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