ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2014 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::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even);
 247                 case info_flags::has_indices: {
 248                         if (flags & status_flags::has_indices)
 249                                 return true;
 250                         else if (flags & status_flags::has_no_indices)
 251                                 return false;
 252                         else if (basis.info(info_flags::has_indices)) {
 253                                 setflag(status_flags::has_indices);
 254                                 clearflag(status_flags::has_no_indices);
 255                                 return true;
 256                         } else {
 257                                 clearflag(status_flags::has_indices);
 258                                 setflag(status_flags::has_no_indices);
 259                                 return false;
 260                         }
 261                 }
 262         }
 263         return inherited::info(inf);
 264 }
 265
 266 size_t power::nops() const
 267 {
 268         return 2;
 269 }
 270
 271 ex power::op(size_t i) const
 272 {
 273         GINAC_ASSERT(i<2);
 274
 275         return i==0 ? basis : exponent;
 276 }
 277
 278 ex power::map(map_function & f) const
 279 {
 280         const ex &mapped_basis = f(basis);
 281         const ex &mapped_exponent = f(exponent);
 282
 283         if (!are_ex_trivially_equal(basis, mapped_basis)
 284          || !are_ex_trivially_equal(exponent, mapped_exponent))
 285                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 286         else
 287                 return *this;
 288 }
 289
 290 bool power::is_polynomial(const ex & var) const
 291 {
 292         if (basis.is_polynomial(var)) {
 293                 if (basis.has(var))
 294                         // basis is non-constant polynomial in var
 295                         return exponent.info(info_flags::nonnegint);
 296                 else
 297                         // basis is constant in var
 298                         return !exponent.has(var);
 299         }
 300         // basis is a non-polynomial function of var
 301         return false;
 302 }
 303
 304 int power::degree(const ex & s) const
 305 {
 306         if (is_equal(ex_to<basic>(s)))
 307                 return 1;
 308         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 309                 if (basis.is_equal(s))
 310                         return ex_to<numeric>(exponent).to_int();
 311                 else
 312                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 313         } else if (basis.has(s))
 314                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 315         else
 316                 return 0;
 317 }
 318
 319 int power::ldegree(const ex & s) const
 320 {
 321         if (is_equal(ex_to<basic>(s)))
 322                 return 1;
 323         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 324                 if (basis.is_equal(s))
 325                         return ex_to<numeric>(exponent).to_int();
 326                 else
 327                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 328         } else if (basis.has(s))
 329                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 330         else
 331                 return 0;
 332 }
 333
 334 ex power::coeff(const ex & s, int n) const
 335 {
 336         if (is_equal(ex_to<basic>(s)))
 337                 return n==1 ? _ex1 : _ex0;
 338         else if (!basis.is_equal(s)) {
 339                 // basis not equal to s
 340                 if (n == 0)
 341                         return *this;
 342                 else
 343                         return _ex0;
 344         } else {
 345                 // basis equal to s
 346                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 347                         // integer exponent
 348                         int int_exp = ex_to<numeric>(exponent).to_int();
 349                         if (n == int_exp)
 350                                 return _ex1;
 351                         else
 352                                 return _ex0;
 353                 } else {
 354                         // non-integer exponents are treated as zero
 355                         if (n == 0)
 356                                 return *this;
 357                         else
 358                                 return _ex0;
 359                 }
 360         }
 361 }
 362
 363 /** Perform automatic term rewriting rules in this class.  In the following
 364  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 365  *  stand for such expressions that contain a plain number.
 366  *  - ^(x,0) -> 1  (also handles ^(0,0))
 367  *  - ^(x,1) -> x
 368  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 369  *  - ^(1,x) -> 1
 370  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 371  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 372  *  - ^(^(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!)
 373  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 374  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 375  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 376  *
 377  *  @param level cut-off in recursive evaluation */
 378 ex power::eval(int level) const
 379 {
 380         if ((level==1) && (flags & status_flags::evaluated))
 381                 return *this;
 382         else if (level == -max_recursion_level)
 383                 throw(std::runtime_error("max recursion level reached"));
 384
 385         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 386         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 387
 388         const numeric *num_basis = NULL;
 389         const numeric *num_exponent = NULL;
 390
 391         if (is_exactly_a<numeric>(ebasis)) {
 392                 num_basis = &ex_to<numeric>(ebasis);
 393         }
 394         if (is_exactly_a<numeric>(eexponent)) {
 395                 num_exponent = &ex_to<numeric>(eexponent);
 396         }
 397
 398         // ^(x,0) -> 1  (0^0 also handled here)
 399         if (eexponent.is_zero()) {
 400                 if (ebasis.is_zero())
 401                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 402                 else
 403                         return _ex1;
 404         }
 405
 406         // ^(x,1) -> x
 407         if (eexponent.is_equal(_ex1))
 408                 return ebasis;
 409
 410         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 411         if ( ebasis.is_zero() && num_exponent ) {
 412                 if ((num_exponent->real()).is_zero())
 413                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 414                 else if ((num_exponent->real()).is_negative())
 415                         throw (pole_error("power::eval(): division by zero",1));
 416                 else
 417                         return _ex0;
 418         }
 419
 420         // ^(1,x) -> 1
 421         if (ebasis.is_equal(_ex1))
 422                 return _ex1;
 423
 424         // power of a function calculated by separate rules defined for this function
 425         if (is_exactly_a<function>(ebasis))
 426                 return ex_to<function>(ebasis).power(eexponent);
 427
 428         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 429         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 430                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 431
 432         if ( num_exponent ) {
 433
 434                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 435                 // except if c1,c2 are rational, but c1^c2 is not)
 436                 if ( num_basis ) {
 437                         const bool basis_is_crational = num_basis->is_crational();
 438                         const bool exponent_is_crational = num_exponent->is_crational();
 439                         if (!basis_is_crational || !exponent_is_crational) {
 440                                 // return a plain float
 441                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 442                                                                                                status_flags::evaluated |
 443                                                                                                status_flags::expanded);
 444                         }
 445
 446                         const numeric res = num_basis->power(*num_exponent);
 447                         if (res.is_crational()) {
 448                                 return res;
 449                         }
 450                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 451
 452                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 453                         if (basis_is_crational && exponent_is_crational
 454                             && num_exponent->is_real()
 455                             && !num_exponent->is_integer()) {
 456                                 const numeric n = num_exponent->numer();
 457                                 const numeric m = num_exponent->denom();
 458                                 numeric r;
 459                                 numeric q = iquo(n, m, r);
 460                                 if (r.is_negative()) {
 461                                         r += m;
 462                                         --q;
 463                                 }
 464                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 465                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 466                                                 // try it for numerator and denominator separately, in order to
 467                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 468                                                 const numeric bnum = num_basis->numer();
 469                                                 const numeric bden = num_basis->denom();
 470                                                 const numeric res_bnum = bnum.power(*num_exponent);
 471                                                 const numeric res_bden = bden.power(*num_exponent);
 472                                                 if (res_bnum.is_integer())
 473                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 474                                                 if (res_bden.is_integer())
 475                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 476                                         }
 477                                         return this->hold();
 478                                 } else {
 479                                         // assemble resulting product, but allowing for a re-evaluation,
 480                                         // because otherwise we'll end up with something like
 481                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 482                                         // instead of 7/16*7^(1/3).
 483                                         ex prod = power(*num_basis,r.div(m));
 484                                         return prod*power(*num_basis,q);
 485                                 }
 486                         }
 487                 }
 488
 489                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 490                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
 491                 // case c1==1 should not happen, see below!)
 492                 if (is_exactly_a<power>(ebasis)) {
 493                         const power & sub_power = ex_to<power>(ebasis);
 494                         const ex & sub_basis = sub_power.basis;
 495                         const ex & sub_exponent = sub_power.exponent;
 496                         if (is_exactly_a<numeric>(sub_exponent)) {
 497                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 498                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 499                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()
 500                                                 || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
 501                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 502                                 }
 503                         }
 504                 }
 505
 506                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 507                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 508                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 509                 }
 510
 511                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 512                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 513                         numeric icont = ebasis.integer_content();
 514                         const numeric lead_coeff =
 515                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 516
 517                         const bool canonicalizable = lead_coeff.is_integer();
 518                         const bool unit_normal = lead_coeff.is_pos_integer();
 519                         if (canonicalizable && (! unit_normal))
 520                                 icont = icont.mul(*_num_1_p);
 521
 522                         if (canonicalizable && (icont != *_num1_p)) {
 523                                 const add& addref = ex_to<add>(ebasis);
 524                                 add* addp = new add(addref);
 525                                 addp->setflag(status_flags::dynallocated);
 526                                 addp->clearflag(status_flags::hash_calculated);
 527                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 528                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 529                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 530
 531                                 const numeric c = icont.power(*num_exponent);
 532                                 if (likely(c != *_num1_p))
 533                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 534                                 else
 535                                         return power(*addp, *num_exponent);
 536                         }
 537                 }
 538
 539                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 540                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 541                 if (is_exactly_a<mul>(ebasis)) {
 542                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 543                         const mul & mulref = ex_to<mul>(ebasis);
 544                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 545                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 546                                 if (num_coeff.is_real()) {
 547                                         if (num_coeff.is_positive()) {
 548                                                 mul *mulp = new mul(mulref);
 549                                                 mulp->overall_coeff = _ex1;
 550                                                 mulp->setflag(status_flags::dynallocated);
 551                                                 mulp->clearflag(status_flags::evaluated);
 552                                                 mulp->clearflag(status_flags::hash_calculated);
 553                                                 return (new mul(power(*mulp,exponent),
 554                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 555                                         } else {
 556                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 557                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 558                                                         mul *mulp = new mul(mulref);
 559                                                         mulp->overall_coeff = _ex_1;
 560                                                         mulp->setflag(status_flags::dynallocated);
 561                                                         mulp->clearflag(status_flags::evaluated);
 562                                                         mulp->clearflag(status_flags::hash_calculated);
 563                                                         return (new mul(power(*mulp,exponent),
 564                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 565                                                 }
 566                                         }
 567                                 }
 568                         }
 569                 }
 570
 571                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 572                 if (num_exponent->is_pos_integer() &&
 573                     ebasis.return_type() != return_types::commutative &&
 574                     !is_a<matrix>(ebasis)) {
 575                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 576                 }
 577         }
 578
 579         if (are_ex_trivially_equal(ebasis,basis) &&
 580             are_ex_trivially_equal(eexponent,exponent)) {
 581                 return this->hold();
 582         }
 583         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 584                                                        status_flags::evaluated);
 585 }
 586
 587 ex power::evalf(int level) const
 588 {
 589         ex ebasis;
 590         ex eexponent;
 591
 592         if (level==1) {
 593                 ebasis = basis;
 594                 eexponent = exponent;
 595         } else if (level == -max_recursion_level) {
 596                 throw(std::runtime_error("max recursion level reached"));
 597         } else {
 598                 ebasis = basis.evalf(level-1);
 599                 if (!is_exactly_a<numeric>(exponent))
 600                         eexponent = exponent.evalf(level-1);
 601                 else
 602                         eexponent = exponent;
 603         }
 604
 605         return power(ebasis,eexponent);
 606 }
 607
 608 ex power::evalm() const
 609 {
 610         const ex ebasis = basis.evalm();
 611         const ex eexponent = exponent.evalm();
 612         if (is_a<matrix>(ebasis)) {
 613                 if (is_exactly_a<numeric>(eexponent)) {
 614                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 615                 }
 616         }
 617         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 618 }
 619
 620 bool power::has(const ex & other, unsigned options) const
 621 {
 622         if (!(options & has_options::algebraic))
 623                 return basic::has(other, options);
 624         if (!is_a<power>(other))
 625                 return basic::has(other, options);
 626         if (!exponent.info(info_flags::integer)
 627                         || !other.op(1).info(info_flags::integer))
 628                 return basic::has(other, options);
 629         if (exponent.info(info_flags::posint)
 630                         && other.op(1).info(info_flags::posint)
 631                         && ex_to<numeric>(exponent).to_int()
 632                                         > ex_to<numeric>(other.op(1)).to_int()
 633                         && basis.match(other.op(0)))
 634                 return true;
 635         if (exponent.info(info_flags::negint)
 636                         && other.op(1).info(info_flags::negint)
 637                         && ex_to<numeric>(exponent).to_int()
 638                                         < ex_to<numeric>(other.op(1)).to_int()
 639                         && basis.match(other.op(0)))
 640                 return true;
 641         return basic::has(other, options);
 642 }
 643
 644 // from mul.cpp
 645 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 646
 647 ex power::subs(const exmap & m, unsigned options) const
 648 {
 649         const ex &subsed_basis = basis.subs(m, options);
 650         const ex &subsed_exponent = exponent.subs(m, options);
 651
 652         if (!are_ex_trivially_equal(basis, subsed_basis)
 653          || !are_ex_trivially_equal(exponent, subsed_exponent))
 654                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 655
 656         if (!(options & subs_options::algebraic))
 657                 return subs_one_level(m, options);
 658
 659         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 660                 int nummatches = std::numeric_limits<int>::max();
 661                 exmap repls;
 662                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 663                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 664                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 665                         ex result = (*this)*power(anum/aden, nummatches);
 666                         return (ex_to<basic>(result)).subs_one_level(m, options);
 667                 }
 668         }
 669
 670         return subs_one_level(m, options);
 671 }
 672
 673 ex power::eval_ncmul(const exvector & v) const
 674 {
 675         return inherited::eval_ncmul(v);
 676 }
 677
 678 ex power::conjugate() const
 679 {
 680         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 681         // branch cut which runs along the negative real axis.
 682         if (basis.info(info_flags::positive)) {
 683                 ex newexponent = exponent.conjugate();
 684                 if (are_ex_trivially_equal(exponent, newexponent)) {
 685                         return *this;
 686                 }
 687                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
 688         }
 689         if (exponent.info(info_flags::integer)) {
 690                 ex newbasis = basis.conjugate();
 691                 if (are_ex_trivially_equal(basis, newbasis)) {
 692                         return *this;
 693                 }
 694                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
 695         }
 696         return conjugate_function(*this).hold();
 697 }
 698
 699 ex power::real_part() const
 700 {
 701         if (exponent.info(info_flags::integer)) {
 702                 ex basis_real = basis.real_part();
 703                 if (basis_real == basis)
 704                         return *this;
 705                 realsymbol a("a"),b("b");
 706                 ex result;
 707                 if (exponent.info(info_flags::posint))
 708                         result = power(a+I*b,exponent);
 709                 else
 710                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 711                 result = result.expand();
 712                 result = result.real_part();
 713                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 714                 return result;
 715         }
 716
 717         ex a = basis.real_part();
 718         ex b = basis.imag_part();
 719         ex c = exponent.real_part();
 720         ex d = exponent.imag_part();
 721         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 722 }
 723
 724 ex power::imag_part() const
 725 {
 726         if (exponent.info(info_flags::integer)) {
 727                 ex basis_real = basis.real_part();
 728                 if (basis_real == basis)
 729                         return 0;
 730                 realsymbol a("a"),b("b");
 731                 ex result;
 732                 if (exponent.info(info_flags::posint))
 733                         result = power(a+I*b,exponent);
 734                 else
 735                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 736                 result = result.expand();
 737                 result = result.imag_part();
 738                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 739                 return result;
 740         }
 741
 742         ex a=basis.real_part();
 743         ex b=basis.imag_part();
 744         ex c=exponent.real_part();
 745         ex d=exponent.imag_part();
 746         return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 747 }
 748
 749 // protected
 750
 751 // protected
 752
 753 /** Implementation of ex::diff() for a power.
 754  *  @see ex::diff */
 755 ex power::derivative(const symbol & s) const
 756 {
 757         if (is_a<numeric>(exponent)) {
 758                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 759                 epvector newseq;
 760                 newseq.reserve(2);
 761                 newseq.push_back(expair(basis, exponent - _ex1));
 762                 newseq.push_back(expair(basis.diff(s), _ex1));
 763                 return mul(newseq, exponent);
 764         } else {
 765                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 766                 return mul(*this,
 767                            add(mul(exponent.diff(s), log(basis)),
 768                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 769         }
 770 }
 771
 772 int power::compare_same_type(const basic & other) const
 773 {
 774         GINAC_ASSERT(is_exactly_a<power>(other));
 775         const power &o = static_cast<const power &>(other);
 776
 777         int cmpval = basis.compare(o.basis);
 778         if (cmpval)
 779                 return cmpval;
 780         else
 781                 return exponent.compare(o.exponent);
 782 }
 783
 784 unsigned power::return_type() const
 785 {
 786         return basis.return_type();
 787 }
 788
 789 return_type_t power::return_type_tinfo() const
 790 {
 791         return basis.return_type_tinfo();
 792 }
 793
 794 ex power::expand(unsigned options) const
 795 {
 796         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 797                 // A special case worth optimizing.
 798                 setflag(status_flags::expanded);
 799                 return *this;
 800         }
 801
 802         const ex expanded_basis = basis.expand(options);
 803         const ex expanded_exponent = exponent.expand(options);
 804
 805         // x^(a+b) -> x^a * x^b
 806         if (is_exactly_a<add>(expanded_exponent)) {
 807                 const add &a = ex_to<add>(expanded_exponent);
 808                 exvector distrseq;
 809                 distrseq.reserve(a.seq.size() + 1);
 810                 epvector::const_iterator last = a.seq.end();
 811                 epvector::const_iterator cit = a.seq.begin();
 812                 while (cit!=last) {
 813                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 814                         ++cit;
 815                 }
 816
 817                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 818                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 819                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 820                         int int_exponent = num_exponent.to_int();
 821                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 822                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 823                         else
 824                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 825                 } else
 826                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 827
 828                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 829                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 830                 return r.expand(options);
 831         }
 832
 833         if (!is_exactly_a<numeric>(expanded_exponent) ||
 834                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 835                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 836                         return this->hold();
 837                 } else {
 838                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 839                 }
 840         }
 841
 842         // integer numeric exponent
 843         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 844         int int_exponent = num_exponent.to_int();
 845
 846         // (x+y)^n, n>0
 847         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 848                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 849
 850         // (x*y)^n -> x^n * y^n
 851         if (is_exactly_a<mul>(expanded_basis))
 852                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 853
 854         // cannot expand further
 855         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 856                 return this->hold();
 857         else
 858                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 859 }
 860
 861 //////////
 862 // new virtual functions which can be overridden by derived classes
 863 //////////
 864
 865 // none
 866
 867 //////////
 868 // non-virtual functions in this class
 869 //////////
 870
 871 /** expand a^n where a is an add and n is a positive integer.
 872  *  @see power::expand */
 873 ex power::expand_add(const add & a, int n, unsigned options) const
 874 {
 875         if (n==2)
 876                 return expand_add_2(a, options);
 877
 878         const size_t m = a.nops();
 879         exvector result;
 880         // The number of terms will be the number of combinatorial compositions,
 881         // i.e. the number of unordered arrangements of m nonnegative integers
 882         // which sum up to n.  It is frequently written as C_n(m) and directly
 883         // related with binomial coefficients:
 884         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 885         intvector k(m-1);
 886         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 887         intvector upper_limit(m-1);
 888
 889         for (size_t l=0; l<m-1; ++l) {
 890                 k[l] = 0;
 891                 k_cum[l] = 0;
 892                 upper_limit[l] = n;
 893         }
 894
 895         while (true) {
 896                 exvector term;
 897                 term.reserve(m+1);
 898                 for (std::size_t l = 0; l < m - 1; ++l) {
 899                         const ex & b = a.op(l);
 900                         GINAC_ASSERT(!is_exactly_a<add>(b));
 901                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 902                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 903                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 904                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 905                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 906                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 907                         if (is_exactly_a<mul>(b))
 908                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 909                         else
 910                                 term.push_back(power(b,k[l]));
 911                 }
 912
 913                 const ex & b = a.op(m - 1);
 914                 GINAC_ASSERT(!is_exactly_a<add>(b));
 915                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 916                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 917                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 918                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 919                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 920                              !is_exactly_a<power>(ex_to<power>(b).basis));
 921                 if (is_exactly_a<mul>(b))
 922                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 923                 else
 924                         term.push_back(power(b,n-k_cum[m-2]));
 925
 926                 numeric f = binomial(numeric(n),numeric(k[0]));
 927                 for (std::size_t l = 1; l < m - 1; ++l)
 928                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 929
 930                 term.push_back(f);
 931
 932                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 933
 934                 // increment k[]
 935                 bool done = false;
 936                 std::size_t l = m - 2;
 937                 while ((++k[l]) > upper_limit[l]) {
 938                         k[l] = 0;
 939                         if (l != 0)
 940                                 --l;
 941                         else {
 942                                 done = true;
 943                                 break;
 944                         }
 945                 }
 946                 if (done)
 947                         break;
 948
 949                 // recalc k_cum[] and upper_limit[]
 950                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 951
 952                 for (size_t i=l+1; i<m-1; ++i)
 953                         k_cum[i] = k_cum[i-1]+k[i];
 954
 955                 for (size_t i=l+1; i<m-1; ++i)
 956                         upper_limit[i] = n-k_cum[i-1];
 957         }
 958
 959         return (new add(result))->setflag(status_flags::dynallocated |
 960                                           status_flags::expanded);
 961 }
 962
 963
 964 /** Special case of power::expand_add. Expands a^2 where a is an add.
 965  *  @see power::expand_add */
 966 ex power::expand_add_2(const add & a, unsigned options) const
 967 {
 968         epvector sum;
 969         size_t a_nops = a.nops();
 970         sum.reserve((a_nops*(a_nops+1))/2);
 971         epvector::const_iterator last = a.seq.end();
 972
 973         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 974         // first part: ignore overall_coeff and expand other terms
 975         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 976                 const ex & r = cit0->rest;
 977                 const ex & c = cit0->coeff;
 978
 979                 GINAC_ASSERT(!is_exactly_a<add>(r));
 980                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 981                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 982                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 983                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 984                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 985                              !is_exactly_a<power>(ex_to<power>(r).basis));
 986
 987                 if (c.is_equal(_ex1)) {
 988                         if (is_exactly_a<mul>(r)) {
 989                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 990                                                      _ex1));
 991                         } else {
 992                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 993                                                      _ex1));
 994                         }
 995                 } else {
 996                         if (is_exactly_a<mul>(r)) {
 997                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 998                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 999                         } else {
1000                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1001                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
1002                         }
1003                 }
1004
1005                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1006                         const ex & r1 = cit1->rest;
1007                         const ex & c1 = cit1->coeff;
1008                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
1009                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1010                 }
1011         }
1012
1013         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1014
1015         // second part: add terms coming from overall_factor (if != 0)
1016         if (!a.overall_coeff.is_zero()) {
1017                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1018                 while (i != end) {
1019                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1020                         ++i;
1021                 }
1022                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1023         }
1024
1025         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1026
1027         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1028 }
1029
1030 /** Expand factors of m in m^n where m is a mul and n is an integer.
1031  *  @see power::expand */
1032 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1033 {
1034         GINAC_ASSERT(n.is_integer());
1035
1036         if (n.is_zero()) {
1037                 return _ex1;
1038         }
1039
1040         // do not bother to rename indices if there are no any.
1041         if ((!(options & expand_options::expand_rename_idx))
1042                         && m.info(info_flags::has_indices))
1043                 options |= expand_options::expand_rename_idx;
1044         // Leave it to multiplication since dummy indices have to be renamed
1045         if ((options & expand_options::expand_rename_idx) &&
1046                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1047                 ex result = m;
1048                 exvector va = get_all_dummy_indices(m);
1049                 sort(va.begin(), va.end(), ex_is_less());
1050
1051                 for (int i=1; i < n.to_int(); i++)
1052                         result *= rename_dummy_indices_uniquely(va, m);
1053                 return result;
1054         }
1055
1056         epvector distrseq;
1057         distrseq.reserve(m.seq.size());
1058         bool need_reexpand = false;
1059
1060         epvector::const_iterator last = m.seq.end();
1061         epvector::const_iterator cit = m.seq.begin();
1062         while (cit!=last) {
1063                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1064                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1065                         // this happens when e.g. (a+b)^(1/2) gets squared and
1066                         // the resulting product needs to be reexpanded
1067                         need_reexpand = true;
1068                 }
1069                 distrseq.push_back(p);
1070                 ++cit;
1071         }
1072
1073         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1074         if (need_reexpand)
1075                 return ex(result).expand(options);
1076         if (from_expand)
1077                 return result.setflag(status_flags::expanded);
1078         return result;
1079 }
1080
1081 GINAC_BIND_UNARCHIVER(power);
1082
1083 } // namespace GiNaC