ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2005 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "power.h"
  29 #include "expairseq.h"
  30 #include "add.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "numeric.h"
  34 #include "constant.h"
  35 #include "operators.h"
  36 #include "inifcns.h" // for log() in power::derivative()
  37 #include "matrix.h"
  38 #include "indexed.h"
  39 #include "symbol.h"
  40 #include "lst.h"
  41 #include "archive.h"
  42 #include "utils.h"
  43
  44 namespace GiNaC {
  45
  46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  47   print_func<print_dflt>(&power::do_print_dflt).
  48   print_func<print_latex>(&power::do_print_latex).
  49   print_func<print_csrc>(&power::do_print_csrc).
  50   print_func<print_python>(&power::do_print_python).
  51   print_func<print_python_repr>(&power::do_print_python_repr))
  52
  53 typedef std::vector<int> intvector;
  54
  55 //////////
  56 // default constructor
  57 //////////
  58
  59 power::power() : inherited(TINFO_power) { }
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 // all inlined
  66
  67 //////////
  68 // archiving
  69 //////////
  70
  71 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  72 {
  73         n.find_ex("basis", basis, sym_lst);
  74         n.find_ex("exponent", exponent, sym_lst);
  75 }
  76
  77 void power::archive(archive_node &n) const
  78 {
  79         inherited::archive(n);
  80         n.add_ex("basis", basis);
  81         n.add_ex("exponent", exponent);
  82 }
  83
  84 DEFAULT_UNARCHIVE(power)
  85
  86 //////////
  87 // functions overriding virtual functions from base classes
  88 //////////
  89
  90 // public
  91
  92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  93 {
  94         // Ordinary output of powers using '^' or '**'
  95         if (precedence() <= level)
  96                 c.s << openbrace << '(';
  97         basis.print(c, precedence());
  98         c.s << powersymbol;
  99         c.s << openbrace;
 100         exponent.print(c, precedence());
 101         c.s << closebrace;
 102         if (precedence() <= level)
 103                 c.s << ')' << closebrace;
 104 }
 105
 106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 107 {
 108         if (exponent.is_equal(_ex1_2)) {
 109
 110                 // Square roots are printed in a special way
 111                 c.s << "sqrt(";
 112                 basis.print(c);
 113                 c.s << ')';
 114
 115         } else
 116                 print_power(c, "^", "", "", level);
 117 }
 118
 119 void power::do_print_latex(const print_latex & c, unsigned level) const
 120 {
 121         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 122
 123                 // Powers with negative numeric exponents are printed as fractions
 124                 c.s << "\\frac{1}{";
 125                 power(basis, -exponent).eval().print(c);
 126                 c.s << '}';
 127
 128         } else if (exponent.is_equal(_ex1_2)) {
 129
 130                 // Square roots are printed in a special way
 131                 c.s << "\\sqrt{";
 132                 basis.print(c);
 133                 c.s << '}';
 134
 135         } else
 136                 print_power(c, "^", "{", "}", level);
 137 }
 138
 139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 140 {
 141         // Optimal output of integer powers of symbols to aid compiler CSE.
 142         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 143         // to learn why such a parenthesation is really necessary.
 144         if (exp == 1) {
 145                 x.print(c);
 146         } else if (exp == 2) {
 147                 x.print(c);
 148                 c.s << "*";
 149                 x.print(c);
 150         } else if (exp & 1) {
 151                 x.print(c);
 152                 c.s << "*";
 153                 print_sym_pow(c, x, exp-1);
 154         } else {
 155                 c.s << "(";
 156                 print_sym_pow(c, x, exp >> 1);
 157                 c.s << ")*(";
 158                 print_sym_pow(c, x, exp >> 1);
 159                 c.s << ")";
 160         }
 161 }
 162
 163 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 164 {
 165         // Integer powers of symbols are printed in a special, optimized way
 166         if (exponent.info(info_flags::integer)
 167          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 168                 int exp = ex_to<numeric>(exponent).to_int();
 169                 if (exp > 0)
 170                         c.s << '(';
 171                 else {
 172                         exp = -exp;
 173                         if (is_a<print_csrc_cl_N>(c))
 174                                 c.s << "recip(";
 175                         else
 176                                 c.s << "1.0/(";
 177                 }
 178                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 179                 c.s << ')';
 180
 181         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 182         } else if (exponent.is_equal(_ex_1)) {
 183                 if (is_a<print_csrc_cl_N>(c))
 184                         c.s << "recip(";
 185                 else
 186                         c.s << "1.0/(";
 187                 basis.print(c);
 188                 c.s << ')';
 189
 190         // Otherwise, use the pow() or expt() (CLN) functions
 191         } else {
 192                 if (is_a<print_csrc_cl_N>(c))
 193                         c.s << "expt(";
 194                 else
 195                         c.s << "pow(";
 196                 basis.print(c);
 197                 c.s << ',';
 198                 exponent.print(c);
 199                 c.s << ')';
 200         }
 201 }
 202
 203 void power::do_print_python(const print_python & c, unsigned level) const
 204 {
 205         print_power(c, "**", "", "", level);
 206 }
 207
 208 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 209 {
 210         c.s << class_name() << '(';
 211         basis.print(c);
 212         c.s << ',';
 213         exponent.print(c);
 214         c.s << ')';
 215 }
 216
 217 bool power::info(unsigned inf) const
 218 {
 219         switch (inf) {
 220                 case info_flags::polynomial:
 221                 case info_flags::integer_polynomial:
 222                 case info_flags::cinteger_polynomial:
 223                 case info_flags::rational_polynomial:
 224                 case info_flags::crational_polynomial:
 225                         return exponent.info(info_flags::nonnegint) &&
 226                                basis.info(inf);
 227                 case info_flags::rational_function:
 228                         return exponent.info(info_flags::integer) &&
 229                                basis.info(inf);
 230                 case info_flags::algebraic:
 231                         return !exponent.info(info_flags::integer) ||
 232                                basis.info(inf);
 233         }
 234         return inherited::info(inf);
 235 }
 236
 237 size_t power::nops() const
 238 {
 239         return 2;
 240 }
 241
 242 ex power::op(size_t i) const
 243 {
 244         GINAC_ASSERT(i<2);
 245
 246         return i==0 ? basis : exponent;
 247 }
 248
 249 ex power::map(map_function & f) const
 250 {
 251         const ex &mapped_basis = f(basis);
 252         const ex &mapped_exponent = f(exponent);
 253
 254         if (!are_ex_trivially_equal(basis, mapped_basis)
 255          || !are_ex_trivially_equal(exponent, mapped_exponent))
 256                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 257         else
 258                 return *this;
 259 }
 260
 261 int power::degree(const ex & s) const
 262 {
 263         if (is_equal(ex_to<basic>(s)))
 264                 return 1;
 265         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 266                 if (basis.is_equal(s))
 267                         return ex_to<numeric>(exponent).to_int();
 268                 else
 269                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 270         } else if (basis.has(s))
 271                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 272         else
 273                 return 0;
 274 }
 275
 276 int power::ldegree(const ex & s) const
 277 {
 278         if (is_equal(ex_to<basic>(s)))
 279                 return 1;
 280         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 281                 if (basis.is_equal(s))
 282                         return ex_to<numeric>(exponent).to_int();
 283                 else
 284                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 285         } else if (basis.has(s))
 286                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 287         else
 288                 return 0;
 289 }
 290
 291 ex power::coeff(const ex & s, int n) const
 292 {
 293         if (is_equal(ex_to<basic>(s)))
 294                 return n==1 ? _ex1 : _ex0;
 295         else if (!basis.is_equal(s)) {
 296                 // basis not equal to s
 297                 if (n == 0)
 298                         return *this;
 299                 else
 300                         return _ex0;
 301         } else {
 302                 // basis equal to s
 303                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 304                         // integer exponent
 305                         int int_exp = ex_to<numeric>(exponent).to_int();
 306                         if (n == int_exp)
 307                                 return _ex1;
 308                         else
 309                                 return _ex0;
 310                 } else {
 311                         // non-integer exponents are treated as zero
 312                         if (n == 0)
 313                                 return *this;
 314                         else
 315                                 return _ex0;
 316                 }
 317         }
 318 }
 319
 320 /** Perform automatic term rewriting rules in this class.  In the following
 321  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 322  *  stand for such expressions that contain a plain number.
 323  *  - ^(x,0) -> 1  (also handles ^(0,0))
 324  *  - ^(x,1) -> x
 325  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 326  *  - ^(1,x) -> 1
 327  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 328  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 329  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 330  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 331  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 332  *
 333  *  @param level cut-off in recursive evaluation */
 334 ex power::eval(int level) const
 335 {
 336         if ((level==1) && (flags & status_flags::evaluated))
 337                 return *this;
 338         else if (level == -max_recursion_level)
 339                 throw(std::runtime_error("max recursion level reached"));
 340
 341         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 342         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 343
 344         bool basis_is_numerical = false;
 345         bool exponent_is_numerical = false;
 346         const numeric *num_basis;
 347         const numeric *num_exponent;
 348
 349         if (is_exactly_a<numeric>(ebasis)) {
 350                 basis_is_numerical = true;
 351                 num_basis = &ex_to<numeric>(ebasis);
 352         }
 353         if (is_exactly_a<numeric>(eexponent)) {
 354                 exponent_is_numerical = true;
 355                 num_exponent = &ex_to<numeric>(eexponent);
 356         }
 357
 358         // ^(x,0) -> 1  (0^0 also handled here)
 359         if (eexponent.is_zero()) {
 360                 if (ebasis.is_zero())
 361                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 362                 else
 363                         return _ex1;
 364         }
 365
 366         // ^(x,1) -> x
 367         if (eexponent.is_equal(_ex1))
 368                 return ebasis;
 369
 370         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 371         if (ebasis.is_zero() && exponent_is_numerical) {
 372                 if ((num_exponent->real()).is_zero())
 373                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 374                 else if ((num_exponent->real()).is_negative())
 375                         throw (pole_error("power::eval(): division by zero",1));
 376                 else
 377                         return _ex0;
 378         }
 379
 380         // ^(1,x) -> 1
 381         if (ebasis.is_equal(_ex1))
 382                 return _ex1;
 383
 384         if (exponent_is_numerical) {
 385
 386                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 387                 // except if c1,c2 are rational, but c1^c2 is not)
 388                 if (basis_is_numerical) {
 389                         const bool basis_is_crational = num_basis->is_crational();
 390                         const bool exponent_is_crational = num_exponent->is_crational();
 391                         if (!basis_is_crational || !exponent_is_crational) {
 392                                 // return a plain float
 393                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 394                                                                                                status_flags::evaluated |
 395                                                                                                status_flags::expanded);
 396                         }
 397
 398                         const numeric res = num_basis->power(*num_exponent);
 399                         if (res.is_crational()) {
 400                                 return res;
 401                         }
 402                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 403
 404                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 405                         if (basis_is_crational && exponent_is_crational
 406                             && num_exponent->is_real()
 407                             && !num_exponent->is_integer()) {
 408                                 const numeric n = num_exponent->numer();
 409                                 const numeric m = num_exponent->denom();
 410                                 numeric r;
 411                                 numeric q = iquo(n, m, r);
 412                                 if (r.is_negative()) {
 413                                         r += m;
 414                                         --q;
 415                                 }
 416                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 417                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 418                                                 // try it for numerator and denominator separately, in order to
 419                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 420                                                 const numeric bnum = num_basis->numer();
 421                                                 const numeric bden = num_basis->denom();
 422                                                 const numeric res_bnum = bnum.power(*num_exponent);
 423                                                 const numeric res_bden = bden.power(*num_exponent);
 424                                                 if (res_bnum.is_integer())
 425                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 426                                                 if (res_bden.is_integer())
 427                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 428                                         }
 429                                         return this->hold();
 430                                 } else {
 431                                         // assemble resulting product, but allowing for a re-evaluation,
 432                                         // because otherwise we'll end up with something like
 433                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 434                                         // instead of 7/16*7^(1/3).
 435                                         ex prod = power(*num_basis,r.div(m));
 436                                         return prod*power(*num_basis,q);
 437                                 }
 438                         }
 439                 }
 440
 441                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 442                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 443                 // case c1==1 should not happen, see below!)
 444                 if (is_exactly_a<power>(ebasis)) {
 445                         const power & sub_power = ex_to<power>(ebasis);
 446                         const ex & sub_basis = sub_power.basis;
 447                         const ex & sub_exponent = sub_power.exponent;
 448                         if (is_exactly_a<numeric>(sub_exponent)) {
 449                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 450                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 451                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative())
 452                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 453                         }
 454                 }
 455
 456                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 457                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 458                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 459                 }
 460
 461                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 462                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 463                 if (is_exactly_a<mul>(ebasis)) {
 464                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 465                         const mul & mulref = ex_to<mul>(ebasis);
 466                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 467                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 468                                 if (num_coeff.is_real()) {
 469                                         if (num_coeff.is_positive()) {
 470                                                 mul *mulp = new mul(mulref);
 471                                                 mulp->overall_coeff = _ex1;
 472                                                 mulp->clearflag(status_flags::evaluated);
 473                                                 mulp->clearflag(status_flags::hash_calculated);
 474                                                 return (new mul(power(*mulp,exponent),
 475                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 476                                         } else {
 477                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 478                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 479                                                         mul *mulp = new mul(mulref);
 480                                                         mulp->overall_coeff = _ex_1;
 481                                                         mulp->clearflag(status_flags::evaluated);
 482                                                         mulp->clearflag(status_flags::hash_calculated);
 483                                                         return (new mul(power(*mulp,exponent),
 484                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 485                                                 }
 486                                         }
 487                                 }
 488                         }
 489                 }
 490
 491                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 492                 if (num_exponent->is_pos_integer() &&
 493                     ebasis.return_type() != return_types::commutative &&
 494                     !is_a<matrix>(ebasis)) {
 495                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 496                 }
 497         }
 498
 499         if (are_ex_trivially_equal(ebasis,basis) &&
 500             are_ex_trivially_equal(eexponent,exponent)) {
 501                 return this->hold();
 502         }
 503         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 504                                                        status_flags::evaluated);
 505 }
 506
 507 ex power::evalf(int level) const
 508 {
 509         ex ebasis;
 510         ex eexponent;
 511
 512         if (level==1) {
 513                 ebasis = basis;
 514                 eexponent = exponent;
 515         } else if (level == -max_recursion_level) {
 516                 throw(std::runtime_error("max recursion level reached"));
 517         } else {
 518                 ebasis = basis.evalf(level-1);
 519                 if (!is_exactly_a<numeric>(exponent))
 520                         eexponent = exponent.evalf(level-1);
 521                 else
 522                         eexponent = exponent;
 523         }
 524
 525         return power(ebasis,eexponent);
 526 }
 527
 528 ex power::evalm() const
 529 {
 530         const ex ebasis = basis.evalm();
 531         const ex eexponent = exponent.evalm();
 532         if (is_a<matrix>(ebasis)) {
 533                 if (is_exactly_a<numeric>(eexponent)) {
 534                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 535                 }
 536         }
 537         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 538 }
 539
 540 // from mul.cpp
 541 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 542
 543 ex power::subs(const exmap & m, unsigned options) const
 544 {
 545         const ex &subsed_basis = basis.subs(m, options);
 546         const ex &subsed_exponent = exponent.subs(m, options);
 547
 548         if (!are_ex_trivially_equal(basis, subsed_basis)
 549          || !are_ex_trivially_equal(exponent, subsed_exponent))
 550                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 551
 552         if (!(options & subs_options::algebraic))
 553                 return subs_one_level(m, options);
 554
 555         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 556                 int nummatches = std::numeric_limits<int>::max();
 557                 lst repls;
 558                 if (tryfactsubs(*this, it->first, nummatches, repls))
 559                         return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
 560         }
 561
 562         return subs_one_level(m, options);
 563 }
 564
 565 ex power::eval_ncmul(const exvector & v) const
 566 {
 567         return inherited::eval_ncmul(v);
 568 }
 569
 570 ex power::conjugate() const
 571 {
 572         ex newbasis = basis.conjugate();
 573         ex newexponent = exponent.conjugate();
 574         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 575                 return *this;
 576         }
 577         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 578 }
 579
 580 // protected
 581
 582 /** Implementation of ex::diff() for a power.
 583  *  @see ex::diff */
 584 ex power::derivative(const symbol & s) const
 585 {
 586         if (exponent.info(info_flags::real)) {
 587                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 588                 epvector newseq;
 589                 newseq.reserve(2);
 590                 newseq.push_back(expair(basis, exponent - _ex1));
 591                 newseq.push_back(expair(basis.diff(s), _ex1));
 592                 return mul(newseq, exponent);
 593         } else {
 594                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 595                 return mul(*this,
 596                            add(mul(exponent.diff(s), log(basis)),
 597                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 598         }
 599 }
 600
 601 int power::compare_same_type(const basic & other) const
 602 {
 603         GINAC_ASSERT(is_exactly_a<power>(other));
 604         const power &o = static_cast<const power &>(other);
 605
 606         int cmpval = basis.compare(o.basis);
 607         if (cmpval)
 608                 return cmpval;
 609         else
 610                 return exponent.compare(o.exponent);
 611 }
 612
 613 unsigned power::return_type() const
 614 {
 615         return basis.return_type();
 616 }
 617
 618 unsigned power::return_type_tinfo() const
 619 {
 620         return basis.return_type_tinfo();
 621 }
 622
 623 ex power::expand(unsigned options) const
 624 {
 625         if (options == 0 && (flags & status_flags::expanded))
 626                 return *this;
 627
 628         const ex expanded_basis = basis.expand(options);
 629         const ex expanded_exponent = exponent.expand(options);
 630
 631         // x^(a+b) -> x^a * x^b
 632         if (is_exactly_a<add>(expanded_exponent)) {
 633                 const add &a = ex_to<add>(expanded_exponent);
 634                 exvector distrseq;
 635                 distrseq.reserve(a.seq.size() + 1);
 636                 epvector::const_iterator last = a.seq.end();
 637                 epvector::const_iterator cit = a.seq.begin();
 638                 while (cit!=last) {
 639                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 640                         ++cit;
 641                 }
 642
 643                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 644                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 645                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 646                         int int_exponent = num_exponent.to_int();
 647                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 648                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 649                         else
 650                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 651                 } else
 652                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 653
 654                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 655                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 656                 return r.expand(options);
 657         }
 658
 659         if (!is_exactly_a<numeric>(expanded_exponent) ||
 660                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 661                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 662                         return this->hold();
 663                 } else {
 664                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 665                 }
 666         }
 667
 668         // integer numeric exponent
 669         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 670         int int_exponent = num_exponent.to_int();
 671
 672         // (x+y)^n, n>0
 673         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 674                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 675
 676         // (x*y)^n -> x^n * y^n
 677         if (is_exactly_a<mul>(expanded_basis))
 678                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 679
 680         // cannot expand further
 681         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 682                 return this->hold();
 683         else
 684                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 685 }
 686
 687 //////////
 688 // new virtual functions which can be overridden by derived classes
 689 //////////
 690
 691 // none
 692
 693 //////////
 694 // non-virtual functions in this class
 695 //////////
 696
 697 /** expand a^n where a is an add and n is a positive integer.
 698  *  @see power::expand */
 699 ex power::expand_add(const add & a, int n, unsigned options) const
 700 {
 701         if (n==2)
 702                 return expand_add_2(a, options);
 703
 704         const size_t m = a.nops();
 705         exvector result;
 706         // The number of terms will be the number of combinatorial compositions,
 707         // i.e. the number of unordered arrangement of m nonnegative integers
 708         // which sum up to n.  It is frequently written as C_n(m) and directly
 709         // related with binomial coefficients:
 710         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 711         intvector k(m-1);
 712         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 713         intvector upper_limit(m-1);
 714         int l;
 715
 716         for (size_t l=0; l<m-1; ++l) {
 717                 k[l] = 0;
 718                 k_cum[l] = 0;
 719                 upper_limit[l] = n;
 720         }
 721
 722         while (true) {
 723                 exvector term;
 724                 term.reserve(m+1);
 725                 for (l=0; l<m-1; ++l) {
 726                         const ex & b = a.op(l);
 727                         GINAC_ASSERT(!is_exactly_a<add>(b));
 728                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 729                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 730                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 731                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 732                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 733                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 734                         if (is_exactly_a<mul>(b))
 735                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 736                         else
 737                                 term.push_back(power(b,k[l]));
 738                 }
 739
 740                 const ex & b = a.op(l);
 741                 GINAC_ASSERT(!is_exactly_a<add>(b));
 742                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 743                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 744                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 745                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 746                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 747                              !is_exactly_a<power>(ex_to<power>(b).basis));
 748                 if (is_exactly_a<mul>(b))
 749                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 750                 else
 751                         term.push_back(power(b,n-k_cum[m-2]));
 752
 753                 numeric f = binomial(numeric(n),numeric(k[0]));
 754                 for (l=1; l<m-1; ++l)
 755                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 756
 757                 term.push_back(f);
 758
 759                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 760
 761                 // increment k[]
 762                 l = m-2;
 763                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 764                         k[l] = 0;
 765                         --l;
 766                 }
 767                 if (l<0) break;
 768
 769                 // recalc k_cum[] and upper_limit[]
 770                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 771
 772                 for (size_t i=l+1; i<m-1; ++i)
 773                         k_cum[i] = k_cum[i-1]+k[i];
 774
 775                 for (size_t i=l+1; i<m-1; ++i)
 776                         upper_limit[i] = n-k_cum[i-1];
 777         }
 778
 779         return (new add(result))->setflag(status_flags::dynallocated |
 780                                           status_flags::expanded);
 781 }
 782
 783
 784 /** Special case of power::expand_add. Expands a^2 where a is an add.
 785  *  @see power::expand_add */
 786 ex power::expand_add_2(const add & a, unsigned options) const
 787 {
 788         epvector sum;
 789         size_t a_nops = a.nops();
 790         sum.reserve((a_nops*(a_nops+1))/2);
 791         epvector::const_iterator last = a.seq.end();
 792
 793         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 794         // first part: ignore overall_coeff and expand other terms
 795         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 796                 const ex & r = cit0->rest;
 797                 const ex & c = cit0->coeff;
 798
 799                 GINAC_ASSERT(!is_exactly_a<add>(r));
 800                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 801                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 802                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 803                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 804                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 805                              !is_exactly_a<power>(ex_to<power>(r).basis));
 806
 807                 if (c.is_equal(_ex1)) {
 808                         if (is_exactly_a<mul>(r)) {
 809                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 810                                                      _ex1));
 811                         } else {
 812                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 813                                                      _ex1));
 814                         }
 815                 } else {
 816                         if (is_exactly_a<mul>(r)) {
 817                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 818                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 819                         } else {
 820                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 821                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 822                         }
 823                 }
 824
 825                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 826                         const ex & r1 = cit1->rest;
 827                         const ex & c1 = cit1->coeff;
 828                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 829                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 830                 }
 831         }
 832
 833         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 834
 835         // second part: add terms coming from overall_factor (if != 0)
 836         if (!a.overall_coeff.is_zero()) {
 837                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 838                 while (i != end) {
 839                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 840                         ++i;
 841                 }
 842                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 843         }
 844
 845         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 846
 847         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 848 }
 849
 850 /** Expand factors of m in m^n where m is a mul and n is and integer.
 851  *  @see power::expand */
 852 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 853 {
 854         GINAC_ASSERT(n.is_integer());
 855
 856         if (n.is_zero()) {
 857                 return _ex1;
 858         }
 859
 860         // Leave it to multiplication since dummy indices have to be renamed
 861         if (get_all_dummy_indices(m).size() > 0) {
 862                 ex result = m;
 863                 for (int i=1; i < n.to_int(); i++)
 864                         result *= rename_dummy_indices_uniquely(m,m);
 865                 return result;
 866         }
 867
 868         epvector distrseq;
 869         distrseq.reserve(m.seq.size());
 870         bool need_reexpand = false;
 871
 872         epvector::const_iterator last = m.seq.end();
 873         epvector::const_iterator cit = m.seq.begin();
 874         while (cit!=last) {
 875                 if (is_exactly_a<numeric>(cit->rest)) {
 876                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
 877                 } else {
 878                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 879                         // since n is an integer
 880                         numeric new_coeff = ex_to<numeric>(cit->coeff).mul(n);
 881                         if (from_expand && is_exactly_a<add>(cit->rest) && new_coeff.is_pos_integer()) {
 882                                 // this happens when e.g. (a+b)^(1/2) gets squared and
 883                                 // the resulting product needs to be reexpanded
 884                                 need_reexpand = true;
 885                         }
 886                         distrseq.push_back(expair(cit->rest, new_coeff));
 887                 }
 888                 ++cit;
 889         }
 890
 891         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
 892         if (need_reexpand)
 893                 return ex(result).expand(options);
 894         if (from_expand)
 895                 return result.setflag(status_flags::expanded);
 896         return result;
 897 }
 898
 899 } // namespace GiNaC