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