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 "symbol.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
  42
  43
  44 /*
  45  *  Default ctor, dtor, copy ctor, assignment operator and helpers
  46  */
  47
  48 pseries::pseries() : inherited(TINFO_pseries) { }
  49
  50 void pseries::copy(const pseries &other)
  51 {
  52         inherited::copy(other);
  53         seq = other.seq;
  54         var = other.var;
  55         point = other.point;
  56 }
  57
  58 DEFAULT_DESTROY(pseries)
  59
  60
  61 /*
  62  *  Other ctors
  63  */
  64
  65 /** Construct pseries from a vector of coefficients and powers.
  66  *  expair.rest holds the coefficient, expair.coeff holds the power.
  67  *  The powers must be integers (positive or negative) and in ascending order;
  68  *  the last coefficient can be Order(_ex1) to represent a truncated,
  69  *  non-terminating series.
  70  *
  71  *  @param rel_  expansion variable and point (must hold a relational)
  72  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  73  *  @return newly constructed pseries */
  74 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
  75 {
  76         GINAC_ASSERT(is_exactly_a<relational>(rel_));
  77         GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
  78         point = rel_.rhs();
  79         var = rel_.lhs();
  80 }
  81
  82
  83 /*
  84  *  Archiving
  85  */
  86
  87 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  88 {
  89         for (unsigned int i=0; true; ++i) {
  90                 ex rest;
  91                 ex coeff;
  92                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
  93                         seq.push_back(expair(rest, coeff));
  94                 else
  95                         break;
  96         }
  97         n.find_ex("var", var, sym_lst);
  98         n.find_ex("point", point, sym_lst);
  99 }
 100
 101 void pseries::archive(archive_node &n) const
 102 {
 103         inherited::archive(n);
 104         epvector::const_iterator i = seq.begin(), iend = seq.end();
 105         while (i != iend) {
 106                 n.add_ex("coeff", i->rest);
 107                 n.add_ex("power", i->coeff);
 108                 ++i;
 109         }
 110         n.add_ex("var", var);
 111         n.add_ex("point", point);
 112 }
 113
 114 DEFAULT_UNARCHIVE(pseries)
 115
 116 //////////
 117 // functions overriding virtual functions from base classes
 118 //////////
 119
 120 void pseries::print(const print_context & c, unsigned level) const
 121 {
 122         if (is_a<print_tree>(c)) {
 123
 124                 c.s << std::string(level, ' ') << class_name()
 125                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 126                     << std::endl;
 127                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 128                 unsigned num = seq.size();
 129                 for (unsigned i=0; i<num; ++i) {
 130                         seq[i].rest.print(c, level + delta_indent);
 131                         seq[i].coeff.print(c, level + delta_indent);
 132                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
 133                 }
 134                 var.print(c, level + delta_indent);
 135                 point.print(c, level + delta_indent);
 136
 137         } else if (is_a<print_python_repr>(c)) {
 138                 c.s << class_name() << "(relational(";
 139                 var.print(c);
 140                 c.s << ',';
 141                 point.print(c);
 142                 c.s << "),[";
 143                 unsigned num = seq.size();
 144                 for (unsigned i=0; i<num; ++i) {
 145                         if (i)
 146                                 c.s << ',';
 147                         c.s << '(';
 148                         seq[i].rest.print(c);
 149                         c.s << ',';
 150                         seq[i].coeff.print(c);
 151                         c.s << ')';
 152                 }
 153                 c.s << "])";
 154         } else {
 155
 156                 if (precedence() <= level)
 157                         c.s << "(";
 158
 159                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
 160                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
 161
 162                 // objects of type pseries must not have any zero entries, so the
 163                 // trivial (zero) pseries needs a special treatment here:
 164                 if (seq.empty())
 165                         c.s << '0';
 166                 epvector::const_iterator i = seq.begin(), end = seq.end();
 167                 while (i != end) {
 168                         // print a sign, if needed
 169                         if (i != seq.begin())
 170                                 c.s << '+';
 171                         if (!is_order_function(i->rest)) {
 172                                 // print 'rest', i.e. the expansion coefficient
 173                                 if (i->rest.info(info_flags::numeric) &&
 174                                         i->rest.info(info_flags::positive)) {
 175                                         i->rest.print(c);
 176                                 } else {
 177                                         c.s << par_open;
 178                                         i->rest.print(c);
 179                                         c.s << par_close;
 180                                 }
 181                                 // print 'coeff', something like (x-1)^42
 182                                 if (!i->coeff.is_zero()) {
 183                                         if (is_a<print_latex>(c))
 184                                                 c.s << ' ';
 185                                         else
 186                                                 c.s << '*';
 187                                         if (!point.is_zero()) {
 188                                                 c.s << par_open;
 189                                                 (var-point).print(c);
 190                                                 c.s << par_close;
 191                                         } else
 192                                                 var.print(c);
 193                                         if (i->coeff.compare(_ex1)) {
 194                                                 if (is_a<print_python>(c))
 195                                                         c.s << "**";
 196                                                 else
 197                                                         c.s << '^';
 198                                                 if (i->coeff.info(info_flags::negative)) {
 199                                                         c.s << par_open;
 200                                                         i->coeff.print(c);
 201                                                         c.s << par_close;
 202                                                 } else {
 203                                                         if (is_a<print_latex>(c)) {
 204                                                                 c.s << '{';
 205                                                                 i->coeff.print(c);
 206                                                                 c.s << '}';
 207                                                         } else
 208                                                                 i->coeff.print(c);
 209                                                 }
 210                                         }
 211                                 }
 212                         } else
 213                                 Order(power(var-point,i->coeff)).print(c);
 214                         ++i;
 215                 }
 216
 217                 if (precedence() <= level)
 218                         c.s << ")";
 219         }
 220 }
 221
 222 int pseries::compare_same_type(const basic & other) const
 223 {
 224         GINAC_ASSERT(is_a<pseries>(other));
 225         const pseries &o = static_cast<const pseries &>(other);
 226
 227         // first compare the lengths of the series...
 228         if (seq.size()>o.seq.size())
 229                 return 1;
 230         if (seq.size()<o.seq.size())
 231                 return -1;
 232
 233         // ...then the expansion point...
 234         int cmpval = var.compare(o.var);
 235         if (cmpval)
 236                 return cmpval;
 237         cmpval = point.compare(o.point);
 238         if (cmpval)
 239                 return cmpval;
 240
 241         // ...and if that failed the individual elements
 242         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
 243         while (it!=seq.end() && o_it!=o.seq.end()) {
 244                 cmpval = it->compare(*o_it);
 245                 if (cmpval)
 246                         return cmpval;
 247                 ++it;
 248                 ++o_it;
 249         }
 250
 251         // so they are equal.
 252         return 0;
 253 }
 254
 255 /** Return the number of operands including a possible order term. */
 256 unsigned pseries::nops(void) const
 257 {
 258         return seq.size();
 259 }
 260
 261 /** Return the ith term in the series when represented as a sum. */
 262 ex pseries::op(int i) const
 263 {
 264         if (i < 0 || unsigned(i) >= seq.size())
 265                 throw (std::out_of_range("op() out of range"));
 266         return seq[i].rest * power(var - point, seq[i].coeff);
 267 }
 268
 269 ex &pseries::let_op(int i)
 270 {
 271         throw (std::logic_error("let_op not defined for pseries"));
 272 }
 273
 274 /** Return degree of highest power of the series.  This is usually the exponent
 275  *  of the Order term.  If s is not the expansion variable of the series, the
 276  *  series is examined termwise. */
 277 int pseries::degree(const ex &s) const
 278 {
 279         if (var.is_equal(s)) {
 280                 // Return last exponent
 281                 if (seq.size())
 282                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
 283                 else
 284                         return 0;
 285         } else {
 286                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 287                 if (it == itend)
 288                         return 0;
 289                 int max_pow = INT_MIN;
 290                 while (it != itend) {
 291                         int pow = it->rest.degree(s);
 292                         if (pow > max_pow)
 293                                 max_pow = pow;
 294                         ++it;
 295                 }
 296                 return max_pow;
 297         }
 298 }
 299
 300 /** Return degree of lowest power of the series.  This is usually the exponent
 301  *  of the leading term.  If s is not the expansion variable of the series, the
 302  *  series is examined termwise.  If s is the expansion variable but the
 303  *  expansion point is not zero the series is not expanded to find the degree.
 304  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 305 int pseries::ldegree(const ex &s) const
 306 {
 307         if (var.is_equal(s)) {
 308                 // Return first exponent
 309                 if (seq.size())
 310                         return ex_to<numeric>((seq.begin())->coeff).to_int();
 311                 else
 312                         return 0;
 313         } else {
 314                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 315                 if (it == itend)
 316                         return 0;
 317                 int min_pow = INT_MAX;
 318                 while (it != itend) {
 319                         int pow = it->rest.ldegree(s);
 320                         if (pow < min_pow)
 321                                 min_pow = pow;
 322                         ++it;
 323                 }
 324                 return min_pow;
 325         }
 326 }
 327
 328 /** Return coefficient of degree n in power series if s is the expansion
 329  *  variable.  If the expansion point is nonzero, by definition the n=1
 330  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
 331  *  the expansion took place in the s in the first place).
 332  *  If s is not the expansion variable, an attempt is made to convert the
 333  *  series to a polynomial and return the corresponding coefficient from
 334  *  there. */
 335 ex pseries::coeff(const ex &s, int n) const
 336 {
 337         if (var.is_equal(s)) {
 338                 if (seq.empty())
 339                         return _ex0;
 340
 341                 // Binary search in sequence for given power
 342                 numeric looking_for = numeric(n);
 343                 int lo = 0, hi = seq.size() - 1;
 344                 while (lo <= hi) {
 345                         int mid = (lo + hi) / 2;
 346                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
 347                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
 348                         switch (cmp) {
 349                                 case -1:
 350                                         lo = mid + 1;
 351                                         break;
 352                                 case 0:
 353                                         return seq[mid].rest;
 354                                 case 1:
 355                                         hi = mid - 1;
 356                                         break;
 357                                 default:
 358                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
 359                         }
 360                 }
 361                 return _ex0;
 362         } else
 363                 return convert_to_poly().coeff(s, n);
 364 }
 365
 366 /** Does nothing. */
 367 ex pseries::collect(const ex &s, bool distributed) const
 368 {
 369         return *this;
 370 }
 371
 372 /** Perform coefficient-wise automatic term rewriting rules in this class. */
 373 ex pseries::eval(int level) const
 374 {
 375         if (level == 1)
 376                 return this->hold();
 377
 378         if (level == -max_recursion_level)
 379                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
 380
 381         // Construct a new series with evaluated coefficients
 382         epvector new_seq;
 383         new_seq.reserve(seq.size());
 384         epvector::const_iterator it = seq.begin(), itend = seq.end();
 385         while (it != itend) {
 386                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
 387                 ++it;
 388         }
 389         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 390 }
 391
 392 /** Evaluate coefficients numerically. */
 393 ex pseries::evalf(int level) const
 394 {
 395         if (level == 1)
 396                 return *this;
 397
 398         if (level == -max_recursion_level)
 399                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
 400
 401         // Construct a new series with evaluated coefficients
 402         epvector new_seq;
 403         new_seq.reserve(seq.size());
 404         epvector::const_iterator it = seq.begin(), itend = seq.end();
 405         while (it != itend) {
 406                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
 407                 ++it;
 408         }
 409         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 410 }
 411
 412 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
 413 {
 414         // If expansion variable is being substituted, convert the series to a
 415         // polynomial and do the substitution there because the result might
 416         // no longer be a power series
 417         if (ls.has(var))
 418                 return convert_to_poly(true).subs(ls, lr, no_pattern);
 419
 420         // Otherwise construct a new series with substituted coefficients and
 421         // expansion point
 422         epvector newseq;
 423         newseq.reserve(seq.size());
 424         epvector::const_iterator it = seq.begin(), itend = seq.end();
 425         while (it != itend) {
 426                 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
 427                 ++it;
 428         }
 429         return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
 430 }
 431
 432 /** Implementation of ex::expand() for a power series.  It expands all the
 433  *  terms individually and returns the resulting series as a new pseries. */
 434 ex pseries::expand(unsigned options) const
 435 {
 436         epvector newseq;
 437         epvector::const_iterator i = seq.begin(), end = seq.end();
 438         while (i != end) {
 439                 ex restexp = i->rest.expand();
 440                 if (!restexp.is_zero())
 441                         newseq.push_back(expair(restexp, i->coeff));
 442                 ++i;
 443         }
 444         return (new pseries(relational(var,point), newseq))
 445                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 446 }
 447
 448 /** Implementation of ex::diff() for a power series.  It treats the series as a
 449  *  polynomial.
 450  *  @see ex::diff */
 451 ex pseries::derivative(const symbol & s) const
 452 {
 453         if (s == var) {
 454                 epvector new_seq;
 455                 epvector::const_iterator it = seq.begin(), itend = seq.end();
 456
 457                 // FIXME: coeff might depend on var
 458                 while (it != itend) {
 459                         if (is_order_function(it->rest)) {
 460                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
 461                         } else {
 462                                 ex c = it->rest * it->coeff;
 463                                 if (!c.is_zero())
 464                                         new_seq.push_back(expair(c, it->coeff - 1));
 465                         }
 466                         ++it;
 467                 }
 468                 return pseries(relational(var,point), new_seq);
 469         } else {
 470                 return *this;
 471         }
 472 }
 473
 474 ex pseries::convert_to_poly(bool no_order) const
 475 {
 476         ex e;
 477         epvector::const_iterator it = seq.begin(), itend = seq.end();
 478
 479         while (it != itend) {
 480                 if (is_order_function(it->rest)) {
 481                         if (!no_order)
 482                                 e += Order(power(var - point, it->coeff));
 483                 } else
 484                         e += it->rest * power(var - point, it->coeff);
 485                 ++it;
 486         }
 487         return e;
 488 }
 489
 490 bool pseries::is_terminating(void) const
 491 {
 492         return seq.empty() || !is_order_function((seq.end()-1)->rest);
 493 }
 494
 495
 496 /*
 497  *  Implementations of series expansion
 498  */
 499
 500 /** Default implementation of ex::series(). This performs Taylor expansion.
 501  *  @see ex::series */
 502 ex basic::series(const relational & r, int order, unsigned options) const
 503 {
 504         epvector seq;
 505         numeric fac = 1;
 506         ex deriv = *this;
 507         ex coeff = deriv.subs(r);
 508         const symbol &s = ex_to<symbol>(r.lhs());
 509
 510         if (!coeff.is_zero())
 511                 seq.push_back(expair(coeff, _ex0));
 512
 513         int n;
 514         for (n=1; n<order; ++n) {
 515                 fac = fac.mul(n);
 516                 // We need to test for zero in order to see if the series terminates.
 517                 // The problem is that there is no such thing as a perfect test for
 518                 // zero.  Expanding the term occasionally helps a little...
 519                 deriv = deriv.diff(s).expand();
 520                 if (deriv.is_zero())  // Series terminates
 521                         return pseries(r, seq);
 522
 523                 coeff = deriv.subs(r);
 524                 if (!coeff.is_zero())
 525                         seq.push_back(expair(fac.inverse() * coeff, n));
 526         }
 527
 528         // Higher-order terms, if present
 529         deriv = deriv.diff(s);
 530         if (!deriv.expand().is_zero())
 531                 seq.push_back(expair(Order(_ex1), n));
 532         return pseries(r, seq);
 533 }
 534
 535
 536 /** Implementation of ex::series() for symbols.
 537  *  @see ex::series */
 538 ex symbol::series(const relational & r, int order, unsigned options) const
 539 {
 540         epvector seq;
 541         const ex point = r.rhs();
 542         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 543
 544         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
 545                 if (order > 0 && !point.is_zero())
 546                         seq.push_back(expair(point, _ex0));
 547                 if (order > 1)
 548                         seq.push_back(expair(_ex1, _ex1));
 549                 else
 550                         seq.push_back(expair(Order(_ex1), numeric(order)));
 551         } else
 552                 seq.push_back(expair(*this, _ex0));
 553         return pseries(r, seq);
 554 }
 555
 556
 557 /** Add one series object to another, producing a pseries object that
 558  *  represents the sum.
 559  *
 560  *  @param other  pseries object to add with
 561  *  @return the sum as a pseries */
 562 ex pseries::add_series(const pseries &other) const
 563 {
 564         // Adding two series with different variables or expansion points
 565         // results in an empty (constant) series
 566         if (!is_compatible_to(other)) {
 567                 epvector nul;
 568                 nul.push_back(expair(Order(_ex1), _ex0));
 569                 return pseries(relational(var,point), nul);
 570         }
 571
 572         // Series addition
 573         epvector new_seq;
 574         epvector::const_iterator a = seq.begin();
 575         epvector::const_iterator b = other.seq.begin();
 576         epvector::const_iterator a_end = seq.end();
 577         epvector::const_iterator b_end = other.seq.end();
 578         int pow_a = INT_MAX, pow_b = INT_MAX;
 579         for (;;) {
 580                 // If a is empty, fill up with elements from b and stop
 581                 if (a == a_end) {
 582                         while (b != b_end) {
 583                                 new_seq.push_back(*b);
 584                                 ++b;
 585                         }
 586                         break;
 587                 } else
 588                         pow_a = ex_to<numeric>((*a).coeff).to_int();
 589
 590                 // If b is empty, fill up with elements from a and stop
 591                 if (b == b_end) {
 592                         while (a != a_end) {
 593                                 new_seq.push_back(*a);
 594                                 ++a;
 595                         }
 596                         break;
 597                 } else
 598                         pow_b = ex_to<numeric>((*b).coeff).to_int();
 599
 600                 // a and b are non-empty, compare powers
 601                 if (pow_a < pow_b) {
 602                         // a has lesser power, get coefficient from a
 603                         new_seq.push_back(*a);
 604                         if (is_order_function((*a).rest))
 605                                 break;
 606                         ++a;
 607                 } else if (pow_b < pow_a) {
 608                         // b has lesser power, get coefficient from b
 609                         new_seq.push_back(*b);
 610                         if (is_order_function((*b).rest))
 611                                 break;
 612                         ++b;
 613                 } else {
 614                         // Add coefficient of a and b
 615                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
 616                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
 617                                 break;  // Order term ends the sequence
 618                         } else {
 619                                 ex sum = (*a).rest + (*b).rest;
 620                                 if (!(sum.is_zero()))
 621                                         new_seq.push_back(expair(sum, numeric(pow_a)));
 622                                 ++a;
 623                                 ++b;
 624                         }
 625                 }
 626         }
 627         return pseries(relational(var,point), new_seq);
 628 }
 629
 630
 631 /** Implementation of ex::series() for sums. This performs series addition when
 632  *  adding pseries objects.
 633  *  @see ex::series */
 634 ex add::series(const relational & r, int order, unsigned options) const
 635 {
 636         ex acc; // Series accumulator
 637
 638         // Get first term from overall_coeff
 639         acc = overall_coeff.series(r, order, options);
 640
 641         // Add remaining terms
 642         epvector::const_iterator it = seq.begin();
 643         epvector::const_iterator itend = seq.end();
 644         for (; it!=itend; ++it) {
 645                 ex op;
 646                 if (is_ex_exactly_of_type(it->rest, pseries))
 647                         op = it->rest;
 648                 else
 649                         op = it->rest.series(r, order, options);
 650                 if (!it->coeff.is_equal(_ex1))
 651                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
 652
 653                 // Series addition
 654                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
 655         }
 656         return acc;
 657 }
 658
 659
 660 /** Multiply a pseries object with a numeric constant, producing a pseries
 661  *  object that represents the product.
 662  *
 663  *  @param other  constant to multiply with
 664  *  @return the product as a pseries */
 665 ex pseries::mul_const(const numeric &other) const
 666 {
 667         epvector new_seq;
 668         new_seq.reserve(seq.size());
 669
 670         epvector::const_iterator it = seq.begin(), itend = seq.end();
 671         while (it != itend) {
 672                 if (!is_order_function(it->rest))
 673                         new_seq.push_back(expair(it->rest * other, it->coeff));
 674                 else
 675                         new_seq.push_back(*it);
 676                 ++it;
 677         }
 678         return pseries(relational(var,point), new_seq);
 679 }
 680
 681
 682 /** Multiply one pseries object to another, producing a pseries object that
 683  *  represents the product.
 684  *
 685  *  @param other  pseries object to multiply with
 686  *  @return the product as a pseries */
 687 ex pseries::mul_series(const pseries &other) const
 688 {
 689         // Multiplying two series with different variables or expansion points
 690         // results in an empty (constant) series
 691         if (!is_compatible_to(other)) {
 692                 epvector nul;
 693                 nul.push_back(expair(Order(_ex1), _ex0));
 694                 return pseries(relational(var,point), nul);
 695         }
 696
 697         // Series multiplication
 698         epvector new_seq;
 699         int a_max = degree(var);
 700         int b_max = other.degree(var);
 701         int a_min = ldegree(var);
 702         int b_min = other.ldegree(var);
 703         int cdeg_min = a_min + b_min;
 704         int cdeg_max = a_max + b_max;
 705
 706         int higher_order_a = INT_MAX;
 707         int higher_order_b = INT_MAX;
 708         if (is_order_function(coeff(var, a_max)))
 709                 higher_order_a = a_max + b_min;
 710         if (is_order_function(other.coeff(var, b_max)))
 711                 higher_order_b = b_max + a_min;
 712         int higher_order_c = std::min(higher_order_a, higher_order_b);
 713         if (cdeg_max >= higher_order_c)
 714                 cdeg_max = higher_order_c - 1;
 715
 716         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
 717                 ex co = _ex0;
 718                 // c(i)=a(0)b(i)+...+a(i)b(0)
 719                 for (int i=a_min; cdeg-i>=b_min; ++i) {
 720                         ex a_coeff = coeff(var, i);
 721                         ex b_coeff = other.coeff(var, cdeg-i);
 722                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
 723                                 co += a_coeff * b_coeff;
 724                 }
 725                 if (!co.is_zero())
 726                         new_seq.push_back(expair(co, numeric(cdeg)));
 727         }
 728         if (higher_order_c < INT_MAX)
 729                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
 730         return pseries(relational(var, point), new_seq);
 731 }
 732
 733
 734 /** Implementation of ex::series() for product. This performs series
 735  *  multiplication when multiplying series.
 736  *  @see ex::series */
 737 ex mul::series(const relational & r, int order, unsigned options) const
 738 {
 739         pseries acc; // Series accumulator
 740
 741         // Multiply with remaining terms
 742         const epvector::const_iterator itbeg = seq.begin();
 743         const epvector::const_iterator itend = seq.end();
 744         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
 745                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
 746
 747                 // Series multiplication
 748                 if (it==itbeg)
 749                         acc = ex_to<pseries>(op);
 750                 else
 751                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
 752         }
 753         return acc.mul_const(ex_to<numeric>(overall_coeff));
 754 }
 755
 756
 757 /** Compute the p-th power of a series.
 758  *
 759  *  @param p  power to compute
 760  *  @param deg  truncation order of series calculation */
 761 ex pseries::power_const(const numeric &p, int deg) const
 762 {
 763         // method:
 764         // (due to Leonhard Euler)
 765         // let A(x) be this series and for the time being let it start with a
 766         // constant (later we'll generalize):
 767         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
 768         // We want to compute
 769         //     C(x) = A(x)^p
 770         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
 771         // Taking the derivative on both sides and multiplying with A(x) one
 772         // immediately arrives at
 773         //     C'(x)*A(x) = p*C(x)*A'(x)
 774         // Multiplying this out and comparing coefficients we get the recurrence
 775         // formula
 776         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
 777         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
 778         // which can easily be solved given the starting value c_0 = (a_0)^p.
 779         // For the more general case where the leading coefficient of A(x) is not
 780         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
 781         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
 782         // then of course x^(p*m) but the recurrence formula still holds.
 783
 784         if (seq.empty()) {
 785                 // as a special case, handle the empty (zero) series honoring the
 786                 // usual power laws such as implemented in power::eval()
 787                 if (p.real().is_zero())
 788                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
 789                 else if (p.real().is_negative())
 790                         throw pole_error("pseries::power_const(): division by zero",1);
 791                 else
 792                         return *this;
 793         }
 794
 795         const int ldeg = ldegree(var);
 796         if (!(p*ldeg).is_integer())
 797                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
 798
 799         // O(x^n)^(-m) is undefined
 800         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
 801                 throw pole_error("pseries::power_const(): division by zero",1);
 802
 803         // Compute coefficients of the powered series
 804         exvector co;
 805         co.reserve(deg);
 806         co.push_back(power(coeff(var, ldeg), p));
 807         bool all_sums_zero = true;
 808         for (int i=1; i<deg; ++i) {
 809                 ex sum = _ex0;
 810                 for (int j=1; j<=i; ++j) {
 811                         ex c = coeff(var, j + ldeg);
 812                         if (is_order_function(c)) {
 813                                 co.push_back(Order(_ex1));
 814                                 break;
 815                         } else
 816                                 sum += (p * j - (i - j)) * co[i - j] * c;
 817                 }
 818                 if (!sum.is_zero())
 819                         all_sums_zero = false;
 820                 co.push_back(sum / coeff(var, ldeg) / i);
 821         }
 822
 823         // Construct new series (of non-zero coefficients)
 824         epvector new_seq;
 825         bool higher_order = false;
 826         for (int i=0; i<deg; ++i) {
 827                 if (!co[i].is_zero())
 828                         new_seq.push_back(expair(co[i], p * ldeg + i));
 829                 if (is_order_function(co[i])) {
 830                         higher_order = true;
 831                         break;
 832                 }
 833         }
 834         if (!higher_order && !all_sums_zero)
 835                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
 836         return pseries(relational(var,point), new_seq);
 837 }
 838
 839
 840 /** Return a new pseries object with the powers shifted by deg. */
 841 pseries pseries::shift_exponents(int deg) const
 842 {
 843         epvector newseq = seq;
 844         epvector::iterator i = newseq.begin(), end  = newseq.end();
 845         while (i != end) {
 846                 i->coeff += deg;
 847                 ++i;
 848         }
 849         return pseries(relational(var, point), newseq);
 850 }
 851
 852
 853 /** Implementation of ex::series() for powers. This performs Laurent expansion
 854  *  of reciprocals of series at singularities.
 855  *  @see ex::series */
 856 ex power::series(const relational & r, int order, unsigned options) const
 857 {
 858         // If basis is already a series, just power it
 859         if (is_ex_exactly_of_type(basis, pseries))
 860                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
 861
 862         // Basis is not a series, may there be a singularity?
 863         bool must_expand_basis = false;
 864         try {
 865                 basis.subs(r);
 866         } catch (pole_error) {
 867                 must_expand_basis = true;
 868         }
 869
 870         // Is the expression of type something^(-int)?
 871         if (!must_expand_basis && !exponent.info(info_flags::negint))
 872                 return basic::series(r, order, options);
 873
 874         // Is the expression of type 0^something?
 875         if (!must_expand_basis && !basis.subs(r).is_zero())
 876                 return basic::series(r, order, options);
 877
 878         // Singularity encountered, is the basis equal to (var - point)?
 879         if (basis.is_equal(r.lhs() - r.rhs())) {
 880                 epvector new_seq;
 881                 if (ex_to<numeric>(exponent).to_int() < order)
 882                         new_seq.push_back(expair(_ex1, exponent));
 883                 else
 884                         new_seq.push_back(expair(Order(_ex1), exponent));
 885                 return pseries(r, new_seq);
 886         }
 887
 888         // No, expand basis into series
 889         ex e = basis.series(r, order, options);
 890         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
 891 }
 892
 893
 894 /** Re-expansion of a pseries object. */
 895 ex pseries::series(const relational & r, int order, unsigned options) const
 896 {
 897         const ex p = r.rhs();
 898         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
 899         const symbol &s = ex_to<symbol>(r.lhs());
 900
 901         if (var.is_equal(s) && point.is_equal(p)) {
 902                 if (order > degree(s))
 903                         return *this;
 904                 else {
 905                         epvector new_seq;
 906                         epvector::const_iterator it = seq.begin(), itend = seq.end();
 907                         while (it != itend) {
 908                                 int o = ex_to<numeric>(it->coeff).to_int();
 909                                 if (o >= order) {
 910                                         new_seq.push_back(expair(Order(_ex1), o));
 911                                         break;
 912                                 }
 913                                 new_seq.push_back(*it);
 914                                 ++it;
 915                         }
 916                         return pseries(r, new_seq);
 917                 }
 918         } else
 919                 return convert_to_poly().series(r, order, options);
 920 }
 921
 922
 923 /** Compute the truncated series expansion of an expression.
 924  *  This function returns an expression containing an object of class pseries
 925  *  to represent the series. If the series does not terminate within the given
 926  *  truncation order, the last term of the series will be an order term.
 927  *
 928  *  @param r  expansion relation, lhs holds variable and rhs holds point
 929  *  @param order  truncation order of series calculations
 930  *  @param options  of class series_options
 931  *  @return an expression holding a pseries object */
 932 ex ex::series(const ex & r, int order, unsigned options) const
 933 {
 934         GINAC_ASSERT(bp!=0);
 935         ex e;
 936         relational rel_;
 937
 938         if (is_ex_exactly_of_type(r,relational))
 939                 rel_ = ex_to<relational>(r);
 940         else if (is_ex_exactly_of_type(r,symbol))
 941                 rel_ = relational(r,_ex0);
 942         else
 943                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
 944
 945         try {
 946                 e = bp->series(rel_, order, options);
 947         } catch (std::exception &x) {
 948                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
 949         }
 950         return e;
 951 }
 952
 953 } // namespace GiNaC