ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2007 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 #include "relational.h"
  44 #include "compiler.h"
  45
  46 namespace GiNaC {
  47
  48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  49   print_func<print_dflt>(&power::do_print_dflt).
  50   print_func<print_latex>(&power::do_print_latex).
  51   print_func<print_csrc>(&power::do_print_csrc).
  52   print_func<print_python>(&power::do_print_python).
  53   print_func<print_python_repr>(&power::do_print_python_repr))
  54
  55 typedef std::vector<int> intvector;
  56
  57 //////////
  58 // default constructor
  59 //////////
  60
  61 power::power() : inherited(&power::tinfo_static) { }
  62
  63 //////////
  64 // other constructors
  65 //////////
  66
  67 // all inlined
  68
  69 //////////
  70 // archiving
  71 //////////
  72
  73 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  74 {
  75         n.find_ex("basis", basis, sym_lst);
  76         n.find_ex("exponent", exponent, sym_lst);
  77 }
  78
  79 void power::archive(archive_node &n) const
  80 {
  81         inherited::archive(n);
  82         n.add_ex("basis", basis);
  83         n.add_ex("exponent", exponent);
  84 }
  85
  86 DEFAULT_UNARCHIVE(power)
  87
  88 //////////
  89 // functions overriding virtual functions from base classes
  90 //////////
  91
  92 // public
  93
  94 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  95 {
  96         // Ordinary output of powers using '^' or '**'
  97         if (precedence() <= level)
  98                 c.s << openbrace << '(';
  99         basis.print(c, precedence());
 100         c.s << powersymbol;
 101         c.s << openbrace;
 102         exponent.print(c, precedence());
 103         c.s << closebrace;
 104         if (precedence() <= level)
 105                 c.s << ')' << closebrace;
 106 }
 107
 108 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 109 {
 110         if (exponent.is_equal(_ex1_2)) {
 111
 112                 // Square roots are printed in a special way
 113                 c.s << "sqrt(";
 114                 basis.print(c);
 115                 c.s << ')';
 116
 117         } else
 118                 print_power(c, "^", "", "", level);
 119 }
 120
 121 void power::do_print_latex(const print_latex & c, unsigned level) const
 122 {
 123         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 124
 125                 // Powers with negative numeric exponents are printed as fractions
 126                 c.s << "\\frac{1}{";
 127                 power(basis, -exponent).eval().print(c);
 128                 c.s << '}';
 129
 130         } else if (exponent.is_equal(_ex1_2)) {
 131
 132                 // Square roots are printed in a special way
 133                 c.s << "\\sqrt{";
 134                 basis.print(c);
 135                 c.s << '}';
 136
 137         } else
 138                 print_power(c, "^", "{", "}", level);
 139 }
 140
 141 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 142 {
 143         // Optimal output of integer powers of symbols to aid compiler CSE.
 144         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 145         // to learn why such a parenthesation is really necessary.
 146         if (exp == 1) {
 147                 x.print(c);
 148         } else if (exp == 2) {
 149                 x.print(c);
 150                 c.s << "*";
 151                 x.print(c);
 152         } else if (exp & 1) {
 153                 x.print(c);
 154                 c.s << "*";
 155                 print_sym_pow(c, x, exp-1);
 156         } else {
 157                 c.s << "(";
 158                 print_sym_pow(c, x, exp >> 1);
 159                 c.s << ")*(";
 160                 print_sym_pow(c, x, exp >> 1);
 161                 c.s << ")";
 162         }
 163 }
 164
 165 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 166 {
 167         if (is_a<print_csrc_cl_N>(c)) {
 168                 if (exponent.is_equal(_ex_1)) {
 169                         c.s << "recip(";
 170                         basis.print(c);
 171                         c.s << ')';
 172                         return;
 173                 }
 174                 c.s << "expt(";
 175                 basis.print(c);
 176                 c.s << ", ";
 177                 exponent.print(c);
 178                 c.s << ')';
 179                 return;
 180         }
 181
 182         // Integer powers of symbols are printed in a special, optimized way
 183         if (exponent.info(info_flags::integer)
 184          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 185                 int exp = ex_to<numeric>(exponent).to_int();
 186                 if (exp > 0)
 187                         c.s << '(';
 188                 else {
 189                         exp = -exp;
 190                         c.s << "1.0/(";
 191                 }
 192                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 193                 c.s << ')';
 194
 195         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 196         } else if (exponent.is_equal(_ex_1)) {
 197                 c.s << "1.0/(";
 198                 basis.print(c);
 199                 c.s << ')';
 200
 201         // Otherwise, use the pow() function
 202         } else {
 203                 c.s << "pow(";
 204                 basis.print(c);
 205                 c.s << ',';
 206                 exponent.print(c);
 207                 c.s << ')';
 208         }
 209 }
 210
 211 void power::do_print_python(const print_python & c, unsigned level) const
 212 {
 213         print_power(c, "**", "", "", level);
 214 }
 215
 216 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 217 {
 218         c.s << class_name() << '(';
 219         basis.print(c);
 220         c.s << ',';
 221         exponent.print(c);
 222         c.s << ')';
 223 }
 224
 225 bool power::info(unsigned inf) const
 226 {
 227         switch (inf) {
 228                 case info_flags::polynomial:
 229                 case info_flags::integer_polynomial:
 230                 case info_flags::cinteger_polynomial:
 231                 case info_flags::rational_polynomial:
 232                 case info_flags::crational_polynomial:
 233                         return exponent.info(info_flags::nonnegint) &&
 234                                basis.info(inf);
 235                 case info_flags::rational_function:
 236                         return exponent.info(info_flags::integer) &&
 237                                basis.info(inf);
 238                 case info_flags::algebraic:
 239                         return !exponent.info(info_flags::integer) ||
 240                                basis.info(inf);
 241                 case info_flags::expanded:
 242                         return (flags & status_flags::expanded);
 243         }
 244         return inherited::info(inf);
 245 }
 246
 247 size_t power::nops() const
 248 {
 249         return 2;
 250 }
 251
 252 ex power::op(size_t i) const
 253 {
 254         GINAC_ASSERT(i<2);
 255
 256         return i==0 ? basis : exponent;
 257 }
 258
 259 ex power::map(map_function & f) const
 260 {
 261         const ex &mapped_basis = f(basis);
 262         const ex &mapped_exponent = f(exponent);
 263
 264         if (!are_ex_trivially_equal(basis, mapped_basis)
 265          || !are_ex_trivially_equal(exponent, mapped_exponent))
 266                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 267         else
 268                 return *this;
 269 }
 270
 271 bool power::is_polynomial(const ex & var) const
 272 {
 273         if (exponent.has(var))
 274                 return false;
 275         if (!exponent.info(info_flags::nonnegint))
 276                 return false;
 277         return basis.is_polynomial(var);
 278 }
 279
 280 int power::degree(const ex & s) const
 281 {
 282         if (is_equal(ex_to<basic>(s)))
 283                 return 1;
 284         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 285                 if (basis.is_equal(s))
 286                         return ex_to<numeric>(exponent).to_int();
 287                 else
 288                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 289         } else if (basis.has(s))
 290                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 291         else
 292                 return 0;
 293 }
 294
 295 int power::ldegree(const ex & s) const
 296 {
 297         if (is_equal(ex_to<basic>(s)))
 298                 return 1;
 299         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 300                 if (basis.is_equal(s))
 301                         return ex_to<numeric>(exponent).to_int();
 302                 else
 303                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 304         } else if (basis.has(s))
 305                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 306         else
 307                 return 0;
 308 }
 309
 310 ex power::coeff(const ex & s, int n) const
 311 {
 312         if (is_equal(ex_to<basic>(s)))
 313                 return n==1 ? _ex1 : _ex0;
 314         else if (!basis.is_equal(s)) {
 315                 // basis not equal to s
 316                 if (n == 0)
 317                         return *this;
 318                 else
 319                         return _ex0;
 320         } else {
 321                 // basis equal to s
 322                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 323                         // integer exponent
 324                         int int_exp = ex_to<numeric>(exponent).to_int();
 325                         if (n == int_exp)
 326                                 return _ex1;
 327                         else
 328                                 return _ex0;
 329                 } else {
 330                         // non-integer exponents are treated as zero
 331                         if (n == 0)
 332                                 return *this;
 333                         else
 334                                 return _ex0;
 335                 }
 336         }
 337 }
 338
 339 /** Perform automatic term rewriting rules in this class.  In the following
 340  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 341  *  stand for such expressions that contain a plain number.
 342  *  - ^(x,0) -> 1  (also handles ^(0,0))
 343  *  - ^(x,1) -> x
 344  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 345  *  - ^(1,x) -> 1
 346  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 347  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 348  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 349  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 350  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 351  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 352  *
 353  *  @param level cut-off in recursive evaluation */
 354 ex power::eval(int level) const
 355 {
 356         if ((level==1) && (flags & status_flags::evaluated))
 357                 return *this;
 358         else if (level == -max_recursion_level)
 359                 throw(std::runtime_error("max recursion level reached"));
 360
 361         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 362         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 363
 364         bool basis_is_numerical = false;
 365         bool exponent_is_numerical = false;
 366         const numeric *num_basis;
 367         const numeric *num_exponent;
 368
 369         if (is_exactly_a<numeric>(ebasis)) {
 370                 basis_is_numerical = true;
 371                 num_basis = &ex_to<numeric>(ebasis);
 372         }
 373         if (is_exactly_a<numeric>(eexponent)) {
 374                 exponent_is_numerical = true;
 375                 num_exponent = &ex_to<numeric>(eexponent);
 376         }
 377
 378         // ^(x,0) -> 1  (0^0 also handled here)
 379         if (eexponent.is_zero()) {
 380                 if (ebasis.is_zero())
 381                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 382                 else
 383                         return _ex1;
 384         }
 385
 386         // ^(x,1) -> x
 387         if (eexponent.is_equal(_ex1))
 388                 return ebasis;
 389
 390         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 391         if (ebasis.is_zero() && exponent_is_numerical) {
 392                 if ((num_exponent->real()).is_zero())
 393                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 394                 else if ((num_exponent->real()).is_negative())
 395                         throw (pole_error("power::eval(): division by zero",1));
 396                 else
 397                         return _ex0;
 398         }
 399
 400         // ^(1,x) -> 1
 401         if (ebasis.is_equal(_ex1))
 402                 return _ex1;
 403
 404         // power of a function calculated by separate rules defined for this function
 405         if (is_exactly_a<function>(ebasis))
 406                 return ex_to<function>(ebasis).power(eexponent);
 407
 408         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 409         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 410                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 411
 412         if (exponent_is_numerical) {
 413
 414                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 415                 // except if c1,c2 are rational, but c1^c2 is not)
 416                 if (basis_is_numerical) {
 417                         const bool basis_is_crational = num_basis->is_crational();
 418                         const bool exponent_is_crational = num_exponent->is_crational();
 419                         if (!basis_is_crational || !exponent_is_crational) {
 420                                 // return a plain float
 421                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 422                                                                                                status_flags::evaluated |
 423                                                                                                status_flags::expanded);
 424                         }
 425
 426                         const numeric res = num_basis->power(*num_exponent);
 427                         if (res.is_crational()) {
 428                                 return res;
 429                         }
 430                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 431
 432                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 433                         if (basis_is_crational && exponent_is_crational
 434                             && num_exponent->is_real()
 435                             && !num_exponent->is_integer()) {
 436                                 const numeric n = num_exponent->numer();
 437                                 const numeric m = num_exponent->denom();
 438                                 numeric r;
 439                                 numeric q = iquo(n, m, r);
 440                                 if (r.is_negative()) {
 441                                         r += m;
 442                                         --q;
 443                                 }
 444                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 445                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 446                                                 // try it for numerator and denominator separately, in order to
 447                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 448                                                 const numeric bnum = num_basis->numer();
 449                                                 const numeric bden = num_basis->denom();
 450                                                 const numeric res_bnum = bnum.power(*num_exponent);
 451                                                 const numeric res_bden = bden.power(*num_exponent);
 452                                                 if (res_bnum.is_integer())
 453                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 454                                                 if (res_bden.is_integer())
 455                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 456                                         }
 457                                         return this->hold();
 458                                 } else {
 459                                         // assemble resulting product, but allowing for a re-evaluation,
 460                                         // because otherwise we'll end up with something like
 461                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 462                                         // instead of 7/16*7^(1/3).
 463                                         ex prod = power(*num_basis,r.div(m));
 464                                         return prod*power(*num_basis,q);
 465                                 }
 466                         }
 467                 }
 468
 469                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 470                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 471                 // case c1==1 should not happen, see below!)
 472                 if (is_exactly_a<power>(ebasis)) {
 473                         const power & sub_power = ex_to<power>(ebasis);
 474                         const ex & sub_basis = sub_power.basis;
 475                         const ex & sub_exponent = sub_power.exponent;
 476                         if (is_exactly_a<numeric>(sub_exponent)) {
 477                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 478                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 479                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
 480                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 481                                 }
 482                         }
 483                 }
 484
 485                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 486                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 487                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 488                 }
 489
 490                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 491                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 492                         numeric icont = ebasis.integer_content();
 493                         const numeric& lead_coeff =
 494                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div_dyn(icont);
 495
 496                         const bool canonicalizable = lead_coeff.is_integer();
 497                         const bool unit_normal = lead_coeff.is_pos_integer();
 498                         if (canonicalizable && (! unit_normal))
 499                                 icont = icont.mul(*_num_1_p);
 500
 501                         if (canonicalizable && (icont != *_num1_p)) {
 502                                 const add& addref = ex_to<add>(ebasis);
 503                                 add* addp = new add(addref);
 504                                 addp->setflag(status_flags::dynallocated);
 505                                 addp->clearflag(status_flags::hash_calculated);
 506                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 507                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 508                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 509
 510                                 const numeric c = icont.power(*num_exponent);
 511                                 if (likely(c != *_num1_p))
 512                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 513                                 else
 514                                         return power(*addp, *num_exponent);
 515                         }
 516                 }
 517
 518                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 519                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 520                 if (is_exactly_a<mul>(ebasis)) {
 521                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 522                         const mul & mulref = ex_to<mul>(ebasis);
 523                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 524                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 525                                 if (num_coeff.is_real()) {
 526                                         if (num_coeff.is_positive()) {
 527                                                 mul *mulp = new mul(mulref);
 528                                                 mulp->overall_coeff = _ex1;
 529                                                 mulp->clearflag(status_flags::evaluated);
 530                                                 mulp->clearflag(status_flags::hash_calculated);
 531                                                 return (new mul(power(*mulp,exponent),
 532                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 533                                         } else {
 534                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 535                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 536                                                         mul *mulp = new mul(mulref);
 537                                                         mulp->overall_coeff = _ex_1;
 538                                                         mulp->clearflag(status_flags::evaluated);
 539                                                         mulp->clearflag(status_flags::hash_calculated);
 540                                                         return (new mul(power(*mulp,exponent),
 541                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 542                                                 }
 543                                         }
 544                                 }
 545                         }
 546                 }
 547
 548                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 549                 if (num_exponent->is_pos_integer() &&
 550                     ebasis.return_type() != return_types::commutative &&
 551                     !is_a<matrix>(ebasis)) {
 552                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 553                 }
 554         }
 555
 556         if (are_ex_trivially_equal(ebasis,basis) &&
 557             are_ex_trivially_equal(eexponent,exponent)) {
 558                 return this->hold();
 559         }
 560         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 561                                                        status_flags::evaluated);
 562 }
 563
 564 ex power::evalf(int level) const
 565 {
 566         ex ebasis;
 567         ex eexponent;
 568
 569         if (level==1) {
 570                 ebasis = basis;
 571                 eexponent = exponent;
 572         } else if (level == -max_recursion_level) {
 573                 throw(std::runtime_error("max recursion level reached"));
 574         } else {
 575                 ebasis = basis.evalf(level-1);
 576                 if (!is_exactly_a<numeric>(exponent))
 577                         eexponent = exponent.evalf(level-1);
 578                 else
 579                         eexponent = exponent;
 580         }
 581
 582         return power(ebasis,eexponent);
 583 }
 584
 585 ex power::evalm() const
 586 {
 587         const ex ebasis = basis.evalm();
 588         const ex eexponent = exponent.evalm();
 589         if (is_a<matrix>(ebasis)) {
 590                 if (is_exactly_a<numeric>(eexponent)) {
 591                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 592                 }
 593         }
 594         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 595 }
 596
 597 bool power::has(const ex & other, unsigned options) const
 598 {
 599         if (!(options & has_options::algebraic))
 600                 return basic::has(other, options);
 601         if (!is_a<power>(other))
 602                 return basic::has(other, options);
 603         if (!exponent.info(info_flags::integer)
 604                         || !other.op(1).info(info_flags::integer))
 605                 return basic::has(other, options);
 606         if (exponent.info(info_flags::posint)
 607                         && other.op(1).info(info_flags::posint)
 608                         && ex_to<numeric>(exponent).to_int()
 609                                         > ex_to<numeric>(other.op(1)).to_int()
 610                         && basis.match(other.op(0)))
 611                 return true;
 612         if (exponent.info(info_flags::negint)
 613                         && other.op(1).info(info_flags::negint)
 614                         && ex_to<numeric>(exponent).to_int()
 615                                         < ex_to<numeric>(other.op(1)).to_int()
 616                         && basis.match(other.op(0)))
 617                 return true;
 618         return basic::has(other, options);
 619 }
 620
 621 // from mul.cpp
 622 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 623
 624 ex power::subs(const exmap & m, unsigned options) const
 625 {
 626         const ex &subsed_basis = basis.subs(m, options);
 627         const ex &subsed_exponent = exponent.subs(m, options);
 628
 629         if (!are_ex_trivially_equal(basis, subsed_basis)
 630          || !are_ex_trivially_equal(exponent, subsed_exponent))
 631                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 632
 633         if (!(options & subs_options::algebraic))
 634                 return subs_one_level(m, options);
 635
 636         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 637                 int nummatches = std::numeric_limits<int>::max();
 638                 lst repls;
 639                 if (tryfactsubs(*this, it->first, nummatches, repls))
 640                         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);
 641         }
 642
 643         return subs_one_level(m, options);
 644 }
 645
 646 ex power::eval_ncmul(const exvector & v) const
 647 {
 648         return inherited::eval_ncmul(v);
 649 }
 650
 651 ex power::conjugate() const
 652 {
 653         ex newbasis = basis.conjugate();
 654         ex newexponent = exponent.conjugate();
 655         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 656                 return *this;
 657         }
 658         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 659 }
 660
 661 ex power::real_part() const
 662 {
 663         if (exponent.info(info_flags::integer)) {
 664                 ex basis_real = basis.real_part();
 665                 if (basis_real == basis)
 666                         return *this;
 667                 realsymbol a("a"),b("b");
 668                 ex result;
 669                 if (exponent.info(info_flags::posint))
 670                         result = power(a+I*b,exponent);
 671                 else
 672                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 673                 result = result.expand();
 674                 result = result.real_part();
 675                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 676                 return result;
 677         }
 678
 679         ex a = basis.real_part();
 680         ex b = basis.imag_part();
 681         ex c = exponent.real_part();
 682         ex d = exponent.imag_part();
 683         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 684 }
 685
 686 ex power::imag_part() const
 687 {
 688         if (exponent.info(info_flags::integer)) {
 689                 ex basis_real = basis.real_part();
 690                 if (basis_real == basis)
 691                         return 0;
 692                 realsymbol a("a"),b("b");
 693                 ex result;
 694                 if (exponent.info(info_flags::posint))
 695                         result = power(a+I*b,exponent);
 696                 else
 697                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 698                 result = result.expand();
 699                 result = result.imag_part();
 700                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 701                 return result;
 702         }
 703
 704         ex a=basis.real_part();
 705         ex b=basis.imag_part();
 706         ex c=exponent.real_part();
 707         ex d=exponent.imag_part();
 708         return
 709                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 710 }
 711
 712 // protected
 713
 714 // protected
 715
 716 /** Implementation of ex::diff() for a power.
 717  *  @see ex::diff */
 718 ex power::derivative(const symbol & s) const
 719 {
 720         if (is_a<numeric>(exponent)) {
 721                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 722                 epvector newseq;
 723                 newseq.reserve(2);
 724                 newseq.push_back(expair(basis, exponent - _ex1));
 725                 newseq.push_back(expair(basis.diff(s), _ex1));
 726                 return mul(newseq, exponent);
 727         } else {
 728                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 729                 return mul(*this,
 730                            add(mul(exponent.diff(s), log(basis)),
 731                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 732         }
 733 }
 734
 735 int power::compare_same_type(const basic & other) const
 736 {
 737         GINAC_ASSERT(is_exactly_a<power>(other));
 738         const power &o = static_cast<const power &>(other);
 739
 740         int cmpval = basis.compare(o.basis);
 741         if (cmpval)
 742                 return cmpval;
 743         else
 744                 return exponent.compare(o.exponent);
 745 }
 746
 747 unsigned power::return_type() const
 748 {
 749         return basis.return_type();
 750 }
 751
 752 tinfo_t power::return_type_tinfo() const
 753 {
 754         return basis.return_type_tinfo();
 755 }
 756
 757 ex power::expand(unsigned options) const
 758 {
 759         if (options == 0 && (flags & status_flags::expanded))
 760                 return *this;
 761
 762         const ex expanded_basis = basis.expand(options);
 763         const ex expanded_exponent = exponent.expand(options);
 764
 765         // x^(a+b) -> x^a * x^b
 766         if (is_exactly_a<add>(expanded_exponent)) {
 767                 const add &a = ex_to<add>(expanded_exponent);
 768                 exvector distrseq;
 769                 distrseq.reserve(a.seq.size() + 1);
 770                 epvector::const_iterator last = a.seq.end();
 771                 epvector::const_iterator cit = a.seq.begin();
 772                 while (cit!=last) {
 773                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 774                         ++cit;
 775                 }
 776
 777                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 778                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 779                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 780                         int int_exponent = num_exponent.to_int();
 781                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 782                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 783                         else
 784                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 785                 } else
 786                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 787
 788                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 789                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 790                 return r.expand(options);
 791         }
 792
 793         if (!is_exactly_a<numeric>(expanded_exponent) ||
 794                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 795                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 796                         return this->hold();
 797                 } else {
 798                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 799                 }
 800         }
 801
 802         // integer numeric exponent
 803         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 804         int int_exponent = num_exponent.to_int();
 805
 806         // (x+y)^n, n>0
 807         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 808                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 809
 810         // (x*y)^n -> x^n * y^n
 811         if (is_exactly_a<mul>(expanded_basis))
 812                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 813
 814         // cannot expand further
 815         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 816                 return this->hold();
 817         else
 818                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 819 }
 820
 821 //////////
 822 // new virtual functions which can be overridden by derived classes
 823 //////////
 824
 825 // none
 826
 827 //////////
 828 // non-virtual functions in this class
 829 //////////
 830
 831 /** expand a^n where a is an add and n is a positive integer.
 832  *  @see power::expand */
 833 ex power::expand_add(const add & a, int n, unsigned options) const
 834 {
 835         if (n==2)
 836                 return expand_add_2(a, options);
 837
 838         const size_t m = a.nops();
 839         exvector result;
 840         // The number of terms will be the number of combinatorial compositions,
 841         // i.e. the number of unordered arrangements of m nonnegative integers
 842         // which sum up to n.  It is frequently written as C_n(m) and directly
 843         // related with binomial coefficients:
 844         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 845         intvector k(m-1);
 846         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 847         intvector upper_limit(m-1);
 848         int l;
 849
 850         for (size_t l=0; l<m-1; ++l) {
 851                 k[l] = 0;
 852                 k_cum[l] = 0;
 853                 upper_limit[l] = n;
 854         }
 855
 856         while (true) {
 857                 exvector term;
 858                 term.reserve(m+1);
 859                 for (l=0; l<m-1; ++l) {
 860                         const ex & b = a.op(l);
 861                         GINAC_ASSERT(!is_exactly_a<add>(b));
 862                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 863                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 864                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 865                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 866                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 867                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 868                         if (is_exactly_a<mul>(b))
 869                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 870                         else
 871                                 term.push_back(power(b,k[l]));
 872                 }
 873
 874                 const ex & b = a.op(l);
 875                 GINAC_ASSERT(!is_exactly_a<add>(b));
 876                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 877                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 878                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 879                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 880                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 881                              !is_exactly_a<power>(ex_to<power>(b).basis));
 882                 if (is_exactly_a<mul>(b))
 883                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 884                 else
 885                         term.push_back(power(b,n-k_cum[m-2]));
 886
 887                 numeric f = binomial(numeric(n),numeric(k[0]));
 888                 for (l=1; l<m-1; ++l)
 889                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 890
 891                 term.push_back(f);
 892
 893                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 894
 895                 // increment k[]
 896                 l = m-2;
 897                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 898                         k[l] = 0;
 899                         --l;
 900                 }
 901                 if (l<0) break;
 902
 903                 // recalc k_cum[] and upper_limit[]
 904                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 905
 906                 for (size_t i=l+1; i<m-1; ++i)
 907                         k_cum[i] = k_cum[i-1]+k[i];
 908
 909                 for (size_t i=l+1; i<m-1; ++i)
 910                         upper_limit[i] = n-k_cum[i-1];
 911         }
 912
 913         return (new add(result))->setflag(status_flags::dynallocated |
 914                                           status_flags::expanded);
 915 }
 916
 917
 918 /** Special case of power::expand_add. Expands a^2 where a is an add.
 919  *  @see power::expand_add */
 920 ex power::expand_add_2(const add & a, unsigned options) const
 921 {
 922         epvector sum;
 923         size_t a_nops = a.nops();
 924         sum.reserve((a_nops*(a_nops+1))/2);
 925         epvector::const_iterator last = a.seq.end();
 926
 927         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 928         // first part: ignore overall_coeff and expand other terms
 929         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 930                 const ex & r = cit0->rest;
 931                 const ex & c = cit0->coeff;
 932
 933                 GINAC_ASSERT(!is_exactly_a<add>(r));
 934                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 935                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 936                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 937                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 938                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 939                              !is_exactly_a<power>(ex_to<power>(r).basis));
 940
 941                 if (c.is_equal(_ex1)) {
 942                         if (is_exactly_a<mul>(r)) {
 943                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 944                                                      _ex1));
 945                         } else {
 946                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 947                                                      _ex1));
 948                         }
 949                 } else {
 950                         if (is_exactly_a<mul>(r)) {
 951                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 952                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 953                         } else {
 954                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 955                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 956                         }
 957                 }
 958
 959                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 960                         const ex & r1 = cit1->rest;
 961                         const ex & c1 = cit1->coeff;
 962                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 963                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 964                 }
 965         }
 966
 967         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 968
 969         // second part: add terms coming from overall_factor (if != 0)
 970         if (!a.overall_coeff.is_zero()) {
 971                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 972                 while (i != end) {
 973                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 974                         ++i;
 975                 }
 976                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 977         }
 978
 979         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 980
 981         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 982 }
 983
 984 /** Expand factors of m in m^n where m is a mul and n is an integer.
 985  *  @see power::expand */
 986 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 987 {
 988         GINAC_ASSERT(n.is_integer());
 989
 990         if (n.is_zero()) {
 991                 return _ex1;
 992         }
 993
 994         // Leave it to multiplication since dummy indices have to be renamed
 995         if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
 996                 ex result = m;
 997                 exvector va = get_all_dummy_indices(m);
 998                 sort(va.begin(), va.end(), ex_is_less());
 999
1000                 for (int i=1; i < n.to_int(); i++)
1001                         result *= rename_dummy_indices_uniquely(va, m);
1002                 return result;
1003         }
1004
1005         epvector distrseq;
1006         distrseq.reserve(m.seq.size());
1007         bool need_reexpand = false;
1008
1009         epvector::const_iterator last = m.seq.end();
1010         epvector::const_iterator cit = m.seq.begin();
1011         while (cit!=last) {
1012                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1013                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1014                         // this happens when e.g. (a+b)^(1/2) gets squared and
1015                         // the resulting product needs to be reexpanded
1016                         need_reexpand = true;
1017                 }
1018                 distrseq.push_back(p);
1019                 ++cit;
1020         }
1021
1022         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1023         if (need_reexpand)
1024                 return ex(result).expand(options);
1025         if (from_expand)
1026                 return result.setflag(status_flags::expanded);
1027         return result;
1028 }
1029
1030 } // namespace GiNaC