ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "power.h"
  29 #include "expairseq.h"
  30 #include "add.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "numeric.h"
  34 #include "constant.h"
  35 #include "operators.h"
  36 #include "inifcns.h" // for log() in power::derivative()
  37 #include "matrix.h"
  38 #include "indexed.h"
  39 #include "symbol.h"
  40 #include "lst.h"
  41 #include "archive.h"
  42 #include "utils.h"
  43
  44 namespace GiNaC {
  45
  46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  47   print_func<print_dflt>(&power::do_print_dflt).
  48   print_func<print_latex>(&power::do_print_latex).
  49   print_func<print_csrc>(&power::do_print_csrc).
  50   print_func<print_python>(&power::do_print_python).
  51   print_func<print_python_repr>(&power::do_print_python_repr))
  52
  53 typedef std::vector<int> intvector;
  54
  55 //////////
  56 // default constructor
  57 //////////
  58
  59 power::power() : inherited(TINFO_power) { }
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 // all inlined
  66
  67 //////////
  68 // archiving
  69 //////////
  70
  71 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  72 {
  73         n.find_ex("basis", basis, sym_lst);
  74         n.find_ex("exponent", exponent, sym_lst);
  75 }
  76
  77 void power::archive(archive_node &n) const
  78 {
  79         inherited::archive(n);
  80         n.add_ex("basis", basis);
  81         n.add_ex("exponent", exponent);
  82 }
  83
  84 DEFAULT_UNARCHIVE(power)
  85
  86 //////////
  87 // functions overriding virtual functions from base classes
  88 //////////
  89
  90 // public
  91
  92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  93 {
  94         // Ordinary output of powers using '^' or '**'
  95         if (precedence() <= level)
  96                 c.s << openbrace << '(';
  97         basis.print(c, precedence());
  98         c.s << powersymbol;
  99         c.s << openbrace;
 100         exponent.print(c, precedence());
 101         c.s << closebrace;
 102         if (precedence() <= level)
 103                 c.s << ')' << closebrace;
 104 }
 105
 106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 107 {
 108         if (exponent.is_equal(_ex1_2)) {
 109
 110                 // Square roots are printed in a special way
 111                 c.s << "sqrt(";
 112                 basis.print(c);
 113                 c.s << ')';
 114
 115         } else
 116                 print_power(c, "^", "", "", level);
 117 }
 118
 119 void power::do_print_latex(const print_latex & c, unsigned level) const
 120 {
 121         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 122
 123                 // Powers with negative numeric exponents are printed as fractions
 124                 c.s << "\\frac{1}{";
 125                 power(basis, -exponent).eval().print(c);
 126                 c.s << '}';
 127
 128         } else if (exponent.is_equal(_ex1_2)) {
 129
 130                 // Square roots are printed in a special way
 131                 c.s << "\\sqrt{";
 132                 basis.print(c);
 133                 c.s << '}';
 134
 135         } else
 136                 print_power(c, "^", "{", "}", level);
 137 }
 138
 139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 140 {
 141         // Optimal output of integer powers of symbols to aid compiler CSE.
 142         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 143         // to learn why such a parenthesation is really necessary.
 144         if (exp == 1) {
 145                 x.print(c);
 146         } else if (exp == 2) {
 147                 x.print(c);
 148                 c.s << "*";
 149                 x.print(c);
 150         } else if (exp & 1) {
 151                 x.print(c);
 152                 c.s << "*";
 153                 print_sym_pow(c, x, exp-1);
 154         } else {
 155                 c.s << "(";
 156                 print_sym_pow(c, x, exp >> 1);
 157                 c.s << ")*(";
 158                 print_sym_pow(c, x, exp >> 1);
 159                 c.s << ")";
 160         }
 161 }
 162
 163 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 164 {
 165         // Integer powers of symbols are printed in a special, optimized way
 166         if (exponent.info(info_flags::integer)
 167          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 168                 int exp = ex_to<numeric>(exponent).to_int();
 169                 if (exp > 0)
 170                         c.s << '(';
 171                 else {
 172                         exp = -exp;
 173                         if (is_a<print_csrc_cl_N>(c))
 174                                 c.s << "recip(";
 175                         else
 176                                 c.s << "1.0/(";
 177                 }
 178                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 179                 c.s << ')';
 180
 181         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 182         } else if (exponent.is_equal(_ex_1)) {
 183                 if (is_a<print_csrc_cl_N>(c))
 184                         c.s << "recip(";
 185                 else
 186                         c.s << "1.0/(";
 187                 basis.print(c);
 188                 c.s << ')';
 189
 190         // Otherwise, use the pow() or expt() (CLN) functions
 191         } else {
 192                 if (is_a<print_csrc_cl_N>(c))
 193                         c.s << "expt(";
 194                 else
 195                         c.s << "pow(";
 196                 basis.print(c);
 197                 c.s << ',';
 198                 exponent.print(c);
 199                 c.s << ')';
 200         }
 201 }
 202
 203 void power::do_print_python(const print_python & c, unsigned level) const
 204 {
 205         print_power(c, "**", "", "", level);
 206 }
 207
 208 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 209 {
 210         c.s << class_name() << '(';
 211         basis.print(c);
 212         c.s << ',';
 213         exponent.print(c);
 214         c.s << ')';
 215 }
 216
 217 bool power::info(unsigned inf) const
 218 {
 219         switch (inf) {
 220                 case info_flags::polynomial:
 221                 case info_flags::integer_polynomial:
 222                 case info_flags::cinteger_polynomial:
 223                 case info_flags::rational_polynomial:
 224                 case info_flags::crational_polynomial:
 225                         return exponent.info(info_flags::nonnegint);
 226                 case info_flags::rational_function:
 227                         return exponent.info(info_flags::integer);
 228                 case info_flags::algebraic:
 229                         return (!exponent.info(info_flags::integer) ||
 230                                         basis.info(inf));
 231         }
 232         return inherited::info(inf);
 233 }
 234
 235 size_t power::nops() const
 236 {
 237         return 2;
 238 }
 239
 240 ex power::op(size_t i) const
 241 {
 242         GINAC_ASSERT(i<2);
 243
 244         return i==0 ? basis : exponent;
 245 }
 246
 247 ex power::map(map_function & f) const
 248 {
 249         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 250 }
 251
 252 int power::degree(const ex & s) const
 253 {
 254         if (is_equal(ex_to<basic>(s)))
 255                 return 1;
 256         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 257                 if (basis.is_equal(s))
 258                         return ex_to<numeric>(exponent).to_int();
 259                 else
 260                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 261         } else if (basis.has(s))
 262                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 263         else
 264                 return 0;
 265 }
 266
 267 int power::ldegree(const ex & s) const
 268 {
 269         if (is_equal(ex_to<basic>(s)))
 270                 return 1;
 271         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 272                 if (basis.is_equal(s))
 273                         return ex_to<numeric>(exponent).to_int();
 274                 else
 275                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 276         } else if (basis.has(s))
 277                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 278         else
 279                 return 0;
 280 }
 281
 282 ex power::coeff(const ex & s, int n) const
 283 {
 284         if (is_equal(ex_to<basic>(s)))
 285                 return n==1 ? _ex1 : _ex0;
 286         else if (!basis.is_equal(s)) {
 287                 // basis not equal to s
 288                 if (n == 0)
 289                         return *this;
 290                 else
 291                         return _ex0;
 292         } else {
 293                 // basis equal to s
 294                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 295                         // integer exponent
 296                         int int_exp = ex_to<numeric>(exponent).to_int();
 297                         if (n == int_exp)
 298                                 return _ex1;
 299                         else
 300                                 return _ex0;
 301                 } else {
 302                         // non-integer exponents are treated as zero
 303                         if (n == 0)
 304                                 return *this;
 305                         else
 306                                 return _ex0;
 307                 }
 308         }
 309 }
 310
 311 /** Perform automatic term rewriting rules in this class.  In the following
 312  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 313  *  stand for such expressions that contain a plain number.
 314  *  - ^(x,0) -> 1  (also handles ^(0,0))
 315  *  - ^(x,1) -> x
 316  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 317  *  - ^(1,x) -> 1
 318  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 319  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 320  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 321  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 322  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 323  *
 324  *  @param level cut-off in recursive evaluation */
 325 ex power::eval(int level) const
 326 {
 327         if ((level==1) && (flags & status_flags::evaluated))
 328                 return *this;
 329         else if (level == -max_recursion_level)
 330                 throw(std::runtime_error("max recursion level reached"));
 331
 332         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 333         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 334
 335         bool basis_is_numerical = false;
 336         bool exponent_is_numerical = false;
 337         const numeric *num_basis;
 338         const numeric *num_exponent;
 339
 340         if (is_exactly_a<numeric>(ebasis)) {
 341                 basis_is_numerical = true;
 342                 num_basis = &ex_to<numeric>(ebasis);
 343         }
 344         if (is_exactly_a<numeric>(eexponent)) {
 345                 exponent_is_numerical = true;
 346                 num_exponent = &ex_to<numeric>(eexponent);
 347         }
 348
 349         // ^(x,0) -> 1  (0^0 also handled here)
 350         if (eexponent.is_zero()) {
 351                 if (ebasis.is_zero())
 352                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 353                 else
 354                         return _ex1;
 355         }
 356
 357         // ^(x,1) -> x
 358         if (eexponent.is_equal(_ex1))
 359                 return ebasis;
 360
 361         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 362         if (ebasis.is_zero() && exponent_is_numerical) {
 363                 if ((num_exponent->real()).is_zero())
 364                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 365                 else if ((num_exponent->real()).is_negative())
 366                         throw (pole_error("power::eval(): division by zero",1));
 367                 else
 368                         return _ex0;
 369         }
 370
 371         // ^(1,x) -> 1
 372         if (ebasis.is_equal(_ex1))
 373                 return _ex1;
 374
 375         if (exponent_is_numerical) {
 376
 377                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 378                 // except if c1,c2 are rational, but c1^c2 is not)
 379                 if (basis_is_numerical) {
 380                         const bool basis_is_crational = num_basis->is_crational();
 381                         const bool exponent_is_crational = num_exponent->is_crational();
 382                         if (!basis_is_crational || !exponent_is_crational) {
 383                                 // return a plain float
 384                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 385                                                                                                status_flags::evaluated |
 386                                                                                                status_flags::expanded);
 387                         }
 388
 389                         const numeric res = num_basis->power(*num_exponent);
 390                         if (res.is_crational()) {
 391                                 return res;
 392                         }
 393                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 394
 395                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 396                         if (basis_is_crational && exponent_is_crational
 397                             && num_exponent->is_real()
 398                             && !num_exponent->is_integer()) {
 399                                 const numeric n = num_exponent->numer();
 400                                 const numeric m = num_exponent->denom();
 401                                 numeric r;
 402                                 numeric q = iquo(n, m, r);
 403                                 if (r.is_negative()) {
 404                                         r += m;
 405                                         --q;
 406                                 }
 407                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 408                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 409                                                 // try it for numerator and denominator separately, in order to
 410                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 411                                                 const numeric bnum = num_basis->numer();
 412                                                 const numeric bden = num_basis->denom();
 413                                                 const numeric res_bnum = bnum.power(*num_exponent);
 414                                                 const numeric res_bden = bden.power(*num_exponent);
 415                                                 if (res_bnum.is_integer())
 416                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 417                                                 if (res_bden.is_integer())
 418                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 419                                         }
 420                                         return this->hold();
 421                                 } else {
 422                                         // assemble resulting product, but allowing for a re-evaluation,
 423                                         // because otherwise we'll end up with something like
 424                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 425                                         // instead of 7/16*7^(1/3).
 426                                         ex prod = power(*num_basis,r.div(m));
 427                                         return prod*power(*num_basis,q);
 428                                 }
 429                         }
 430                 }
 431
 432                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 433                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 434                 // case c1==1 should not happen, see below!)
 435                 if (is_exactly_a<power>(ebasis)) {
 436                         const power & sub_power = ex_to<power>(ebasis);
 437                         const ex & sub_basis = sub_power.basis;
 438                         const ex & sub_exponent = sub_power.exponent;
 439                         if (is_exactly_a<numeric>(sub_exponent)) {
 440                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 441                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 442                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
 443                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 444                         }
 445                 }
 446
 447                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 448                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 449                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 450                 }
 451
 452                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 453                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 454                 if (is_exactly_a<mul>(ebasis)) {
 455                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 456                         const mul & mulref = ex_to<mul>(ebasis);
 457                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 458                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 459                                 if (num_coeff.is_real()) {
 460                                         if (num_coeff.is_positive()) {
 461                                                 mul *mulp = new mul(mulref);
 462                                                 mulp->overall_coeff = _ex1;
 463                                                 mulp->clearflag(status_flags::evaluated);
 464                                                 mulp->clearflag(status_flags::hash_calculated);
 465                                                 return (new mul(power(*mulp,exponent),
 466                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 467                                         } else {
 468                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
 469                                                 if (!num_coeff.is_equal(_num_1)) {
 470                                                         mul *mulp = new mul(mulref);
 471                                                         mulp->overall_coeff = _ex_1;
 472                                                         mulp->clearflag(status_flags::evaluated);
 473                                                         mulp->clearflag(status_flags::hash_calculated);
 474                                                         return (new mul(power(*mulp,exponent),
 475                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 476                                                 }
 477                                         }
 478                                 }
 479                         }
 480                 }
 481
 482                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 483                 if (num_exponent->is_pos_integer() &&
 484                     ebasis.return_type() != return_types::commutative &&
 485                     !is_a<matrix>(ebasis)) {
 486                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 487                 }
 488         }
 489
 490         if (are_ex_trivially_equal(ebasis,basis) &&
 491             are_ex_trivially_equal(eexponent,exponent)) {
 492                 return this->hold();
 493         }
 494         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 495                                                        status_flags::evaluated);
 496 }
 497
 498 ex power::evalf(int level) const
 499 {
 500         ex ebasis;
 501         ex eexponent;
 502
 503         if (level==1) {
 504                 ebasis = basis;
 505                 eexponent = exponent;
 506         } else if (level == -max_recursion_level) {
 507                 throw(std::runtime_error("max recursion level reached"));
 508         } else {
 509                 ebasis = basis.evalf(level-1);
 510                 if (!is_exactly_a<numeric>(exponent))
 511                         eexponent = exponent.evalf(level-1);
 512                 else
 513                         eexponent = exponent;
 514         }
 515
 516         return power(ebasis,eexponent);
 517 }
 518
 519 ex power::evalm() const
 520 {
 521         const ex ebasis = basis.evalm();
 522         const ex eexponent = exponent.evalm();
 523         if (is_a<matrix>(ebasis)) {
 524                 if (is_exactly_a<numeric>(eexponent)) {
 525                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 526                 }
 527         }
 528         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 529 }
 530
 531 // from mul.cpp
 532 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 533
 534 ex power::subs(const exmap & m, unsigned options) const
 535 {
 536         const ex &subsed_basis = basis.subs(m, options);
 537         const ex &subsed_exponent = exponent.subs(m, options);
 538
 539         if (!are_ex_trivially_equal(basis, subsed_basis)
 540          || !are_ex_trivially_equal(exponent, subsed_exponent))
 541                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 542
 543         if (!(options & subs_options::algebraic))
 544                 return subs_one_level(m, options);
 545
 546         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 547                 int nummatches = std::numeric_limits<int>::max();
 548                 lst repls;
 549                 if (tryfactsubs(*this, it->first, nummatches, repls))
 550                         return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
 551         }
 552
 553         return subs_one_level(m, options);
 554 }
 555
 556 ex power::eval_ncmul(const exvector & v) const
 557 {
 558         return inherited::eval_ncmul(v);
 559 }
 560
 561 // protected
 562
 563 /** Implementation of ex::diff() for a power.
 564  *  @see ex::diff */
 565 ex power::derivative(const symbol & s) const
 566 {
 567         if (exponent.info(info_flags::real)) {
 568                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 569                 epvector newseq;
 570                 newseq.reserve(2);
 571                 newseq.push_back(expair(basis, exponent - _ex1));
 572                 newseq.push_back(expair(basis.diff(s), _ex1));
 573                 return mul(newseq, exponent);
 574         } else {
 575                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 576                 return mul(*this,
 577                            add(mul(exponent.diff(s), log(basis)),
 578                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 579         }
 580 }
 581
 582 int power::compare_same_type(const basic & other) const
 583 {
 584         GINAC_ASSERT(is_exactly_a<power>(other));
 585         const power &o = static_cast<const power &>(other);
 586
 587         int cmpval = basis.compare(o.basis);
 588         if (cmpval)
 589                 return cmpval;
 590         else
 591                 return exponent.compare(o.exponent);
 592 }
 593
 594 unsigned power::return_type() const
 595 {
 596         return basis.return_type();
 597 }
 598
 599 unsigned power::return_type_tinfo() const
 600 {
 601         return basis.return_type_tinfo();
 602 }
 603
 604 ex power::expand(unsigned options) const
 605 {
 606         if (options == 0 && (flags & status_flags::expanded))
 607                 return *this;
 608
 609         const ex expanded_basis = basis.expand(options);
 610         const ex expanded_exponent = exponent.expand(options);
 611
 612         // x^(a+b) -> x^a * x^b
 613         if (is_exactly_a<add>(expanded_exponent)) {
 614                 const add &a = ex_to<add>(expanded_exponent);
 615                 exvector distrseq;
 616                 distrseq.reserve(a.seq.size() + 1);
 617                 epvector::const_iterator last = a.seq.end();
 618                 epvector::const_iterator cit = a.seq.begin();
 619                 while (cit!=last) {
 620                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 621                         ++cit;
 622                 }
 623
 624                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 625                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 626                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 627                         int int_exponent = num_exponent.to_int();
 628                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 629                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 630                         else
 631                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 632                 } else
 633                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 634
 635                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 636                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 637                 return r.expand();
 638         }
 639
 640         if (!is_exactly_a<numeric>(expanded_exponent) ||
 641                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 642                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 643                         return this->hold();
 644                 } else {
 645                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 646                 }
 647         }
 648
 649         // integer numeric exponent
 650         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 651         int int_exponent = num_exponent.to_int();
 652
 653         // (x+y)^n, n>0
 654         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 655                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 656
 657         // (x*y)^n -> x^n * y^n
 658         if (is_exactly_a<mul>(expanded_basis))
 659                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 660
 661         // cannot expand further
 662         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 663                 return this->hold();
 664         else
 665                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 666 }
 667
 668 //////////
 669 // new virtual functions which can be overridden by derived classes
 670 //////////
 671
 672 // none
 673
 674 //////////
 675 // non-virtual functions in this class
 676 //////////
 677
 678 /** expand a^n where a is an add and n is a positive integer.
 679  *  @see power::expand */
 680 ex power::expand_add(const add & a, int n) const
 681 {
 682         if (n==2)
 683                 return expand_add_2(a);
 684
 685         const size_t m = a.nops();
 686         exvector result;
 687         // The number of terms will be the number of combinatorial compositions,
 688         // i.e. the number of unordered arrangement of m nonnegative integers
 689         // which sum up to n.  It is frequently written as C_n(m) and directly
 690         // related with binomial coefficients:
 691         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 692         intvector k(m-1);
 693         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 694         intvector upper_limit(m-1);
 695         int l;
 696
 697         for (size_t l=0; l<m-1; ++l) {
 698                 k[l] = 0;
 699                 k_cum[l] = 0;
 700                 upper_limit[l] = n;
 701         }
 702
 703         while (true) {
 704                 exvector term;
 705                 term.reserve(m+1);
 706                 for (l=0; l<m-1; ++l) {
 707                         const ex & b = a.op(l);
 708                         GINAC_ASSERT(!is_exactly_a<add>(b));
 709                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 710                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 711                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 712                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 713                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 714                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 715                         if (is_exactly_a<mul>(b))
 716                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 717                         else
 718                                 term.push_back(power(b,k[l]));
 719                 }
 720
 721                 const ex & b = a.op(l);
 722                 GINAC_ASSERT(!is_exactly_a<add>(b));
 723                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 724                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 725                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 726                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 727                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 728                              !is_exactly_a<power>(ex_to<power>(b).basis));
 729                 if (is_exactly_a<mul>(b))
 730                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 731                 else
 732                         term.push_back(power(b,n-k_cum[m-2]));
 733
 734                 numeric f = binomial(numeric(n),numeric(k[0]));
 735                 for (l=1; l<m-1; ++l)
 736                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 737
 738                 term.push_back(f);
 739
 740                 result.push_back((new mul(term))->setflag(status_flags::dynallocated));
 741
 742                 // increment k[]
 743                 l = m-2;
 744                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 745                         k[l] = 0;
 746                         --l;
 747                 }
 748                 if (l<0) break;
 749
 750                 // recalc k_cum[] and upper_limit[]
 751                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 752
 753                 for (size_t i=l+1; i<m-1; ++i)
 754                         k_cum[i] = k_cum[i-1]+k[i];
 755
 756                 for (size_t i=l+1; i<m-1; ++i)
 757                         upper_limit[i] = n-k_cum[i-1];
 758         }
 759
 760         return (new add(result))->setflag(status_flags::dynallocated |
 761                                           status_flags::expanded);
 762 }
 763
 764
 765 /** Special case of power::expand_add. Expands a^2 where a is an add.
 766  *  @see power::expand_add */
 767 ex power::expand_add_2(const add & a) const
 768 {
 769         epvector sum;
 770         size_t a_nops = a.nops();
 771         sum.reserve((a_nops*(a_nops+1))/2);
 772         epvector::const_iterator last = a.seq.end();
 773
 774         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 775         // first part: ignore overall_coeff and expand other terms
 776         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 777                 const ex & r = cit0->rest;
 778                 const ex & c = cit0->coeff;
 779
 780                 GINAC_ASSERT(!is_exactly_a<add>(r));
 781                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 782                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 783                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 784                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 785                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 786                              !is_exactly_a<power>(ex_to<power>(r).basis));
 787
 788                 if (c.is_equal(_ex1)) {
 789                         if (is_exactly_a<mul>(r)) {
 790                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 791                                                      _ex1));
 792                         } else {
 793                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 794                                                      _ex1));
 795                         }
 796                 } else {
 797                         if (is_exactly_a<mul>(r)) {
 798                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r),_num2),
 799                                                      ex_to<numeric>(c).power_dyn(_num2)));
 800                         } else {
 801                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 802                                                      ex_to<numeric>(c).power_dyn(_num2)));
 803                         }
 804                 }
 805
 806                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 807                         const ex & r1 = cit1->rest;
 808                         const ex & c1 = cit1->coeff;
 809                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 810                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 811                 }
 812         }
 813
 814         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 815
 816         // second part: add terms coming from overall_factor (if != 0)
 817         if (!a.overall_coeff.is_zero()) {
 818                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 819                 while (i != end) {
 820                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
 821                         ++i;
 822                 }
 823                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
 824         }
 825
 826         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 827
 828         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 829 }
 830
 831 /** Expand factors of m in m^n where m is a mul and n is and integer.
 832  *  @see power::expand */
 833 ex power::expand_mul(const mul & m, const numeric & n) const
 834 {
 835         GINAC_ASSERT(n.is_integer());
 836
 837         if (n.is_zero())
 838                 return _ex1;
 839
 840         epvector distrseq;
 841         distrseq.reserve(m.seq.size());
 842         epvector::const_iterator last = m.seq.end();
 843         epvector::const_iterator cit = m.seq.begin();
 844         while (cit!=last) {
 845                 if (is_exactly_a<numeric>(cit->rest)) {
 846                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
 847                 } else {
 848                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 849                         // since n is an integer
 850                         distrseq.push_back(expair(cit->rest, ex_to<numeric>(cit->coeff).mul(n)));
 851                 }
 852                 ++cit;
 853         }
 854         return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 855 }
 856
 857 } // namespace GiNaC