must be integers (positive or negative) and in ascending order;
  75  *  the last coefficient can be Order(_ex1()) to represent a truncated,
  76  *  non-terminating series.
  77  *
  78  *  @param rel_  expansion variable and point (must hold a relational)
  79  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  80  *  @return newly constructed pseries */
  81 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  82 {
  83         debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
  84         GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
  85         GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
  86         point = rel_.rhs();
  87         var = *static_cast<symbol *>(rel_.lhs().bp);
  88 }
  89
  90
  91 /*
  92  *  Archiving
  93  */
  94
  95 /** Construct object from archive_node. */
  96 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  97 {
  98         debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
  99         for (unsigned int i=0; true; ++i) {
 100                 ex rest;
 101                 ex coeff;
 102                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
 103                         seq.push_back(expair(rest, coeff));
 104                 else
 105                         break;
 106         }
 107         n.find_ex("var", var, sym_lst);
 108         n.find_ex("point", point, sym_lst);
 109 }
 110
 111 /** Unarchive the object. */
 112 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
 113 {
 114         return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
 115 }
 116
 117 /** Archive the object. */
 118 void pseries::archive(archive_node &n) const
 119 {
 120         inherited::archive(n);
 121         epvector::const_iterator i = seq.begin(), iend = seq.end();
 122         while (i != iend) {
 123                 n.add_ex("coeff", i->rest);
 124                 n.add_ex("power", i->coeff);
 125                 ++i;
 126         }
 127         n.add_ex("var", var);
 128         n.add_ex("point", point);
 129 }
 130
 131 //////////
 132 // functions overriding virtual functions from bases classes
 133 //////////
 134
 135 void pseries::print(std::ostream &os, unsigned upper_precedence) const
 136 {
 137         debugmsg("pseries print", LOGLEVEL_PRINT);
 138         if (precedence<=upper_precedence) os << "(";
 139         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 140                 // omit zero terms
 141                 if (i->rest.is_zero())
 142                         continue;
 143                 // print a sign, if needed
 144                 if (i!=seq.begin())
 145                         os << '+';
 146                 if (!is_order_function(i->rest)) {
 147                         // print 'rest', i.e. the expansion coefficient
 148                         if (i->rest.info(info_flags::numeric) &&
 149                                 i->rest.info(info_flags::positive)) {
 150                                 os << i->rest;
 151                         } else
 152                                 os << "(" << i->rest << ')';
 153                         // print 'coeff', something like (x-1)^42
 154                         if (!i->coeff.is_zero()) {
 155                                 os << '*';
 156                                 if (!point.is_zero())
 157                                         os << '(' << var-point << ')';
 158                                 else
 159                                         os << var;
 160                                 if (i->coeff.compare(_ex1())) {
 161                                         os << '^';
 162                                         if (i->coeff.info(info_flags::negative))
 163                                                 os << '(' << i->coeff << ')';
 164                                         else
 165                                                 os << i->coeff;
 166                                 }
 167                         }
 168                 } else {
 169                         os << Order(power(var-point,i->coeff));
 170                 }
 171         }
 172         if (precedence<=upper_precedence) os << ")";
 173 }
 174
 175
 176 void pseries::printraw(std::ostream &os) const
 177 {
 178         debugmsg("pseries printraw", LOGLEVEL_PRINT);
 179         os << "pseries(" << var << ";" << point << ";";
 180         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 181                 os << "(" << (*i).rest << "," << (*i).coeff << "),";
 182         }
 183         os << ")";
 184 }
 185
 186
 187 void pseries::printtree(std::ostream & os, unsigned indent) const
 188 {
 189         debugmsg("pseries printtree",LOGLEVEL_PRINT);
 190         os << std::string(indent,' ') << "pseries "
 191            << ", hash=" << hashvalue
 192            << " (0x" << std::hex << hashvalue << std::dec << ")"
 193            << ", flags=" << flags << std::endl;
 194         for (unsigned i=0; i<seq.size(); ++i) {
 195                 seq[i].rest.printtree(os,indent+delta_indent);
 196                 seq[i].coeff.printtree(os,indent+delta_indent);
 197                 if (i!=seq.size()-1)
 198                         os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
 199         }
 200         var.printtree(os, indent+delta_indent);
 201         point.printtree(os, indent+delta_indent);
 202 }
 203
 204 int pseries::compare_same_type(const basic & other) const
 205 {
 206         GINAC_ASSERT(is_of_type(other, pseries));
 207         const pseries &o = static_cast<const pseries &>(other);
 208
 209         int cmpval = var.compare(o.var);
 210         if (cmpval)
 211                 return cmpval;
 212         cmpval = point.compare(o.point);
 213         if (cmpval)
 214                 return cmpval;
 215
 216         epvector::const_iterator it1 = seq.begin(), it2 = o.seq.begin(), it1end = seq.end(), it2end = o.seq.end();
 217         while ((it1 != it1end) && (it2 != it2end)) {
 218                 cmpval = it1->compare(*it2);
 219                 if (cmpval)
 220                         return cmpval;
 221                 it1++; it2++;
 222         }
 223         if (it1 == it1end)
 224                 return it2 == it2end ? 0 : -1;
 225
 226         return 0;
 227 }
 228
 229 /** Return the number of operands including a possible order term. */
 230 unsigned pseries::nops(void) const
 231 {
 232         return seq.size();
 233 }
 234
 235
 236 /** Return the ith term in the series when represented as a sum. */
 237 ex pseries::op(int i) const
 238 {
 239         if (i < 0 || unsigned(i) >= seq.size())
 240                 throw (std::out_of_range("op() out of range"));
 241         return seq[i].rest * power(var - point, seq[i].coeff);
 242 }
 243
 244
 245 ex &pseries::let_op(int i)
 246 {
 247         throw (std::logic_error("let_op not defined for pseries"));
 248 }
 249
 250
 251 /** Return degree of highest power of the series.  This is usually the exponent
 252  *  of the Order term.  If s is not the expansion variable of the series, the
 253  *  series is examined termwise. */
 254 int pseries::degree(const symbol &s) const
 255 {
 256         if (var.is_equal(s)) {
 257                 // Return last exponent
 258                 if (seq.size())
 259                         return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
 260                 else
 261                         return 0;
 262         } else {
 263                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 264                 if (it == itend)
 265                         return 0;
 266                 int max_pow = INT_MIN;
 267                 while (it != itend) {
 268                         int pow = it->rest.degree(s);
 269                         if (pow > max_pow)
 270                                 max_pow = pow;
 271                         ++it;
 272                 }
 273                 return max_pow;
 274         }
 275 }
 276
 277 /** Return degree of lowest power of the series.  This is usually the exponent
 278  *  of the leading term.  If s is not the expansion variable of the series, the
 279  *  series is examined termwise.  If s is the expansion variable but the
 280  *  expansion point is not zero the series is not expanded to find the degree.
 281  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 282 int pseries::ldegree(const symbol &s) const
 283 {
 284         if (var.is_equal(s)) {
 285                 // Return first exponent
 286                 if (seq.size())
 287                         return ex_to_numeric((*(seq.begin())).coeff).to_int();
 288                 else
 289                         return 0;
 290         } else {
 291                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 292                 if (it == itend)
 293                         return 0;
 294                 int min_pow = INT_MAX;
 295                 while (it != itend) {
 296                         int pow = it->rest.ldegree(s);
 297                         if (pow < min_pow)
 298                                 min_pow = pow;
 299                         ++it;
 300                 }
 301                 return min_pow;
 302         }
 303 }
 304
 305 ex pseries::coeff(const symbol &s, int n) const
 306 {
 307         if (var.is_equal(s)) {
 308                 if (seq.size() == 0)
 309                         return _ex0();
 310
 311                 // Binary search in sequence for given power
 312                 numeric looking_for = numeric(n);
 313                 int lo = 0, hi = seq.size() - 1;
 314                 while (lo <= hi) {
 315                         int mid = (lo + hi) / 2;
 316                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
 317                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
 318                         switch (cmp) {
 319                                 case -1:
 320                                         lo = mid + 1;
 321                                         break;
 322                                 case 0:
 323                                         return seq[mid].rest;
 324                                 case 1:
 325                                         hi = mid - 1;
 326                                         break;
 327                                 default:
 328                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 329                         }
 330                 }
 331                 return _ex0();
 332         } else
 333                 return convert_to_poly().coeff(s, n);
 334 }
 335
 336
 337 ex pseries::collect(const symbol &s) const
 338 {
 339         return *this;
 340 }
 341
 342
 343 /** Evaluate coefficients. */
 344 ex pseries::eval(int level) const
 345 {
 346         if (level == 1)
 347                 return this->hold();
 348
 349         if (level == -max_recursion_level)
 350                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 351
 352         // Construct a new series with evaluated coefficients
 353         epvector new_seq;
 354         new_seq.reserve(seq.size());
 355         epvector::const_iterator it = seq.begin(), itend = seq.end();
 356         while (it != itend) {
 357                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 358                 ++it;
 359         }
 360         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 361 }
 362
 363
 364 /** Evaluate coefficients numerically. */
 365 ex pseries::evalf(int level) const
 366 {
 367         if (level == 1)
 368                 return *this;
 369
 370         if (level == -max_recursion_level)
 371                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 372
 373         // Construct a new series with evaluated coefficients
 374         epvector new_seq;
 375         new_seq.reserve(seq.size());
 376         epvector::const_iterator it = seq.begin(), itend = seq.end();
 377         while (it != itend) {
 378                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 379                 ++it;
 380         }
 381         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 382 }
 383
 384
 385 ex pseries::subs(const lst & ls, const lst & lr) const
 386 {
 387         // If expansion variable is being substituted, convert the series to a
 388         // polynomial and do the substitution there because the result might
 389         // no longer be a power series
 390         if (ls.has(var))
 391                 return convert_to_poly(true).subs(ls, lr);
 392
 393         // Otherwise construct a new series with substituted coefficients and
 394         // expansion point
 395         epvector newseq;
 396         newseq.reserve(seq.size());
 397         epvector::const_iterator it = seq.begin(), itend = seq.end();
 398         while (it != itend) {
 399                 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
 400                 ++it;
 401         }
 402         return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
 403 }
 404
 405
 406 /** Implementation of ex::expand() for a power series.  It expands all the
 407  *  terms individually and returns the resulting series as a new pseries.
 408  *  @see ex::diff */
 409 ex pseries::expand(unsigned options) const
 410 {
 411         epvector newseq;
 412         newseq.reserve(seq.size());
 413         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
 414                 newseq.push_back(expair(i->rest.expand(), i->coeff));
 415         return (new pseries(relational(var,point), newseq))
 416                 ->setflag(status_flags::dynallocated | status_flags::expanded);
 417 }
 418
 419
 420 /** Implementation of ex::diff() for a power series.  It treats the series as a
 421  *  polynomial.
 422  *  @see ex::diff */
 423 ex pseries::derivative(const symbol & s) const
 424 {
 425         if (s == var) {
 426                 epvector new_seq;
 427                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 428
 429                 // FIXME: coeff might depend on var
 430                 while (it != itend) {
 431                         if (is_order_function(it->rest)) {
 432                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 433                         } else {
 434                                 ex c = it->rest * it->coeff;
 435                                 if (!c.is_zero())
 436                                         new_seq.push_back(expair(c, it->coeff - 1));
 437                         }
 438                         ++it;
 439                 }
 440                 return pseries(relational(var,point), new_seq);
 441         } else {
 442                 return *this;
 443         }
 444 }
 445
 446
 447 /*
 448  *  Construct ordinary polynomial out of series
 449  */
 450
 451 /** Convert a pseries object to an ordinary polynomial.
 452  *
 453  *  @param no_order flag: discard higher order terms */
 454 ex pseries::convert_to_poly(bool no_order) const
 455 {
 456         ex e;
 457         epvector::const_iterator it = seq.begin(), itend = seq.end();
 458
 459         while (it != itend) {
 460                 if (is_order_function(it->rest)) {
 461                         if (!no_order)
 462                                 e += Order(power(var - point, it->coeff));
 463                 } else
 464                         e += it->rest * power(var - point, it->coeff);
 465                 ++it;
 466         }
 467         return e;
 468 }
 469
 470 /** Returns true if there is no order term, i.e. the series terminates and
 471  *  false otherwise. */
 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  *  Implementation 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, numeric(0)));
 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         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 524
 525         if (this->is_equal(*s)) {
 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         const symbol *s = static_cast<symbol *>(var.bp);
 682         int a_max = degree(*s);
 683         int b_max = other.degree(*s);
 684         int a_min = ldegree(*s);
 685         int b_min = other.ldegree(*s);
 686         int cdeg_min = a_min + b_min;
 687         int cdeg_max = a_max + b_max;
 688
 689         int higher_order_a = INT_MAX;
 690         int higher_order_b = INT_MAX;
 691         if (is_order_function(coeff(*s, a_max)))
 692                 higher_order_a = a_max + b_min;
 693         if (is_order_function(other.coeff(*s, b_max)))
 694                 higher_order_b = b_max + a_min;
 695         int higher_order_c = std::min(higher_order_a, higher_order_b);
 696         if (cdeg_max >= higher_order_c)
 697                 cdeg_max = higher_order_c - 1;
 698
 699         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 700                 ex co = _ex0();
 701                 // c(i)=a(0)b(i)+...+a(i)b(0)
 702                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 703                         ex a_coeff = coeff(*s, i);
 704                         ex b_coeff = other.coeff(*s, cdeg-i);
 705                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 706                                 co += a_coeff * b_coeff;
 707                 }
 708                 if (!co.is_zero())
 709                         new_seq.push_back(expair(co, numeric(cdeg)));
 710         }
 711         if (higher_order_c < INT_MAX)
 712                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
 713         return pseries(relational(var,point), new_seq);
 714 }
 715
 716
 717 /** Implementation of ex::series() for product. This performs series
 718  *  multiplication when multiplying series.
 719  *  @see ex::series */
 720 ex mul::series(const relational & r, int order, unsigned options) const
 721 {
 722         ex acc; // Series accumulator
 723
 724         // Get first term from overall_coeff
 725         acc = overall_coeff.series(r, order, options);
 726
 727         // Multiply with remaining terms
 728         epvector::const_iterator it = seq.begin();
 729         epvector::const_iterator itend = seq.end();
 730         for (; it!=itend; ++it) {
 731                 ex op = it->rest;
 732                 if (op.info(info_flags::numeric)) {
 733                         // series * const (special case, faster)
 734                         ex f = power(op, it->coeff);
 735                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
 736                         continue;
 737                 } else if (!is_ex_exactly_of_type(op, pseries))
 738                         op = op.series(r, order, options);
 739                 if (!it->coeff.is_equal(_ex1()))
 740                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
 741
 742                 // Series multiplication
 743                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
 744         }
 745         return acc;
 746 }
 747
 748
 749 /** Compute the p-th power of a series.
 750  *
 751  *  @param p  power to compute
 752  *  @param deg  truncation order of series calculation */
 753 ex pseries::power_const(const numeric &p, int deg) const
 754 {
 755         // method:
 756         // let A(x) be this series and for the time being let it start with a
 757         // constant (later we'll generalize):
 758         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 759         // We want to compute
 760         //     C(x) = A(x)^p
 761         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 762         // Taking the derivative on both sides and multiplying with A(x) one
 763         // immediately arrives at
 764         //     C'(x)*A(x) = p*C(x)*A'(x)
 765         // Multiplying this out and comparing coefficients we get the recurrence
 766         // formula
 767         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 768         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 769         // which can easily be solved given the starting value c_0 = (a_0)^p.
 770         // For the more general case where the leading coefficient of A(x) is not
 771         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 772         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 773         // then of course x^(p*m) but the recurrence formula still holds.
 774         const symbol *s = static_cast<symbol *>(var.bp);
 775         int ldeg = ldegree(*s);
 776
 777         // Compute coefficients of the powered series
 778         exvector co;
 779         co.reserve(deg);
 780         co.push_back(power(coeff(*s, ldeg), p));
 781         bool all_sums_zero = true;
 782         for (int i=1; i<deg; ++i) {
 783                 ex sum = _ex0();
 784                 for (int j=1; j<=i; ++j) {
 785                         ex c = coeff(*s, j + ldeg);
 786                         if (is_order_function(c)) {
 787                                 co.push_back(Order(_ex1()));
 788                                 break;
 789                         } else
 790                                 sum += (p * j - (i - j)) * co[i - j] * c;
 791                 }
 792                 if (!sum.is_zero())
 793                         all_sums_zero = false;
 794                 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
 795         }
 796
 797         // Construct new series (of non-zero coefficients)
 798         epvector new_seq;
 799         bool higher_order = false;
 800         for (int i=0; i<deg; ++i) {
 801                 if (!co[i].is_zero())
 802                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
 803                 if (is_order_function(co[i])) {
 804                         higher_order = true;
 805                         break;
 806                 }
 807         }
 808         if (!higher_order && !all_sums_zero)
 809                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
 810         return pseries(relational(var,point), new_seq);
 811 }
 812
 813
 814 /** Return a new pseries object with the powers shifted by deg. */
 815 pseries pseries::shift_exponents(int deg) const
 816 {
 817         epvector newseq(seq);
 818         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
 819                 i->coeff = i->coeff + deg;
 820         return pseries(relational(var, point), newseq);
 821 }
 822
 823
 824 /** Implementation of ex::series() for powers. This performs Laurent expansion
 825  *  of reciprocals of series at singularities.
 826  *  @see ex::series */
 827 ex power::series(const relational & r, int order, unsigned options) const
 828 {
 829         ex e;
 830         if (!is_ex_exactly_of_type(basis, pseries)) {
 831                 // Basis is not a series, may there be a singularity?
 832                 bool must_expand_basis = false;
 833                 try {
 834                         basis.subs(r);
 835                 } catch (pole_error) {
 836                         must_expand_basis = true;
 837                 }
 838
 839                 // Is the expression of type something^(-int)?
 840                 if (!must_expand_basis && !exponent.info(info_flags::negint))
 841                         return basic::series(r, order, options);
 842
 843                 // Is the expression of type 0^something?
 844                 if (!must_expand_basis && !basis.subs(r).is_zero())
 845                         return basic::series(r, order, options);
 846
 847                 // Singularity encountered, expand basis into series
 848                 e = basis.series(r, order, options);
 849         } else {
 850                 // Basis is a series
 851                 e = basis;
 852         }
 853
 854         // Power e
 855         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 856 }
 857
 858
 859 /** Re-expansion of a pseries object. */
 860 ex pseries::series(const relational & r, int order, unsigned options) const
 861 {
 862         const ex p = r.rhs();
 863         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
 864         const symbol *s = static_cast<symbol *>(r.lhs().bp);
 865
 866         if (var.is_equal(*s) && point.is_equal(p)) {
 867                 if (order > degree(*s))
 868                         return *this;
 869                 else {
 870                         epvector new_seq;
 871                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 872                         while (it != itend) {
 873                                 int o = ex_to_numeric(it->coeff).to_int();
 874                                 if (o >= order) {
 875                                         new_seq.push_back(expair(Order(_ex1()), o));
 876                                         break;
 877                                 }
 878                                 new_seq.push_back(*it);
 879                                 ++it;
 880                         }
 881                         return pseries(r, new_seq);
 882                 }
 883         } else
 884                 return convert_to_poly().series(r, order, options);
 885 }
 886
 887
 888 /** Compute the truncated series expansion of an expression.
 889  *  This function returns an expression containing an object of class pseries
 890  *  to represent the series. If the series does not terminate within the given
 891  *  truncation order, the last term of the series will be an order term.
 892  *
 893  *  @param r  expansion relation, lhs holds variable and rhs holds point
 894  *  @param order  truncation order of series calculations
 895  *  @param options  of class series_options
 896  *  @return an expression holding a pseries object */
 897 ex ex::series(const ex & r, int order, unsigned options) const
 898 {
 899         GINAC_ASSERT(bp!=0);
 900         ex e;
 901         relational rel_;
 902
 903         if (is_ex_exactly_of_type(r,relational))
 904                 rel_ = ex_to_relational(r);
 905         else if (is_ex_exactly_of_type(r,symbol))
 906                 rel_ = relational(r,_ex0());
 907         else
 908                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 909
 910         try {
 911                 e = bp->series(rel_, order, options);
 912         } catch (std::exception &x) {
 913                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 914         }
 915         return e;
 916 }
 917
 918 //////////
 919 // static member variables
 920 //////////
 921
 922 // protected
 923
 924 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
 925
 926 #ifndef NO_NAMESPACE_GINAC
 927 } // namespace GiNaC
 928 #endif // ndef NO_NAMESPACE_GINAC