ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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 "power.h"
  24 #include "expairseq.h"
  25 #include "add.h"
  26 #include "mul.h"
  27 #include "ncmul.h"
  28 #include "numeric.h"
  29 #include "constant.h"
  30 #include "operators.h"
  31 #include "inifcns.h" // for log() in power::derivative()
  32 #include "matrix.h"
  33 #include "indexed.h"
  34 #include "symbol.h"
  35 #include "lst.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38 #include "relational.h"
  39 #include "compiler.h"
  40
  41 #include <iostream>
  42 #include <limits>
  43 #include <stdexcept>
  44 #include <vector>
  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   print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
  55
  56 typedef std::vector<int> intvector;
  57
  58 //////////
  59 // default constructor
  60 //////////
  61
  62 power::power() { }
  63
  64 //////////
  65 // other constructors
  66 //////////
  67
  68 // all inlined
  69
  70 //////////
  71 // archiving
  72 //////////
  73
  74 void power::read_archive(const archive_node &n, lst &sym_lst)
  75 {
  76         inherited::read_archive(n, sym_lst);
  77         n.find_ex("basis", basis, sym_lst);
  78         n.find_ex("exponent", exponent, sym_lst);
  79 }
  80
  81 void power::archive(archive_node &n) const
  82 {
  83         inherited::archive(n);
  84         n.add_ex("basis", basis);
  85         n.add_ex("exponent", exponent);
  86 }
  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_cl_N(const print_csrc_cl_N& c, unsigned level) const
 166 {
 167         if (exponent.is_equal(_ex_1)) {
 168                 c.s << "recip(";
 169                 basis.print(c);
 170                 c.s << ')';
 171                 return;
 172         }
 173         c.s << "expt(";
 174         basis.print(c);
 175         c.s << ", ";
 176         exponent.print(c);
 177         c.s << ')';
 178 }
 179
 180 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 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                 case info_flags::positive:
 244                         return basis.info(info_flags::positive) && exponent.info(info_flags::real);
 245                 case info_flags::nonnegative:
 246                         return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
 247                                (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
 248                 case info_flags::has_indices: {
 249                         if (flags & status_flags::has_indices)
 250                                 return true;
 251                         else if (flags & status_flags::has_no_indices)
 252                                 return false;
 253                         else if (basis.info(info_flags::has_indices)) {
 254                                 setflag(status_flags::has_indices);
 255                                 clearflag(status_flags::has_no_indices);
 256                                 return true;
 257                         } else {
 258                                 clearflag(status_flags::has_indices);
 259                                 setflag(status_flags::has_no_indices);
 260                                 return false;
 261                         }
 262                 }
 263         }
 264         return inherited::info(inf);
 265 }
 266
 267 size_t power::nops() const
 268 {
 269         return 2;
 270 }
 271
 272 ex power::op(size_t i) const
 273 {
 274         GINAC_ASSERT(i<2);
 275
 276         return i==0 ? basis : exponent;
 277 }
 278
 279 ex power::map(map_function & f) const
 280 {
 281         const ex &mapped_basis = f(basis);
 282         const ex &mapped_exponent = f(exponent);
 283
 284         if (!are_ex_trivially_equal(basis, mapped_basis)
 285          || !are_ex_trivially_equal(exponent, mapped_exponent))
 286                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 287         else
 288                 return *this;
 289 }
 290
 291 bool power::is_polynomial(const ex & var) const
 292 {
 293         if (basis.is_polynomial(var)) {
 294                 if (basis.has(var))
 295                         // basis is non-constant polynomial in var
 296                         return exponent.info(info_flags::nonnegint);
 297                 else
 298                         // basis is constant in var
 299                         return !exponent.has(var);
 300         }
 301         // basis is a non-polynomial function of var
 302         return false;
 303 }
 304
 305 int power::degree(const ex & s) const
 306 {
 307         if (is_equal(ex_to<basic>(s)))
 308                 return 1;
 309         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 310                 if (basis.is_equal(s))
 311                         return ex_to<numeric>(exponent).to_int();
 312                 else
 313                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 314         } else if (basis.has(s))
 315                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 316         else
 317                 return 0;
 318 }
 319
 320 int power::ldegree(const ex & s) const
 321 {
 322         if (is_equal(ex_to<basic>(s)))
 323                 return 1;
 324         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 325                 if (basis.is_equal(s))
 326                         return ex_to<numeric>(exponent).to_int();
 327                 else
 328                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 329         } else if (basis.has(s))
 330                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 331         else
 332                 return 0;
 333 }
 334
 335 ex power::coeff(const ex & s, int n) const
 336 {
 337         if (is_equal(ex_to<basic>(s)))
 338                 return n==1 ? _ex1 : _ex0;
 339         else if (!basis.is_equal(s)) {
 340                 // basis not equal to s
 341                 if (n == 0)
 342                         return *this;
 343                 else
 344                         return _ex0;
 345         } else {
 346                 // basis equal to s
 347                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 348                         // integer exponent
 349                         int int_exp = ex_to<numeric>(exponent).to_int();
 350                         if (n == int_exp)
 351                                 return _ex1;
 352                         else
 353                                 return _ex0;
 354                 } else {
 355                         // non-integer exponents are treated as zero
 356                         if (n == 0)
 357                                 return *this;
 358                         else
 359                                 return _ex0;
 360                 }
 361         }
 362 }
 363
 364 /** Perform automatic term rewriting rules in this class.  In the following
 365  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 366  *  stand for such expressions that contain a plain number.
 367  *  - ^(x,0) -> 1  (also handles ^(0,0))
 368  *  - ^(x,1) -> x
 369  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 370  *  - ^(1,x) -> 1
 371  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 372  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 373  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0), case c1=1 should not happen, see below!)
 374  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 375  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 376  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 377  *
 378  *  @param level cut-off in recursive evaluation */
 379 ex power::eval(int level) const
 380 {
 381         if ((level==1) && (flags & status_flags::evaluated))
 382                 return *this;
 383         else if (level == -max_recursion_level)
 384                 throw(std::runtime_error("max recursion level reached"));
 385
 386         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 387         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 388
 389         const numeric *num_basis = NULL;
 390         const numeric *num_exponent = NULL;
 391
 392         if (is_exactly_a<numeric>(ebasis)) {
 393                 num_basis = &ex_to<numeric>(ebasis);
 394         }
 395         if (is_exactly_a<numeric>(eexponent)) {
 396                 num_exponent = &ex_to<numeric>(eexponent);
 397         }
 398
 399         // ^(x,0) -> 1  (0^0 also handled here)
 400         if (eexponent.is_zero()) {
 401                 if (ebasis.is_zero())
 402                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 403                 else
 404                         return _ex1;
 405         }
 406
 407         // ^(x,1) -> x
 408         if (eexponent.is_equal(_ex1))
 409                 return ebasis;
 410
 411         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 412         if ( ebasis.is_zero() && num_exponent ) {
 413                 if ((num_exponent->real()).is_zero())
 414                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 415                 else if ((num_exponent->real()).is_negative())
 416                         throw (pole_error("power::eval(): division by zero",1));
 417                 else
 418                         return _ex0;
 419         }
 420
 421         // ^(1,x) -> 1
 422         if (ebasis.is_equal(_ex1))
 423                 return _ex1;
 424
 425         // power of a function calculated by separate rules defined for this function
 426         if (is_exactly_a<function>(ebasis))
 427                 return ex_to<function>(ebasis).power(eexponent);
 428
 429         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 430         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 431                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 432
 433         if ( num_exponent ) {
 434
 435                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 436                 // except if c1,c2 are rational, but c1^c2 is not)
 437                 if ( num_basis ) {
 438                         const bool basis_is_crational = num_basis->is_crational();
 439                         const bool exponent_is_crational = num_exponent->is_crational();
 440                         if (!basis_is_crational || !exponent_is_crational) {
 441                                 // return a plain float
 442                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 443                                                                                                status_flags::evaluated |
 444                                                                                                status_flags::expanded);
 445                         }
 446
 447                         const numeric res = num_basis->power(*num_exponent);
 448                         if (res.is_crational()) {
 449                                 return res;
 450                         }
 451                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 452
 453                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 454                         if (basis_is_crational && exponent_is_crational
 455                             && num_exponent->is_real()
 456                             && !num_exponent->is_integer()) {
 457                                 const numeric n = num_exponent->numer();
 458                                 const numeric m = num_exponent->denom();
 459                                 numeric r;
 460                                 numeric q = iquo(n, m, r);
 461                                 if (r.is_negative()) {
 462                                         r += m;
 463                                         --q;
 464                                 }
 465                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 466                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 467                                                 // try it for numerator and denominator separately, in order to
 468                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 469                                                 const numeric bnum = num_basis->numer();
 470                                                 const numeric bden = num_basis->denom();
 471                                                 const numeric res_bnum = bnum.power(*num_exponent);
 472                                                 const numeric res_bden = bden.power(*num_exponent);
 473                                                 if (res_bnum.is_integer())
 474                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 475                                                 if (res_bden.is_integer())
 476                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 477                                         }
 478                                         return this->hold();
 479                                 } else {
 480                                         // assemble resulting product, but allowing for a re-evaluation,
 481                                         // because otherwise we'll end up with something like
 482                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 483                                         // instead of 7/16*7^(1/3).
 484                                         ex prod = power(*num_basis,r.div(m));
 485                                         return prod*power(*num_basis,q);
 486                                 }
 487                         }
 488                 }
 489
 490                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 491                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
 492                 // case c1==1 should not happen, see below!)
 493                 if (is_exactly_a<power>(ebasis)) {
 494                         const power & sub_power = ex_to<power>(ebasis);
 495                         const ex & sub_basis = sub_power.basis;
 496                         const ex & sub_exponent = sub_power.exponent;
 497                         if (is_exactly_a<numeric>(sub_exponent)) {
 498                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 499                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 500                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()
 501                                                 || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
 502                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 503                                 }
 504                         }
 505                 }
 506
 507                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 508                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 509                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 510                 }
 511
 512                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 513                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 514                         numeric icont = ebasis.integer_content();
 515                         const numeric lead_coeff =
 516                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 517
 518                         const bool canonicalizable = lead_coeff.is_integer();
 519                         const bool unit_normal = lead_coeff.is_pos_integer();
 520                         if (canonicalizable && (! unit_normal))
 521                                 icont = icont.mul(*_num_1_p);
 522
 523                         if (canonicalizable && (icont != *_num1_p)) {
 524                                 const add& addref = ex_to<add>(ebasis);
 525                                 add* addp = new add(addref);
 526                                 addp->setflag(status_flags::dynallocated);
 527                                 addp->clearflag(status_flags::hash_calculated);
 528                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 529                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 530                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 531
 532                                 const numeric c = icont.power(*num_exponent);
 533                                 if (likely(c != *_num1_p))
 534                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 535                                 else
 536                                         return power(*addp, *num_exponent);
 537                         }
 538                 }
 539
 540                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 541                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 542                 if (is_exactly_a<mul>(ebasis)) {
 543                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 544                         const mul & mulref = ex_to<mul>(ebasis);
 545                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 546                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 547                                 if (num_coeff.is_real()) {
 548                                         if (num_coeff.is_positive()) {
 549                                                 mul *mulp = new mul(mulref);
 550                                                 mulp->overall_coeff = _ex1;
 551                                                 mulp->setflag(status_flags::dynallocated);
 552                                                 mulp->clearflag(status_flags::evaluated);
 553                                                 mulp->clearflag(status_flags::hash_calculated);
 554                                                 return (new mul(power(*mulp,exponent),
 555                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 556                                         } else {
 557                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 558                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 559                                                         mul *mulp = new mul(mulref);
 560                                                         mulp->overall_coeff = _ex_1;
 561                                                         mulp->setflag(status_flags::dynallocated);
 562                                                         mulp->clearflag(status_flags::evaluated);
 563                                                         mulp->clearflag(status_flags::hash_calculated);
 564                                                         return (new mul(power(*mulp,exponent),
 565                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 566                                                 }
 567                                         }
 568                                 }
 569                         }
 570                 }
 571
 572                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 573                 if (num_exponent->is_pos_integer() &&
 574                     ebasis.return_type() != return_types::commutative &&
 575                     !is_a<matrix>(ebasis)) {
 576                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 577                 }
 578         }
 579
 580         if (are_ex_trivially_equal(ebasis,basis) &&
 581             are_ex_trivially_equal(eexponent,exponent)) {
 582                 return this->hold();
 583         }
 584         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 585                                                        status_flags::evaluated);
 586 }
 587
 588 ex power::evalf(int level) const
 589 {
 590         ex ebasis;
 591         ex eexponent;
 592
 593         if (level==1) {
 594                 ebasis = basis;
 595                 eexponent = exponent;
 596         } else if (level == -max_recursion_level) {
 597                 throw(std::runtime_error("max recursion level reached"));
 598         } else {
 599                 ebasis = basis.evalf(level-1);
 600                 if (!is_exactly_a<numeric>(exponent))
 601                         eexponent = exponent.evalf(level-1);
 602                 else
 603                         eexponent = exponent;
 604         }
 605
 606         return power(ebasis,eexponent);
 607 }
 608
 609 ex power::evalm() const
 610 {
 611         const ex ebasis = basis.evalm();
 612         const ex eexponent = exponent.evalm();
 613         if (is_a<matrix>(ebasis)) {
 614                 if (is_exactly_a<numeric>(eexponent)) {
 615                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 616                 }
 617         }
 618         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 619 }
 620
 621 bool power::has(const ex & other, unsigned options) const
 622 {
 623         if (!(options & has_options::algebraic))
 624                 return basic::has(other, options);
 625         if (!is_a<power>(other))
 626                 return basic::has(other, options);
 627         if (!exponent.info(info_flags::integer)
 628                         || !other.op(1).info(info_flags::integer))
 629                 return basic::has(other, options);
 630         if (exponent.info(info_flags::posint)
 631                         && other.op(1).info(info_flags::posint)
 632                         && ex_to<numeric>(exponent).to_int()
 633                                         > ex_to<numeric>(other.op(1)).to_int()
 634                         && basis.match(other.op(0)))
 635                 return true;
 636         if (exponent.info(info_flags::negint)
 637                         && other.op(1).info(info_flags::negint)
 638                         && ex_to<numeric>(exponent).to_int()
 639                                         < ex_to<numeric>(other.op(1)).to_int()
 640                         && basis.match(other.op(0)))
 641                 return true;
 642         return basic::has(other, options);
 643 }
 644
 645 // from mul.cpp
 646 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 647
 648 ex power::subs(const exmap & m, unsigned options) const
 649 {
 650         const ex &subsed_basis = basis.subs(m, options);
 651         const ex &subsed_exponent = exponent.subs(m, options);
 652
 653         if (!are_ex_trivially_equal(basis, subsed_basis)
 654          || !are_ex_trivially_equal(exponent, subsed_exponent))
 655                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 656
 657         if (!(options & subs_options::algebraic))
 658                 return subs_one_level(m, options);
 659
 660         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 661                 int nummatches = std::numeric_limits<int>::max();
 662                 exmap repls;
 663                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 664                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 665                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 666                         ex result = (*this)*power(anum/aden, nummatches);
 667                         return (ex_to<basic>(result)).subs_one_level(m, options);
 668                 }
 669         }
 670
 671         return subs_one_level(m, options);
 672 }
 673
 674 ex power::eval_ncmul(const exvector & v) const
 675 {
 676         return inherited::eval_ncmul(v);
 677 }
 678
 679 ex power::conjugate() const
 680 {
 681         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 682         // branch cut which runs along the negative real axis.
 683         if (basis.info(info_flags::positive)) {
 684                 ex newexponent = exponent.conjugate();
 685                 if (are_ex_trivially_equal(exponent, newexponent)) {
 686                         return *this;
 687                 }
 688                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
 689         }
 690         if (exponent.info(info_flags::integer)) {
 691                 ex newbasis = basis.conjugate();
 692                 if (are_ex_trivially_equal(basis, newbasis)) {
 693                         return *this;
 694                 }
 695                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
 696         }
 697         return conjugate_function(*this).hold();
 698 }
 699
 700 ex power::real_part() const
 701 {
 702         if (exponent.info(info_flags::integer)) {
 703                 ex basis_real = basis.real_part();
 704                 if (basis_real == basis)
 705                         return *this;
 706                 realsymbol a("a"),b("b");
 707                 ex result;
 708                 if (exponent.info(info_flags::posint))
 709                         result = power(a+I*b,exponent);
 710                 else
 711                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 712                 result = result.expand();
 713                 result = result.real_part();
 714                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 715                 return result;
 716         }
 717
 718         ex a = basis.real_part();
 719         ex b = basis.imag_part();
 720         ex c = exponent.real_part();
 721         ex d = exponent.imag_part();
 722         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 723 }
 724
 725 ex power::imag_part() const
 726 {
 727         if (exponent.info(info_flags::integer)) {
 728                 ex basis_real = basis.real_part();
 729                 if (basis_real == basis)
 730                         return 0;
 731                 realsymbol a("a"),b("b");
 732                 ex result;
 733                 if (exponent.info(info_flags::posint))
 734                         result = power(a+I*b,exponent);
 735                 else
 736                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 737                 result = result.expand();
 738                 result = result.imag_part();
 739                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 740                 return result;
 741         }
 742
 743         ex a=basis.real_part();
 744         ex b=basis.imag_part();
 745         ex c=exponent.real_part();
 746         ex d=exponent.imag_part();
 747         return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 748 }
 749
 750 // protected
 751
 752 // protected
 753
 754 /** Implementation of ex::diff() for a power.
 755  *  @see ex::diff */
 756 ex power::derivative(const symbol & s) const
 757 {
 758         if (is_a<numeric>(exponent)) {
 759                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 760                 epvector newseq;
 761                 newseq.reserve(2);
 762                 newseq.push_back(expair(basis, exponent - _ex1));
 763                 newseq.push_back(expair(basis.diff(s), _ex1));
 764                 return mul(newseq, exponent);
 765         } else {
 766                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 767                 return mul(*this,
 768                            add(mul(exponent.diff(s), log(basis)),
 769                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 770         }
 771 }
 772
 773 int power::compare_same_type(const basic & other) const
 774 {
 775         GINAC_ASSERT(is_exactly_a<power>(other));
 776         const power &o = static_cast<const power &>(other);
 777
 778         int cmpval = basis.compare(o.basis);
 779         if (cmpval)
 780                 return cmpval;
 781         else
 782                 return exponent.compare(o.exponent);
 783 }
 784
 785 unsigned power::return_type() const
 786 {
 787         return basis.return_type();
 788 }
 789
 790 return_type_t power::return_type_tinfo() const
 791 {
 792         return basis.return_type_tinfo();
 793 }
 794
 795 ex power::expand(unsigned options) const
 796 {
 797         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 798                 // A special case worth optimizing.
 799                 setflag(status_flags::expanded);
 800                 return *this;
 801         }
 802
 803         // (x*p)^c -> x^c * p^c, if p>0
 804         // makes sense before expanding the basis
 805         if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
 806                 const mul &m = ex_to<mul>(basis);
 807                 exvector prodseq;
 808                 epvector powseq;
 809                 prodseq.reserve(m.seq.size() + 1);
 810                 powseq.reserve(m.seq.size() + 1);
 811                 epvector::const_iterator last = m.seq.end();
 812                 epvector::const_iterator cit = m.seq.begin();
 813                 bool possign = true;
 814
 815                 // search for positive/negative factors
 816                 while (cit!=last) {
 817                         ex e=m.recombine_pair_to_ex(*cit);
 818                         if (e.info(info_flags::positive))
 819                                 prodseq.push_back(pow(e, exponent).expand(options));
 820                         else if (e.info(info_flags::negative)) {
 821                                 prodseq.push_back(pow(-e, exponent).expand(options));
 822                                 possign = !possign;
 823                         } else
 824                                 powseq.push_back(*cit);
 825                         ++cit;
 826                 }
 827
 828                 // take care on the numeric coefficient
 829                 ex coeff=(possign? _ex1 : _ex_1);
 830                 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
 831                         prodseq.push_back(power(m.overall_coeff, exponent));
 832                 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
 833                         prodseq.push_back(power(-m.overall_coeff, exponent));
 834                 else
 835                         coeff *= m.overall_coeff;
 836
 837                 // If positive/negative factors are found, then extract them.
 838                 // In either case we set a flag to avoid the second run on a part
 839                 // which does not have positive/negative terms.
 840                 if (prodseq.size() > 0) {
 841                         ex newbasis = coeff*mul(powseq);
 842                         ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
 843                         return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
 844                 } else
 845                         ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
 846         }
 847
 848         const ex expanded_basis = basis.expand(options);
 849         const ex expanded_exponent = exponent.expand(options);
 850
 851         // x^(a+b) -> x^a * x^b
 852         if (is_exactly_a<add>(expanded_exponent)) {
 853                 const add &a = ex_to<add>(expanded_exponent);
 854                 exvector distrseq;
 855                 distrseq.reserve(a.seq.size() + 1);
 856                 epvector::const_iterator last = a.seq.end();
 857                 epvector::const_iterator cit = a.seq.begin();
 858                 while (cit!=last) {
 859                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 860                         ++cit;
 861                 }
 862
 863                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 864                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 865                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 866                         int int_exponent = num_exponent.to_int();
 867                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 868                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 869                         else
 870                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 871                 } else
 872                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 873
 874                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 875                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 876                 return r.expand(options);
 877         }
 878
 879         if (!is_exactly_a<numeric>(expanded_exponent) ||
 880                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 881                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 882                         return this->hold();
 883                 } else {
 884                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 885                 }
 886         }
 887
 888         // integer numeric exponent
 889         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 890         int int_exponent = num_exponent.to_int();
 891
 892         // (x+y)^n, n>0
 893         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 894                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 895
 896         // (x*y)^n -> x^n * y^n
 897         if (is_exactly_a<mul>(expanded_basis))
 898                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 899
 900         // cannot expand further
 901         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 902                 return this->hold();
 903         else
 904                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 905 }
 906
 907 //////////
 908 // new virtual functions which can be overridden by derived classes
 909 //////////
 910
 911 // none
 912
 913 //////////
 914 // non-virtual functions in this class
 915 //////////
 916
 917 /** expand a^n where a is an add and n is a positive integer.
 918  *  @see power::expand */
 919 ex power::expand_add(const add & a, int n, unsigned options) const
 920 {
 921         if (n==2)
 922                 return expand_add_2(a, options);
 923
 924         const size_t m = a.nops();
 925         exvector result;
 926         // The number of terms will be the number of combinatorial compositions,
 927         // i.e. the number of unordered arrangements of m nonnegative integers
 928         // which sum up to n.  It is frequently written as C_n(m) and directly
 929         // related with binomial coefficients:
 930         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 931         intvector k(m-1);
 932         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 933         intvector upper_limit(m-1);
 934
 935         for (size_t l=0; l<m-1; ++l) {
 936                 k[l] = 0;
 937                 k_cum[l] = 0;
 938                 upper_limit[l] = n;
 939         }
 940
 941         while (true) {
 942                 exvector term;
 943                 term.reserve(m+1);
 944                 for (std::size_t l = 0; l < m - 1; ++l) {
 945                         const ex & b = a.op(l);
 946                         GINAC_ASSERT(!is_exactly_a<add>(b));
 947                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 948                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 949                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 950                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 951                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 952                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 953                         if (is_exactly_a<mul>(b))
 954                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 955                         else
 956                                 term.push_back(power(b,k[l]));
 957                 }
 958
 959                 const ex & b = a.op(m - 1);
 960                 GINAC_ASSERT(!is_exactly_a<add>(b));
 961                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 962                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 963                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 964                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 965                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 966                              !is_exactly_a<power>(ex_to<power>(b).basis));
 967                 if (is_exactly_a<mul>(b))
 968                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 969                 else
 970                         term.push_back(power(b,n-k_cum[m-2]));
 971
 972                 numeric f = binomial(numeric(n),numeric(k[0]));
 973                 for (std::size_t l = 1; l < m - 1; ++l)
 974                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 975
 976                 term.push_back(f);
 977
 978                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 979
 980                 // increment k[]
 981                 bool done = false;
 982                 std::size_t l = m - 2;
 983                 while ((++k[l]) > upper_limit[l]) {
 984                         k[l] = 0;
 985                         if (l != 0)
 986                                 --l;
 987                         else {
 988                                 done = true;
 989                                 break;
 990                         }
 991                 }
 992                 if (done)
 993                         break;
 994
 995                 // recalc k_cum[] and upper_limit[]
 996                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 997
 998                 for (size_t i=l+1; i<m-1; ++i)
 999                         k_cum[i] = k_cum[i-1]+k[i];
1000
1001                 for (size_t i=l+1; i<m-1; ++i)
1002                         upper_limit[i] = n-k_cum[i-1];
1003         }
1004
1005         return (new add(result))->setflag(status_flags::dynallocated |
1006                                           status_flags::expanded);
1007 }
1008
1009
1010 /** Special case of power::expand_add. Expands a^2 where a is an add.
1011  *  @see power::expand_add */
1012 ex power::expand_add_2(const add & a, unsigned options) const
1013 {
1014         epvector sum;
1015         size_t a_nops = a.nops();
1016         sum.reserve((a_nops*(a_nops+1))/2);
1017         epvector::const_iterator last = a.seq.end();
1018
1019         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1020         // first part: ignore overall_coeff and expand other terms
1021         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1022                 const ex & r = cit0->rest;
1023                 const ex & c = cit0->coeff;
1024
1025                 GINAC_ASSERT(!is_exactly_a<add>(r));
1026                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1027                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1028                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1029                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
1030                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1031                              !is_exactly_a<power>(ex_to<power>(r).basis));
1032
1033                 if (c.is_equal(_ex1)) {
1034                         if (is_exactly_a<mul>(r)) {
1035                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1036                                                      _ex1));
1037                         } else {
1038                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1039                                                      _ex1));
1040                         }
1041                 } else {
1042                         if (is_exactly_a<mul>(r)) {
1043                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1044                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
1045                         } else {
1046                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1047                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
1048                         }
1049                 }
1050
1051                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1052                         const ex & r1 = cit1->rest;
1053                         const ex & c1 = cit1->coeff;
1054                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
1055                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1056                 }
1057         }
1058
1059         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1060
1061         // second part: add terms coming from overall_factor (if != 0)
1062         if (!a.overall_coeff.is_zero()) {
1063                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1064                 while (i != end) {
1065                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1066                         ++i;
1067                 }
1068                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1069         }
1070
1071         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1072
1073         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1074 }
1075
1076 /** Expand factors of m in m^n where m is a mul and n is an integer.
1077  *  @see power::expand */
1078 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1079 {
1080         GINAC_ASSERT(n.is_integer());
1081
1082         if (n.is_zero()) {
1083                 return _ex1;
1084         }
1085
1086         // do not bother to rename indices if there are no any.
1087         if ((!(options & expand_options::expand_rename_idx))
1088                         && m.info(info_flags::has_indices))
1089                 options |= expand_options::expand_rename_idx;
1090         // Leave it to multiplication since dummy indices have to be renamed
1091         if ((options & expand_options::expand_rename_idx) &&
1092                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1093                 ex result = m;
1094                 exvector va = get_all_dummy_indices(m);
1095                 sort(va.begin(), va.end(), ex_is_less());
1096
1097                 for (int i=1; i < n.to_int(); i++)
1098                         result *= rename_dummy_indices_uniquely(va, m);
1099                 return result;
1100         }
1101
1102         epvector distrseq;
1103         distrseq.reserve(m.seq.size());
1104         bool need_reexpand = false;
1105
1106         epvector::const_iterator last = m.seq.end();
1107         epvector::const_iterator cit = m.seq.begin();
1108         while (cit!=last) {
1109                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1110                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1111                         // this happens when e.g. (a+b)^(1/2) gets squared and
1112                         // the resulting product needs to be reexpanded
1113                         need_reexpand = true;
1114                 }
1115                 distrseq.push_back(p);
1116                 ++cit;
1117         }
1118
1119         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1120         if (need_reexpand)
1121                 return ex(result).expand(options);
1122         if (from_expand)
1123                 return result.setflag(status_flags::expanded);
1124         return result;
1125 }
1126
1127 GINAC_BIND_UNARCHIVER(power);
1128
1129 } // namespace GiNaC