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) 1999-2002 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 <iostream>
  25 #include <stdexcept>
  26
  27 #include "pseries.h"
  28 #include "add.h"
  29 #include "inifcns.h" // for Order function
  30 #include "lst.h"
  31 #include "mul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "operators.h"
  35 #include "symbol.h"
  36 #include "print.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  43
  44
  45 /*
  46  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  47  */
  48
  49 pseries::pseries() : inherited(TINFO_pseries) { }
  50
  51 void pseries::copy(const pseries &other)
  52 {
  53         inherited::copy(other);
  54         seq = other.seq;
  55         var = other.var;
  56         point = other.point;
  57 }
  58
  59 DEFAULT_DESTROY(pseries)
  60
  61
  62 /*
  63  *  Other ctors
  64  */
  65
  66 /** Construct pseries from a vector of coefficients and powers.
  67  *  expair.rest holds the coefficient, expair.coeff holds the power.
  68  *  The powers must be integers (positive or negative) and in ascending order;
  69  *  the last coefficient can be Order(_ex1) to represent a truncated,
  70  *  non-terminating series.
  71  *
  72  *  @param rel_  expansion variable and point (must hold a relational)
  73  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  74  *  @return newly constructed pseries */
  75 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  76 {
  77         GINAC_ASSERT(is_exactly_a<relational>(rel_));
  78         GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
  79         point = rel_.rhs();
  80         var = rel_.lhs();
  81 }
  82
  83
  84 /*
  85  *  Archiving
  86  */
  87
  88 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  89 {
  90         for (unsigned int i=0; true; ++i) {
  91                 ex rest;
  92                 ex coeff;
  93                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  94                         seq.push_back(expair(rest, coeff));
  95                 else
  96                         break;
  97         }
  98         n.find_ex("var", var, sym_lst);
  99         n.find_ex("point", point, sym_lst);
 100 }
 101
 102 void pseries::archive(archive_node &n) const
 103 {
 104         inherited::archive(n);
 105         epvector::const_iterator i = seq.begin(), iend = seq.end();
 106         while (i != iend) {
 107                 n.add_ex("coeff", i->rest);
 108                 n.add_ex("power", i->coeff);
 109                 ++i;
 110         }
 111         n.add_ex("var", var);
 112         n.add_ex("point", point);
 113 }
 114
 115 DEFAULT_UNARCHIVE(pseries)
 116
 117 //////////
 118 // functions overriding virtual functions from base classes
 119 //////////
 120
 121 void pseries::print(const print_context & c, unsigned level) const
 122 {
 123         if (is_a<print_tree>(c)) {
 124
 125                 c.s << std::string(level, ' ') << class_name()
 126                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 127                     << std::endl;
 128                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 129                 unsigned num = seq.size();
 130                 for (unsigned i=0; i<num; ++i) {
 131                         seq[i].rest.print(c, level + delta_indent);
 132                         seq[i].coeff.print(c, level + delta_indent);
 133                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 134                 }
 135                 var.print(c, level + delta_indent);
 136                 point.print(c, level + delta_indent);
 137
 138         } else if (is_a<print_python_repr>(c)) {
 139                 c.s << class_name() << "(relational(";
 140                 var.print(c);
 141                 c.s << ',';
 142                 point.print(c);
 143                 c.s << "),[";
 144                 unsigned num = seq.size();
 145                 for (unsigned i=0; i<num; ++i) {
 146                         if (i)
 147                                 c.s << ',';
 148                         c.s << '(';
 149                         seq[i].rest.print(c);
 150                         c.s << ',';
 151                         seq[i].coeff.print(c);
 152                         c.s << ')';
 153                 }
 154                 c.s << "])";
 155         } else {
 156
 157                 if (precedence() <= level)
 158                         c.s << "(";
 159
 160                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
 161                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
 162
 163                 // objects of type pseries must not have any zero entries, so the
 164                 // trivial (zero) pseries needs a special treatment here:
 165                 if (seq.empty())
 166                         c.s << '0';
 167                 epvector::const_iterator i = seq.begin(), end = seq.end();
 168                 while (i != end) {
 169                         // print a sign, if needed
 170                         if (i != seq.begin())
 171                                 c.s << '+';
 172                         if (!is_order_function(i->rest)) {
 173                                 // print 'rest', i.e. the expansion coefficient
 174                                 if (i->rest.info(info_flags::numeric) &&
 175                                         i->rest.info(info_flags::positive)) {
 176                                         i->rest.print(c);
 177                                 } else {
 178                                         c.s << par_open;
 179                                         i->rest.print(c);
 180                                         c.s << par_close;
 181                                 }
 182                                 // print 'coeff', something like (x-1)^42
 183                                 if (!i->coeff.is_zero()) {
 184                                         if (is_a<print_latex>(c))
 185                                                 c.s << ' ';
 186                                         else
 187                                                 c.s << '*';
 188                                         if (!point.is_zero()) {
 189                                                 c.s << par_open;
 190                                                 (var-point).print(c);
 191                                                 c.s << par_close;
 192                                         } else
 193                                                 var.print(c);
 194                                         if (i->coeff.compare(_ex1)) {
 195                                                 if (is_a<print_python>(c))
 196                                                         c.s << "**";
 197                                                 else
 198                                                         c.s << '^';
 199                                                 if (i->coeff.info(info_flags::negative)) {
 200                                                         c.s << par_open;
 201                                                         i->coeff.print(c);
 202                                                         c.s << par_close;
 203                                                 } else {
 204                                                         if (is_a<print_latex>(c)) {
 205                                                                 c.s << '{';
 206                                                                 i->coeff.print(c);
 207                                                                 c.s << '}';
 208                                                         } else
 209                                                                 i->coeff.print(c);
 210                                                 }
 211                                         }
 212                                 }
 213                         } else
 214                                 Order(power(var-point,i->coeff)).print(c);
 215                         ++i;
 216                 }
 217
 218                 if (precedence() <= level)
 219                         c.s << ")";
 220         }
 221 }
 222
 223 int pseries::compare_same_type(const basic & other) const
 224 {
 225         GINAC_ASSERT(is_a<pseries>(other));
 226         const pseries &o = static_cast<const pseries &>(other);
 227
 228         // first compare the lengths of the series...
 229         if (seq.size()>o.seq.size())
 230                 return 1;
 231         if (seq.size()<o.seq.size())
 232                 return -1;
 233
 234         // ...then the expansion point...
 235         int cmpval = var.compare(o.var);
 236         if (cmpval)
 237                 return cmpval;
 238         cmpval = point.compare(o.point);
 239         if (cmpval)
 240                 return cmpval;
 241
 242         // ...and if that failed the individual elements
 243         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 244         while (it!=seq.end() && o_it!=o.seq.end()) {
 245                 cmpval = it->compare(*o_it);
 246                 if (cmpval)
 247                         return cmpval;
 248                 ++it;
 249                 ++o_it;
 250         }
 251
 252         // so they are equal.
 253         return 0;
 254 }
 255
 256 /** Return the number of operands including a possible order term. */
 257 unsigned pseries::nops(void) const
 258 {
 259         return seq.size();
 260 }
 261
 262 /** Return the ith term in the series when represented as a sum. */
 263 ex pseries::op(int i) const
 264 {
 265         if (i < 0 || unsigned(i) >= seq.size())
 266                 throw (std::out_of_range("op() out of range"));
 267         return seq[i].rest * power(var - point, seq[i].coeff);
 268 }
 269
 270 ex &pseries::let_op(int i)
 271 {
 272         throw (std::logic_error("let_op not defined for pseries"));
 273 }
 274
 275 /** Return degree of highest power of the series.  This is usually the exponent
 276  *  of the Order term.  If s is not the expansion variable of the series, the
 277  *  series is examined termwise. */
 278 int pseries::degree(const ex &s) const
 279 {
 280         if (var.is_equal(s)) {
 281                 // Return last exponent
 282                 if (seq.size())
 283                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 284                 else
 285                         return 0;
 286         } else {
 287                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 288                 if (it == itend)
 289                         return 0;
 290                 int max_pow = INT_MIN;
 291                 while (it != itend) {
 292                         int pow = it->rest.degree(s);
 293                         if (pow > max_pow)
 294                                 max_pow = pow;
 295                         ++it;
 296                 }
 297                 return max_pow;
 298         }
 299 }
 300
 301 /** Return degree of lowest power of the series.  This is usually the exponent
 302  *  of the leading term.  If s is not the expansion variable of the series, the
 303  *  series is examined termwise.  If s is the expansion variable but the
 304  *  expansion point is not zero the series is not expanded to find the degree.
 305  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 306 int pseries::ldegree(const ex &s) const
 307 {
 308         if (var.is_equal(s)) {
 309                 // Return first exponent
 310                 if (seq.size())
 311                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 312                 else
 313                         return 0;
 314         } else {
 315                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 316                 if (it == itend)
 317                         return 0;
 318                 int min_pow = INT_MAX;
 319                 while (it != itend) {
 320                         int pow = it->rest.ldegree(s);
 321                         if (pow < min_pow)
 322                                 min_pow = pow;
 323                         ++it;
 324                 }
 325                 return min_pow;
 326         }
 327 }
 328
 329 /** Return coefficient of degree n in power series if s is the expansion
 330  *  variable.  If the expansion point is nonzero, by definition the n=1
 331  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 332  *  the expansion took place in the s in the first place).
 333  *  If s is not the expansion variable, an attempt is made to convert the
 334  *  series to a polynomial and return the corresponding coefficient from
 335  *  there. */
 336 ex pseries::coeff(const ex &s, int n) const
 337 {
 338         if (var.is_equal(s)) {
 339                 if (seq.empty())
 340                         return _ex0;
 341
 342                 // Binary search in sequence for given power
 343                 numeric looking_for = numeric(n);
 344                 int lo = 0, hi = seq.size() - 1;
 345                 while (lo <= hi) {
 346                         int mid = (lo + hi) / 2;
 347                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 348                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 349                         switch (cmp) {
 350                                 case -1:
 351                                         lo = mid + 1;
 352                                         break;
 353                                 case 0:
 354                                         return seq[mid].rest;
 355                                 case 1:
 356                                         hi = mid - 1;
 357                                         break;
 358                                 default:
 359                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 360                         }
 361                 }
 362                 return _ex0;
 363         } else
 364                 return convert_to_poly().coeff(s, n);
 365 }
 366
 367 /** Does nothing. */
 368 ex pseries::collect(const ex &s, bool distributed) const
 369 {
 370         return *this;
 371 }
 372
 373 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 374 ex pseries::eval(int level) const
 375 {
 376         if (level == 1)
 377                 return this->hold();
 378
 379         if (level == -max_recursion_level)
 380                 throw (std::runtime_error("pseries::eval(): 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.eval(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 /** Evaluate coefficients numerically. */
 394 ex pseries::evalf(int level) const
 395 {
 396         if (level == 1)
 397                 return *this;
 398
 399         if (level == -max_recursion_level)
 400                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 401
 402         // Construct a new series with evaluated coefficients
 403         epvector new_seq;
 404         new_seq.reserve(seq.size());
 405         epvector::const_iterator it = seq.begin(), itend = seq.end();
 406         while (it != itend) {
 407                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 408                 ++it;
 409         }
 410         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 411 }
 412
 413 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
 414 {
 415         // If expansion variable is being substituted, convert the series to a
 416         // polynomial and do the substitution there because the result might
 417         // no longer be a power series
 418         if (ls.has(var))
 419                 return convert_to_poly(true).subs(ls, lr, no_pattern);
 420
 421         // Otherwise construct a new series with substituted coefficients and
 422         // expansion point
 423         epvector newseq;
 424         newseq.reserve(seq.size());
 425         epvector::const_iterator it = seq.begin(), itend = seq.end();
 426         while (it != itend) {
 427                 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
 428                 ++it;
 429         }
 430         return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
 431 }
 432
 433 /** Implementation of ex::expand() for a power series.  It expands all the
 434  *  terms individually and returns the resulting series as a new pseries. */
 435 ex pseries::expand(unsigned options) const
 436 {
 437         epvector newseq;
 438         epvector::const_iterator i = seq.begin(), end = seq.end();
 439         while (i != end) {
 440                 ex restexp = i->rest.expand();
 441                 if (!restexp.is_zero())
 442                         newseq.push_back(expair(restexp, i->coeff));
 443                 ++i;
 444         }
 445         return (new pseries(relational(var,point), newseq))
 446                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 447 }
 448
 449 /** Implementation of ex::diff() for a power series.  It treats the series as a
 450  *  polynomial.
 451  *  @see ex::diff */
 452 ex pseries::derivative(const symbol & s) const
 453 {
 454         if (s == var) {
 455                 epvector new_seq;
 456                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 457
 458                 // FIXME: coeff might depend on var
 459                 while (it != itend) {
 460                         if (is_order_function(it->rest)) {
 461                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 462                         } else {
 463                                 ex c = it->rest * it->coeff;
 464                                 if (!c.is_zero())
 465                                         new_seq.push_back(expair(c, it->coeff - 1));
 466                         }
 467                         ++it;
 468                 }
 469                 return pseries(relational(var,point), new_seq);
 470         } else {
 471                 return *this;
 472         }
 473 }
 474
 475 ex pseries::convert_to_poly(bool no_order) const
 476 {
 477         ex e;
 478         epvector::const_iterator it = seq.begin(), itend = seq.end();
 479
 480         while (it != itend) {
 481                 if (is_order_function(it->rest)) {
 482                         if (!no_order)
 483                                 e += Order(power(var - point, it->coeff));
 484                 } else
 485                         e += it->rest * power(var - point, it->coeff);
 486                 ++it;
 487         }
 488         return e;
 489 }
 490
 491 bool pseries::is_terminating(void) const
 492 {
 493         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 494 }
 495
 496
 497 /*
 498  *  Implementations of series expansion
 499  */
 500
 501 /** Default implementation of ex::series(). This performs Taylor expansion.
 502  *  @see ex::series */
 503 ex basic::series(const relational & r, int order, unsigned options) const
 504 {
 505         epvector seq;
 506         numeric fac = 1;
 507         ex deriv = *this;
 508         ex coeff = deriv.subs(r);
 509         const symbol &s = ex_to<symbol>(r.lhs());
 510
 511         if (!coeff.is_zero())
 512                 seq.push_back(expair(coeff, _ex0));
 513
 514         int n;
 515         for (n=1; n<order; ++n) {
 516                 fac = fac.mul(n);
 517                 // We need to test for zero in order to see if the series terminates.
 518                 // The problem is that there is no such thing as a perfect test for
 519                 // zero.  Expanding the term occasionally helps a little...
 520                 deriv = deriv.diff(s).expand();
 521                 if (deriv.is_zero())  // Series terminates
 522                         return pseries(r, seq);
 523
 524                 coeff = deriv.subs(r);
 525                 if (!coeff.is_zero())
 526                         seq.push_back(expair(fac.inverse() * coeff, n));
 527         }
 528
 529         // Higher-order terms, if present
 530         deriv = deriv.diff(s);
 531         if (!deriv.expand().is_zero())
 532                 seq.push_back(expair(Order(_ex1), n));
 533         return pseries(r, seq);
 534 }
 535
 536
 537 /** Implementation of ex::series() for symbols.
 538  *  @see ex::series */
 539 ex symbol::series(const relational & r, int order, unsigned options) const
 540 {
 541         epvector seq;
 542         const ex point = r.rhs();
 543         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 544
 545         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 546                 if (order > 0 && !point.is_zero())
 547                         seq.push_back(expair(point, _ex0));
 548                 if (order > 1)
 549                         seq.push_back(expair(_ex1, _ex1));
 550                 else
 551                         seq.push_back(expair(Order(_ex1), numeric(order)));
 552         } else
 553                 seq.push_back(expair(*this, _ex0));
 554         return pseries(r, seq);
 555 }
 556
 557
 558 /** Add one series object to another, producing a pseries object that
 559  *  represents the sum.
 560  *
 561  *  @param other  pseries object to add with
 562  *  @return the sum as a pseries */
 563 ex pseries::add_series(const pseries &other) const
 564 {
 565         // Adding two series with different variables or expansion points
 566         // results in an empty (constant) series
 567         if (!is_compatible_to(other)) {
 568                 epvector nul;
 569                 nul.push_back(expair(Order(_ex1), _ex0));
 570                 return pseries(relational(var,point), nul);
 571         }
 572
 573         // Series addition
 574         epvector new_seq;
 575         epvector::const_iterator a = seq.begin();
 576         epvector::const_iterator b = other.seq.begin();
 577         epvector::const_iterator a_end = seq.end();
 578         epvector::const_iterator b_end = other.seq.end();
 579         int pow_a = INT_MAX, pow_b = INT_MAX;
 580         for (;;) {
 581                 // If a is empty, fill up with elements from b and stop
 582                 if (a == a_end) {
 583                         while (b != b_end) {
 584                                 new_seq.push_back(*b);
 585                                 ++b;
 586                         }
 587                         break;
 588                 } else
 589                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 590
 591                 // If b is empty, fill up with elements from a and stop
 592                 if (b == b_end) {
 593                         while (a != a_end) {
 594                                 new_seq.push_back(*a);
 595                                 ++a;
 596                         }
 597                         break;
 598                 } else
 599                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 600
 601                 // a and b are non-empty, compare powers
 602                 if (pow_a < pow_b) {
 603                         // a has lesser power, get coefficient from a
 604                         new_seq.push_back(*a);
 605                         if (is_order_function((*a).rest))
 606                                 break;
 607                         ++a;
 608                 } else if (pow_b < pow_a) {
 609                         // b has lesser power, get coefficient from b
 610                         new_seq.push_back(*b);
 611                         if (is_order_function((*b).rest))
 612                                 break;
 613                         ++b;
 614                 } else {
 615                         // Add coefficient of a and b
 616                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 617                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 618                                 break;  // Order term ends the sequence
 619                         } else {
 620                                 ex sum = (*a).rest + (*b).rest;
 621                                 if (!(sum.is_zero()))
 622                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 623                                 ++a;
 624                                 ++b;
 625                         }
 626                 }
 627         }
 628         return pseries(relational(var,point), new_seq);
 629 }
 630
 631
 632 /** Implementation of ex::series() for sums. This performs series addition when
 633  *  adding pseries objects.
 634  *  @see ex::series */
 635 ex add::series(const relational & r, int order, unsigned options) const
 636 {
 637         ex acc; // Series accumulator
 638
 639         // Get first term from overall_coeff
 640         acc = overall_coeff.series(r, order, options);
 641
 642         // Add remaining terms
 643         epvector::const_iterator it = seq.begin();
 644         epvector::const_iterator itend = seq.end();
 645         for (; it!=itend; ++it) {
 646                 ex op;
 647                 if (is_exactly_a<pseries>(it->rest))
 648                         op = it->rest;
 649                 else
 650                         op = it->rest.series(r, order, options);
 651                 if (!it->coeff.is_equal(_ex1))
 652                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 653
 654                 // Series addition
 655                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 656         }
 657         return acc;
 658 }
 659
 660
 661 /** Multiply a pseries object with a numeric constant, producing a pseries
 662  *  object that represents the product.
 663  *
 664  *  @param other  constant to multiply with
 665  *  @return the product as a pseries */
 666 ex pseries::mul_const(const numeric &other) const
 667 {
 668         epvector new_seq;
 669         new_seq.reserve(seq.size());
 670
 671         epvector::const_iterator it = seq.begin(), itend = seq.end();
 672         while (it != itend) {
 673                 if (!is_order_function(it->rest))
 674                         new_seq.push_back(expair(it->rest * other, it->coeff));
 675                 else
 676                         new_seq.push_back(*it);
 677                 ++it;
 678         }
 679         return pseries(relational(var,point), new_seq);
 680 }
 681
 682
 683 /** Multiply one pseries object to another, producing a pseries object that
 684  *  represents the product.
 685  *
 686  *  @param other  pseries object to multiply with
 687  *  @return the product as a pseries */
 688 ex pseries::mul_series(const pseries &other) const
 689 {
 690         // Multiplying two series with different variables or expansion points
 691         // results in an empty (constant) series
 692         if (!is_compatible_to(other)) {
 693                 epvector nul;
 694                 nul.push_back(expair(Order(_ex1), _ex0));
 695                 return pseries(relational(var,point), nul);
 696         }
 697
 698         // Series multiplication
 699         epvector new_seq;
 700         int a_max = degree(var);
 701         int b_max = other.degree(var);
 702         int a_min = ldegree(var);
 703         int b_min = other.ldegree(var);
 704         int cdeg_min = a_min + b_min;
 705         int cdeg_max = a_max + b_max;
 706
 707         int higher_order_a = INT_MAX;
 708         int higher_order_b = INT_MAX;
 709         if (is_order_function(coeff(var, a_max)))
 710                 higher_order_a = a_max + b_min;
 711         if (is_order_function(other.coeff(var, b_max)))
 712                 higher_order_b = b_max + a_min;
 713         int higher_order_c = std::min(higher_order_a, higher_order_b);
 714         if (cdeg_max >= higher_order_c)
 715                 cdeg_max = higher_order_c - 1;
 716
 717         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 718                 ex co = _ex0;
 719                 // c(i)=a(0)b(i)+...+a(i)b(0)
 720                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 721                         ex a_coeff = coeff(var, i);
 722                         ex b_coeff = other.coeff(var, cdeg-i);
 723                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 724                                 co += a_coeff * b_coeff;
 725                 }
 726                 if (!co.is_zero())
 727                         new_seq.push_back(expair(co, numeric(cdeg)));
 728         }
 729         if (higher_order_c < INT_MAX)
 730                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 731         return pseries(relational(var, point), new_seq);
 732 }
 733
 734
 735 /** Implementation of ex::series() for product. This performs series
 736  *  multiplication when multiplying series.
 737  *  @see ex::series */
 738 ex mul::series(const relational & r, int order, unsigned options) const
 739 {
 740         pseries acc; // Series accumulator
 741
 742         // Multiply with remaining terms
 743         const epvector::const_iterator itbeg = seq.begin();
 744         const epvector::const_iterator itend = seq.end();
 745         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 746                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 747
 748                 // Series multiplication
 749                 if (it==itbeg)
 750                         acc = ex_to<pseries>(op);
 751                 else
 752                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 753         }
 754         return acc.mul_const(ex_to<numeric>(overall_coeff));
 755 }
 756
 757
 758 /** Compute the p-th power of a series.
 759  *
 760  *  @param p  power to compute
 761  *  @param deg  truncation order of series calculation */
 762 ex pseries::power_const(const numeric &p, int deg) const
 763 {
 764         // method:
 765         // (due to Leonhard Euler)
 766         // let A(x) be this series and for the time being let it start with a
 767         // constant (later we'll generalize):
 768         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 769         // We want to compute
 770         //     C(x) = A(x)^p
 771         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 772         // Taking the derivative on both sides and multiplying with A(x) one
 773         // immediately arrives at
 774         //     C'(x)*A(x) = p*C(x)*A'(x)
 775         // Multiplying this out and comparing coefficients we get the recurrence
 776         // formula
 777         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 778         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 779         // which can easily be solved given the starting value c_0 = (a_0)^p.
 780         // For the more general case where the leading coefficient of A(x) is not
 781         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 782         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 783         // then of course x^(p*m) but the recurrence formula still holds.
 784
 785         if (seq.empty()) {
 786                 // as a special case, handle the empty (zero) series honoring the
 787                 // usual power laws such as implemented in power::eval()
 788                 if (p.real().is_zero())
 789                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 790                 else if (p.real().is_negative())
 791                         throw pole_error("pseries::power_const(): division by zero",1);
 792                 else
 793                         return *this;
 794         }
 795
 796         const int ldeg = ldegree(var);
 797         if (!(p*ldeg).is_integer())
 798                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 799
 800         // O(x^n)^(-m) is undefined
 801         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 802                 throw pole_error("pseries::power_const(): division by zero",1);
 803
 804         // Compute coefficients of the powered series
 805         exvector co;
 806         co.reserve(deg);
 807         co.push_back(power(coeff(var, ldeg), p));
 808         bool all_sums_zero = true;
 809         for (int i=1; i<deg; ++i) {
 810                 ex sum = _ex0;
 811                 for (int j=1; j<=i; ++j) {
 812                         ex c = coeff(var, j + ldeg);
 813                         if (is_order_function(c)) {
 814                                 co.push_back(Order(_ex1));
 815                                 break;
 816                         } else
 817                                 sum += (p * j - (i - j)) * co[i - j] * c;
 818                 }
 819                 if (!sum.is_zero())
 820                         all_sums_zero = false;
 821                 co.push_back(sum / coeff(var, ldeg) / i);
 822         }
 823
 824         // Construct new series (of non-zero coefficients)
 825         epvector new_seq;
 826         bool higher_order = false;
 827         for (int i=0; i<deg; ++i) {
 828                 if (!co[i].is_zero())
 829                         new_seq.push_back(expair(co[i], p * ldeg + i));
 830                 if (is_order_function(co[i])) {
 831                         higher_order = true;
 832                         break;
 833                 }
 834         }
 835         if (!higher_order && !all_sums_zero)
 836                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 837         return pseries(relational(var,point), new_seq);
 838 }
 839
 840
 841 /** Return a new pseries object with the powers shifted by deg. */
 842 pseries pseries::shift_exponents(int deg) const
 843 {
 844         epvector newseq = seq;
 845         epvector::iterator i = newseq.begin(), end  = newseq.end();
 846         while (i != end) {
 847                 i->coeff += deg;
 848                 ++i;
 849         }
 850         return pseries(relational(var, point), newseq);
 851 }
 852
 853
 854 /** Implementation of ex::series() for powers. This performs Laurent expansion
 855  *  of reciprocals of series at singularities.
 856  *  @see ex::series */
 857 ex power::series(const relational & r, int order, unsigned options) const
 858 {
 859         // If basis is already a series, just power it
 860         if (is_exactly_a<pseries>(basis))
 861                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 862
 863         // Basis is not a series, may there be a singularity?
 864         bool must_expand_basis = false;
 865         try {
 866                 basis.subs(r);
 867         } catch (pole_error) {
 868                 must_expand_basis = true;
 869         }
 870
 871         // Is the expression of type something^(-int)?
 872         if (!must_expand_basis && !exponent.info(info_flags::negint))
 873                 return basic::series(r, order, options);
 874
 875         // Is the expression of type 0^something?
 876         if (!must_expand_basis && !basis.subs(r).is_zero())
 877                 return basic::series(r, order, options);
 878
 879         // Singularity encountered, is the basis equal to (var - point)?
 880         if (basis.is_equal(r.lhs() - r.rhs())) {
 881                 epvector new_seq;
 882                 if (ex_to<numeric>(exponent).to_int() < order)
 883                         new_seq.push_back(expair(_ex1, exponent));
 884                 else
 885                         new_seq.push_back(expair(Order(_ex1), exponent));
 886                 return pseries(r, new_seq);
 887         }
 888
 889         // No, expand basis into series
 890         ex e = basis.series(r, order, options);
 891         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 892 }
 893
 894
 895 /** Re-expansion of a pseries object. */
 896 ex pseries::series(const relational & r, int order, unsigned options) const
 897 {
 898         const ex p = r.rhs();
 899         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 900         const symbol &s = ex_to<symbol>(r.lhs());
 901
 902         if (var.is_equal(s) && point.is_equal(p)) {
 903                 if (order > degree(s))
 904                         return *this;
 905                 else {
 906                         epvector new_seq;
 907                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 908                         while (it != itend) {
 909                                 int o = ex_to<numeric>(it->coeff).to_int();
 910                                 if (o >= order) {
 911                                         new_seq.push_back(expair(Order(_ex1), o));
 912                                         break;
 913                                 }
 914                                 new_seq.push_back(*it);
 915                                 ++it;
 916                         }
 917                         return pseries(r, new_seq);
 918                 }
 919         } else
 920                 return convert_to_poly().series(r, order, options);
 921 }
 922
 923
 924 /** Compute the truncated series expansion of an expression.
 925  *  This function returns an expression containing an object of class pseries
 926  *  to represent the series. If the series does not terminate within the given
 927  *  truncation order, the last term of the series will be an order term.
 928  *
 929  *  @param r  expansion relation, lhs holds variable and rhs holds point
 930  *  @param order  truncation order of series calculations
 931  *  @param options  of class series_options
 932  *  @return an expression holding a pseries object */
 933 ex ex::series(const ex & r, int order, unsigned options) const
 934 {
 935         GINAC_ASSERT(bp!=0);
 936         ex e;
 937         relational rel_;
 938
 939         if (is_exactly_a<relational>(r))
 940                 rel_ = ex_to<relational>(r);
 941         else if (is_exactly_a<symbol>(r))
 942                 rel_ = relational(r,_ex0);
 943         else
 944                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 945
 946         try {
 947                 e = bp->series(rel_, order, options);
 948         } catch (std::exception &x) {
 949                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 950         }
 951         return e;
 952 }
 953
 954 } // namespace GiNaC