ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 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
  27 #include "power.h"
  28 #include "expairseq.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "numeric.h"
  33 #include "constant.h"
  34 #include "inifcns.h" // for log() in power::derivative()
  35 #include "matrix.h"
  36 #include "indexed.h"
  37 #include "symbol.h"
  38 #include "print.h"
  39 #include "archive.h"
  40 #include "utils.h"
  41
  42 namespace GiNaC {
  43
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  45
  46 typedef std::vector<int> intvector;
  47
  48 //////////
  49 // default ctor, dtor, copy ctor, assignment operator and helpers
  50 //////////
  51
  52 power::power() : inherited(TINFO_power) { }
  53
  54 void power::copy(const power & other)
  55 {
  56         inherited::copy(other);
  57         basis = other.basis;
  58         exponent = other.exponent;
  59 }
  60
  61 DEFAULT_DESTROY(power)
  62
  63 //////////
  64 // other ctors
  65 //////////
  66
  67 // all inlined
  68
  69 //////////
  70 // archiving
  71 //////////
  72
  73 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  74 {
  75         n.find_ex("basis", basis, sym_lst);
  76         n.find_ex("exponent", exponent, sym_lst);
  77 }
  78
  79 void power::archive(archive_node &n) const
  80 {
  81         inherited::archive(n);
  82         n.add_ex("basis", basis);
  83         n.add_ex("exponent", exponent);
  84 }
  85
  86 DEFAULT_UNARCHIVE(power)
  87
  88 //////////
  89 // functions overriding virtual functions from base classes
  90 //////////
  91
  92 // public
  93
  94 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
  95 {
  96         // Optimal output of integer powers of symbols to aid compiler CSE.
  97         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
  98         // to learn why such a parenthisation is really necessary.
  99         if (exp == 1) {
 100                 x.print(c);
 101         } else if (exp == 2) {
 102                 x.print(c);
 103                 c.s << "*";
 104                 x.print(c);
 105         } else if (exp & 1) {
 106                 x.print(c);
 107                 c.s << "*";
 108                 print_sym_pow(c, x, exp-1);
 109         } else {
 110                 c.s << "(";
 111                 print_sym_pow(c, x, exp >> 1);
 112                 c.s << ")*(";
 113                 print_sym_pow(c, x, exp >> 1);
 114                 c.s << ")";
 115         }
 116 }
 117
 118 void power::print(const print_context & c, unsigned level) const
 119 {
 120         if (is_a<print_tree>(c)) {
 121
 122                 inherited::print(c, level);
 123
 124         } else if (is_a<print_csrc>(c)) {
 125
 126                 // Integer powers of symbols are printed in a special, optimized way
 127                 if (exponent.info(info_flags::integer)
 128                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
 129                         int exp = ex_to<numeric>(exponent).to_int();
 130                         if (exp > 0)
 131                                 c.s << '(';
 132                         else {
 133                                 exp = -exp;
 134                                 if (is_a<print_csrc_cl_N>(c))
 135                                         c.s << "recip(";
 136                                 else
 137                                         c.s << "1.0/(";
 138                         }
 139                         print_sym_pow(c, ex_to<symbol>(basis), exp);
 140                         c.s << ')';
 141
 142                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 143                 } else if (exponent.is_equal(_ex_1)) {
 144                         if (is_a<print_csrc_cl_N>(c))
 145                                 c.s << "recip(";
 146                         else
 147                                 c.s << "1.0/(";
 148                         basis.print(c);
 149                         c.s << ')';
 150
 151                 // Otherwise, use the pow() or expt() (CLN) functions
 152                 } else {
 153                         if (is_a<print_csrc_cl_N>(c))
 154                                 c.s << "expt(";
 155                         else
 156                                 c.s << "pow(";
 157                         basis.print(c);
 158                         c.s << ',';
 159                         exponent.print(c);
 160                         c.s << ')';
 161                 }
 162
 163         } else if (is_a<print_python_repr>(c)) {
 164
 165                 c.s << class_name() << '(';
 166                 basis.print(c);
 167                 c.s << ',';
 168                 exponent.print(c);
 169                 c.s << ')';
 170
 171         } else {
 172
 173                 bool is_tex = is_a<print_latex>(c);
 174
 175                 if (exponent.is_equal(_ex1_2)) {
 176                         c.s << (is_tex ? "\\sqrt{" : "sqrt(");
 177                         basis.print(c);
 178                         c.s << (is_tex ? '}' : ')');
 179                 } else {
 180                         if (precedence() <= level)
 181                                 c.s << (is_tex ? "{(" : "(");
 182                         basis.print(c, precedence());
 183                         if (is_a<print_python>(c))
 184                                 c.s << "**";
 185                         else
 186                                 c.s << '^';
 187                         if (is_tex)
 188                                 c.s << '{';
 189                         exponent.print(c, precedence());
 190                         if (is_tex)
 191                                 c.s << '}';
 192                         if (precedence() <= level)
 193                                 c.s << (is_tex ? ")}" : ")");
 194                 }
 195         }
 196 }
 197
 198 bool power::info(unsigned inf) const
 199 {
 200         switch (inf) {
 201                 case info_flags::polynomial:
 202                 case info_flags::integer_polynomial:
 203                 case info_flags::cinteger_polynomial:
 204                 case info_flags::rational_polynomial:
 205                 case info_flags::crational_polynomial:
 206                         return exponent.info(info_flags::nonnegint);
 207                 case info_flags::rational_function:
 208                         return exponent.info(info_flags::integer);
 209                 case info_flags::algebraic:
 210                         return (!exponent.info(info_flags::integer) ||
 211                                         basis.info(inf));
 212         }
 213         return inherited::info(inf);
 214 }
 215
 216 unsigned power::nops() const
 217 {
 218         return 2;
 219 }
 220
 221 ex & power::let_op(int i)
 222 {
 223         GINAC_ASSERT(i>=0);
 224         GINAC_ASSERT(i<2);
 225
 226         return i==0 ? basis : exponent;
 227 }
 228
 229 ex power::map(map_function & f) const
 230 {
 231         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 232 }
 233
 234 int power::degree(const ex & s) const
 235 {
 236         if (is_equal(ex_to<basic>(s)))
 237                 return 1;
 238         else if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 239                 if (basis.is_equal(s))
 240                         return ex_to<numeric>(exponent).to_int();
 241                 else
 242                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 243         } else if (basis.has(s))
 244                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 245         else
 246                 return 0;
 247 }
 248
 249 int power::ldegree(const ex & s) const
 250 {
 251         if (is_equal(ex_to<basic>(s)))
 252                 return 1;
 253         else if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 254                 if (basis.is_equal(s))
 255                         return ex_to<numeric>(exponent).to_int();
 256                 else
 257                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 258         } else if (basis.has(s))
 259                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 260         else
 261                 return 0;
 262 }
 263
 264 ex power::coeff(const ex & s, int n) const
 265 {
 266         if (is_equal(ex_to<basic>(s)))
 267                 return n==1 ? _ex1 : _ex0;
 268         else if (!basis.is_equal(s)) {
 269                 // basis not equal to s
 270                 if (n == 0)
 271                         return *this;
 272                 else
 273                         return _ex0;
 274         } else {
 275                 // basis equal to s
 276                 if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 277                         // integer exponent
 278                         int int_exp = ex_to<numeric>(exponent).to_int();
 279                         if (n == int_exp)
 280                                 return _ex1;
 281                         else
 282                                 return _ex0;
 283                 } else {
 284                         // non-integer exponents are treated as zero
 285                         if (n == 0)
 286                                 return *this;
 287                         else
 288                                 return _ex0;
 289                 }
 290         }
 291 }
 292
 293 /** Perform automatic term rewriting rules in this class.  In the following
 294  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 295  *  stand for such expressions that contain a plain number.
 296  *  - ^(x,0) -> 1  (also handles ^(0,0))
 297  *  - ^(x,1) -> x
 298  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 299  *  - ^(1,x) -> 1
 300  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 301  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 302  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 303  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 304  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 305  *
 306  *  @param level cut-off in recursive evaluation */
 307 ex power::eval(int level) const
 308 {
 309         if ((level==1) && (flags & status_flags::evaluated))
 310                 return *this;
 311         else if (level == -max_recursion_level)
 312                 throw(std::runtime_error("max recursion level reached"));
 313
 314         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 315         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 316
 317         bool basis_is_numerical = false;
 318         bool exponent_is_numerical = false;
 319         const numeric *num_basis;
 320         const numeric *num_exponent;
 321
 322         if (is_ex_exactly_of_type(ebasis, numeric)) {
 323                 basis_is_numerical = true;
 324                 num_basis = &ex_to<numeric>(ebasis);
 325         }
 326         if (is_ex_exactly_of_type(eexponent, numeric)) {
 327                 exponent_is_numerical = true;
 328                 num_exponent = &ex_to<numeric>(eexponent);
 329         }
 330
 331         // ^(x,0) -> 1  (0^0 also handled here)
 332         if (eexponent.is_zero()) {
 333                 if (ebasis.is_zero())
 334                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 335                 else
 336                         return _ex1;
 337         }
 338
 339         // ^(x,1) -> x
 340         if (eexponent.is_equal(_ex1))
 341                 return ebasis;
 342
 343         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 344         if (ebasis.is_zero() && exponent_is_numerical) {
 345                 if ((num_exponent->real()).is_zero())
 346                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 347                 else if ((num_exponent->real()).is_negative())
 348                         throw (pole_error("power::eval(): division by zero",1));
 349                 else
 350                         return _ex0;
 351         }
 352
 353         // ^(1,x) -> 1
 354         if (ebasis.is_equal(_ex1))
 355                 return _ex1;
 356
 357         if (exponent_is_numerical) {
 358
 359                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 360                 // except if c1,c2 are rational, but c1^c2 is not)
 361                 if (basis_is_numerical) {
 362                         const bool basis_is_crational = num_basis->is_crational();
 363                         const bool exponent_is_crational = num_exponent->is_crational();
 364                         if (!basis_is_crational || !exponent_is_crational) {
 365                                 // return a plain float
 366                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 367                                                                                                status_flags::evaluated |
 368                                                                                                status_flags::expanded);
 369                         }
 370
 371                         const numeric res = num_basis->power(*num_exponent);
 372                         if (res.is_crational()) {
 373                                 return res;
 374                         }
 375                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 376
 377                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 378                         if (basis_is_crational && exponent_is_crational
 379                             && num_exponent->is_real()
 380                             && !num_exponent->is_integer()) {
 381                                 const numeric n = num_exponent->numer();
 382                                 const numeric m = num_exponent->denom();
 383                                 numeric r;
 384                                 numeric q = iquo(n, m, r);
 385                                 if (r.is_negative()) {
 386                                         r += m;
 387                                         --q;
 388                                 }
 389                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 390                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 391                                                 // try it for numerator and denominator separately, in order to
 392                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 393                                                 const numeric bnum = num_basis->numer();
 394                                                 const numeric bden = num_basis->denom();
 395                                                 const numeric res_bnum = bnum.power(*num_exponent);
 396                                                 const numeric res_bden = bden.power(*num_exponent);
 397                                                 if (res_bnum.is_integer())
 398                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 399                                                 if (res_bden.is_integer())
 400                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 401                                         }
 402                                         return this->hold();
 403                                 } else {
 404                                         // assemble resulting product, but allowing for a re-evaluation,
 405                                         // because otherwise we'll end up with something like
 406                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 407                                         // instead of 7/16*7^(1/3).
 408                                         ex prod = power(*num_basis,r.div(m));
 409                                         return prod*power(*num_basis,q);
 410                                 }
 411                         }
 412                 }
 413
 414                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 415                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 416                 // case c1==1 should not happen, see below!)
 417                 if (is_ex_exactly_of_type(ebasis,power)) {
 418                         const power & sub_power = ex_to<power>(ebasis);
 419                         const ex & sub_basis = sub_power.basis;
 420                         const ex & sub_exponent = sub_power.exponent;
 421                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 422                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 423                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 424                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
 425                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 426                         }
 427                 }
 428
 429                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 430                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
 431                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 432                 }
 433
 434                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 435                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 436                 if (is_ex_exactly_of_type(ebasis,mul)) {
 437                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 438                         const mul & mulref = ex_to<mul>(ebasis);
 439                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 440                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 441                                 if (num_coeff.is_real()) {
 442                                         if (num_coeff.is_positive()) {
 443                                                 mul *mulp = new mul(mulref);
 444                                                 mulp->overall_coeff = _ex1;
 445                                                 mulp->clearflag(status_flags::evaluated);
 446                                                 mulp->clearflag(status_flags::hash_calculated);
 447                                                 return (new mul(power(*mulp,exponent),
 448                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 449                                         } else {
 450                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
 451                                                 if (!num_coeff.is_equal(_num_1)) {
 452                                                         mul *mulp = new mul(mulref);
 453                                                         mulp->overall_coeff = _ex_1;
 454                                                         mulp->clearflag(status_flags::evaluated);
 455                                                         mulp->clearflag(status_flags::hash_calculated);
 456                                                         return (new mul(power(*mulp,exponent),
 457                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 458                                                 }
 459                                         }
 460                                 }
 461                         }
 462                 }
 463
 464                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 465                 if (num_exponent->is_pos_integer() &&
 466                     ebasis.return_type() != return_types::commutative &&
 467                     !is_ex_of_type(ebasis,matrix)) {
 468                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 469                 }
 470         }
 471
 472         if (are_ex_trivially_equal(ebasis,basis) &&
 473             are_ex_trivially_equal(eexponent,exponent)) {
 474                 return this->hold();
 475         }
 476         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 477                                                        status_flags::evaluated);
 478 }
 479
 480 ex power::evalf(int level) const
 481 {
 482         ex ebasis;
 483         ex eexponent;
 484
 485         if (level==1) {
 486                 ebasis = basis;
 487                 eexponent = exponent;
 488         } else if (level == -max_recursion_level) {
 489                 throw(std::runtime_error("max recursion level reached"));
 490         } else {
 491                 ebasis = basis.evalf(level-1);
 492                 if (!is_exactly_a<numeric>(exponent))
 493                         eexponent = exponent.evalf(level-1);
 494                 else
 495                         eexponent = exponent;
 496         }
 497
 498         return power(ebasis,eexponent);
 499 }
 500
 501 ex power::evalm(void) const
 502 {
 503         const ex ebasis = basis.evalm();
 504         const ex eexponent = exponent.evalm();
 505         if (is_ex_of_type(ebasis,matrix)) {
 506                 if (is_ex_of_type(eexponent,numeric)) {
 507                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 508                 }
 509         }
 510         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 511 }
 512
 513 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 514 {
 515         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 516         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 517
 518         if (are_ex_trivially_equal(basis, subsed_basis)
 519          && are_ex_trivially_equal(exponent, subsed_exponent))
 520                 return basic::subs(ls, lr, no_pattern);
 521         else
 522                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
 523 }
 524
 525 ex power::simplify_ncmul(const exvector & v) const
 526 {
 527         return inherited::simplify_ncmul(v);
 528 }
 529
 530 // protected
 531
 532 /** Implementation of ex::diff() for a power.
 533  *  @see ex::diff */
 534 ex power::derivative(const symbol & s) const
 535 {
 536         if (exponent.info(info_flags::real)) {
 537                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 538                 epvector newseq;
 539                 newseq.reserve(2);
 540                 newseq.push_back(expair(basis, exponent - _ex1));
 541                 newseq.push_back(expair(basis.diff(s), _ex1));
 542                 return mul(newseq, exponent);
 543         } else {
 544                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 545                 return mul(*this,
 546                            add(mul(exponent.diff(s), log(basis)),
 547                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 548         }
 549 }
 550
 551 int power::compare_same_type(const basic & other) const
 552 {
 553         GINAC_ASSERT(is_exactly_a<power>(other));
 554         const power &o = static_cast<const power &>(other);
 555
 556         int cmpval = basis.compare(o.basis);
 557         if (cmpval)
 558                 return cmpval;
 559         else
 560                 return exponent.compare(o.exponent);
 561 }
 562
 563 unsigned power::return_type(void) const
 564 {
 565         return basis.return_type();
 566 }
 567
 568 unsigned power::return_type_tinfo(void) const
 569 {
 570         return basis.return_type_tinfo();
 571 }
 572
 573 ex power::expand(unsigned options) const
 574 {
 575         if (options == 0 && (flags & status_flags::expanded))
 576                 return *this;
 577
 578         const ex expanded_basis = basis.expand(options);
 579         const ex expanded_exponent = exponent.expand(options);
 580
 581         // x^(a+b) -> x^a * x^b
 582         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 583                 const add &a = ex_to<add>(expanded_exponent);
 584                 exvector distrseq;
 585                 distrseq.reserve(a.seq.size() + 1);
 586                 epvector::const_iterator last = a.seq.end();
 587                 epvector::const_iterator cit = a.seq.begin();
 588                 while (cit!=last) {
 589                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 590                         ++cit;
 591                 }
 592
 593                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 594                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 595                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 596                         int int_exponent = num_exponent.to_int();
 597                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 598                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 599                         else
 600                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 601                 } else
 602                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 603
 604                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 605                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 606                 return r.expand();
 607         }
 608
 609         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 610                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 611                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 612                         return this->hold();
 613                 } else {
 614                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 615                 }
 616         }
 617
 618         // integer numeric exponent
 619         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 620         int int_exponent = num_exponent.to_int();
 621
 622         // (x+y)^n, n>0
 623         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 624                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 625
 626         // (x*y)^n -> x^n * y^n
 627         if (is_ex_exactly_of_type(expanded_basis,mul))
 628                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 629
 630         // cannot expand further
 631         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 632                 return this->hold();
 633         else
 634                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 635 }
 636
 637 //////////
 638 // new virtual functions which can be overridden by derived classes
 639 //////////
 640
 641 // none
 642
 643 //////////
 644 // non-virtual functions in this class
 645 //////////
 646
 647 /** expand a^n where a is an add and n is a positive integer.
 648  *  @see power::expand */
 649 ex power::expand_add(const add & a, int n) const
 650 {
 651         if (n==2)
 652                 return expand_add_2(a);
 653
 654         const int m = a.nops();
 655         exvector result;
 656         // The number of terms will be the number of combinatorial compositions,
 657         // i.e. the number of unordered arrangement of m nonnegative integers
 658         // which sum up to n.  It is frequently written as C_n(m) and directly
 659         // related with binomial coefficients:
 660         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 661         intvector k(m-1);
 662         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 663         intvector upper_limit(m-1);
 664         int l;
 665
 666         for (int l=0; l<m-1; ++l) {
 667                 k[l] = 0;
 668                 k_cum[l] = 0;
 669                 upper_limit[l] = n;
 670         }
 671
 672         while (true) {
 673                 exvector term;
 674                 term.reserve(m+1);
 675                 for (l=0; l<m-1; ++l) {
 676                         const ex & b = a.op(l);
 677                         GINAC_ASSERT(!is_exactly_a<add>(b));
 678                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 679                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 680                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 681                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 682                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 683                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 684                         if (is_ex_exactly_of_type(b,mul))
 685                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 686                         else
 687                                 term.push_back(power(b,k[l]));
 688                 }
 689
 690                 const ex & b = a.op(l);
 691                 GINAC_ASSERT(!is_exactly_a<add>(b));
 692                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 693                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 694                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 695                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 696                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 697                              !is_exactly_a<power>(ex_to<power>(b).basis));
 698                 if (is_ex_exactly_of_type(b,mul))
 699                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 700                 else
 701                         term.push_back(power(b,n-k_cum[m-2]));
 702
 703                 numeric f = binomial(numeric(n),numeric(k[0]));
 704                 for (l=1; l<m-1; ++l)
 705                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 706
 707                 term.push_back(f);
 708
 709                 result.push_back((new mul(term))->setflag(status_flags::dynallocated));
 710
 711                 // increment k[]
 712                 l = m-2;
 713                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 714                         k[l] = 0;
 715                         --l;
 716                 }
 717                 if (l<0) break;
 718
 719                 // recalc k_cum[] and upper_limit[]
 720                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 721
 722                 for (int i=l+1; i<m-1; ++i)
 723                         k_cum[i] = k_cum[i-1]+k[i];
 724
 725                 for (int i=l+1; i<m-1; ++i)
 726                         upper_limit[i] = n-k_cum[i-1];
 727         }
 728
 729         return (new add(result))->setflag(status_flags::dynallocated |
 730                                           status_flags::expanded);
 731 }
 732
 733
 734 /** Special case of power::expand_add. Expands a^2 where a is an add.
 735  *  @see power::expand_add */
 736 ex power::expand_add_2(const add & a) const
 737 {
 738         epvector sum;
 739         unsigned a_nops = a.nops();
 740         sum.reserve((a_nops*(a_nops+1))/2);
 741         epvector::const_iterator last = a.seq.end();
 742
 743         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 744         // first part: ignore overall_coeff and expand other terms
 745         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 746                 const ex & r = cit0->rest;
 747                 const ex & c = cit0->coeff;
 748
 749                 GINAC_ASSERT(!is_exactly_a<add>(r));
 750                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 751                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 752                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 753                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 754                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 755                              !is_exactly_a<power>(ex_to<power>(r).basis));
 756
 757                 if (are_ex_trivially_equal(c,_ex1)) {
 758                         if (is_ex_exactly_of_type(r,mul)) {
 759                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 760                                                      _ex1));
 761                         } else {
 762                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 763                                                      _ex1));
 764                         }
 765                 } else {
 766                         if (is_ex_exactly_of_type(r,mul)) {
 767                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 768                                                      ex_to<numeric>(c).power_dyn(_num2)));
 769                         } else {
 770                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 771                                                      ex_to<numeric>(c).power_dyn(_num2)));
 772                         }
 773                 }
 774
 775                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 776                         const ex & r1 = cit1->rest;
 777                         const ex & c1 = cit1->coeff;
 778                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 779                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 780                 }
 781         }
 782
 783         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 784
 785         // second part: add terms coming from overall_factor (if != 0)
 786         if (!a.overall_coeff.is_zero()) {
 787                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 788                 while (i != end) {
 789                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
 790                         ++i;
 791                 }
 792                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
 793         }
 794
 795         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 796
 797         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 798 }
 799
 800 /** Expand factors of m in m^n where m is a mul and n is and integer.
 801  *  @see power::expand */
 802 ex power::expand_mul(const mul & m, const numeric & n) const
 803 {
 804         GINAC_ASSERT(n.is_integer());
 805
 806         if (n.is_zero())
 807                 return _ex1;
 808
 809         epvector distrseq;
 810         distrseq.reserve(m.seq.size());
 811         epvector::const_iterator last = m.seq.end();
 812         epvector::const_iterator cit = m.seq.begin();
 813         while (cit!=last) {
 814                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 815                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 816                 } else {
 817                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 818                         // since n is an integer
 819                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
 820                 }
 821                 ++cit;
 822         }
 823         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 824 }
 825
 826 } // namespace GiNaC