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