ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2011 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 (exponent.has(var))
 293                 return false;
 294         if (!exponent.info(info_flags::nonnegint))
 295                 return false;
 296         return basis.is_polynomial(var);
 297 }
 298
 299 int power::degree(const ex & s) const
 300 {
 301         if (is_equal(ex_to<basic>(s)))
 302                 return 1;
 303         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 304                 if (basis.is_equal(s))
 305                         return ex_to<numeric>(exponent).to_int();
 306                 else
 307                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 308         } else if (basis.has(s))
 309                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 310         else
 311                 return 0;
 312 }
 313
 314 int power::ldegree(const ex & s) const
 315 {
 316         if (is_equal(ex_to<basic>(s)))
 317                 return 1;
 318         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 319                 if (basis.is_equal(s))
 320                         return ex_to<numeric>(exponent).to_int();
 321                 else
 322                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 323         } else if (basis.has(s))
 324                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 325         else
 326                 return 0;
 327 }
 328
 329 ex power::coeff(const ex & s, int n) const
 330 {
 331         if (is_equal(ex_to<basic>(s)))
 332                 return n==1 ? _ex1 : _ex0;
 333         else if (!basis.is_equal(s)) {
 334                 // basis not equal to s
 335                 if (n == 0)
 336                         return *this;
 337                 else
 338                         return _ex0;
 339         } else {
 340                 // basis equal to s
 341                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 342                         // integer exponent
 343                         int int_exp = ex_to<numeric>(exponent).to_int();
 344                         if (n == int_exp)
 345                                 return _ex1;
 346                         else
 347                                 return _ex0;
 348                 } else {
 349                         // non-integer exponents are treated as zero
 350                         if (n == 0)
 351                                 return *this;
 352                         else
 353                                 return _ex0;
 354                 }
 355         }
 356 }
 357
 358 /** Perform automatic term rewriting rules in this class.  In the following
 359  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 360  *  stand for such expressions that contain a plain number.
 361  *  - ^(x,0) -> 1  (also handles ^(0,0))
 362  *  - ^(x,1) -> x
 363  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 364  *  - ^(1,x) -> 1
 365  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 366  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 367  *  - ^(^(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!)
 368  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 369  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 370  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 371  *
 372  *  @param level cut-off in recursive evaluation */
 373 ex power::eval(int level) const
 374 {
 375         if ((level==1) && (flags & status_flags::evaluated))
 376                 return *this;
 377         else if (level == -max_recursion_level)
 378                 throw(std::runtime_error("max recursion level reached"));
 379
 380         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 381         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 382
 383         const numeric *num_basis = NULL;
 384         const numeric *num_exponent = NULL;
 385
 386         if (is_exactly_a<numeric>(ebasis)) {
 387                 num_basis = &ex_to<numeric>(ebasis);
 388         }
 389         if (is_exactly_a<numeric>(eexponent)) {
 390                 num_exponent = &ex_to<numeric>(eexponent);
 391         }
 392
 393         // ^(x,0) -> 1  (0^0 also handled here)
 394         if (eexponent.is_zero()) {
 395                 if (ebasis.is_zero())
 396                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 397                 else
 398                         return _ex1;
 399         }
 400
 401         // ^(x,1) -> x
 402         if (eexponent.is_equal(_ex1))
 403                 return ebasis;
 404
 405         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 406         if ( ebasis.is_zero() && num_exponent ) {
 407                 if ((num_exponent->real()).is_zero())
 408                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 409                 else if ((num_exponent->real()).is_negative())
 410                         throw (pole_error("power::eval(): division by zero",1));
 411                 else
 412                         return _ex0;
 413         }
 414
 415         // ^(1,x) -> 1
 416         if (ebasis.is_equal(_ex1))
 417                 return _ex1;
 418
 419         // power of a function calculated by separate rules defined for this function
 420         if (is_exactly_a<function>(ebasis))
 421                 return ex_to<function>(ebasis).power(eexponent);
 422
 423         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 424         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 425                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 426
 427         if ( num_exponent ) {
 428
 429                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 430                 // except if c1,c2 are rational, but c1^c2 is not)
 431                 if ( num_basis ) {
 432                         const bool basis_is_crational = num_basis->is_crational();
 433                         const bool exponent_is_crational = num_exponent->is_crational();
 434                         if (!basis_is_crational || !exponent_is_crational) {
 435                                 // return a plain float
 436                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 437                                                                                                status_flags::evaluated |
 438                                                                                                status_flags::expanded);
 439                         }
 440
 441                         const numeric res = num_basis->power(*num_exponent);
 442                         if (res.is_crational()) {
 443                                 return res;
 444                         }
 445                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 446
 447                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 448                         if (basis_is_crational && exponent_is_crational
 449                             && num_exponent->is_real()
 450                             && !num_exponent->is_integer()) {
 451                                 const numeric n = num_exponent->numer();
 452                                 const numeric m = num_exponent->denom();
 453                                 numeric r;
 454                                 numeric q = iquo(n, m, r);
 455                                 if (r.is_negative()) {
 456                                         r += m;
 457                                         --q;
 458                                 }
 459                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 460                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 461                                                 // try it for numerator and denominator separately, in order to
 462                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 463                                                 const numeric bnum = num_basis->numer();
 464                                                 const numeric bden = num_basis->denom();
 465                                                 const numeric res_bnum = bnum.power(*num_exponent);
 466                                                 const numeric res_bden = bden.power(*num_exponent);
 467                                                 if (res_bnum.is_integer())
 468                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 469                                                 if (res_bden.is_integer())
 470                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 471                                         }
 472                                         return this->hold();
 473                                 } else {
 474                                         // assemble resulting product, but allowing for a re-evaluation,
 475                                         // because otherwise we'll end up with something like
 476                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 477                                         // instead of 7/16*7^(1/3).
 478                                         ex prod = power(*num_basis,r.div(m));
 479                                         return prod*power(*num_basis,q);
 480                                 }
 481                         }
 482                 }
 483
 484                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 485                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
 486                 // case c1==1 should not happen, see below!)
 487                 if (is_exactly_a<power>(ebasis)) {
 488                         const power & sub_power = ex_to<power>(ebasis);
 489                         const ex & sub_basis = sub_power.basis;
 490                         const ex & sub_exponent = sub_power.exponent;
 491                         if (is_exactly_a<numeric>(sub_exponent)) {
 492                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 493                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 494                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()
 495                                                 || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
 496                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 497                                 }
 498                         }
 499                 }
 500
 501                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 502                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 503                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 504                 }
 505
 506                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 507                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 508                         numeric icont = ebasis.integer_content();
 509                         const numeric lead_coeff =
 510                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 511
 512                         const bool canonicalizable = lead_coeff.is_integer();
 513                         const bool unit_normal = lead_coeff.is_pos_integer();
 514                         if (canonicalizable && (! unit_normal))
 515                                 icont = icont.mul(*_num_1_p);
 516
 517                         if (canonicalizable && (icont != *_num1_p)) {
 518                                 const add& addref = ex_to<add>(ebasis);
 519                                 add* addp = new add(addref);
 520                                 addp->setflag(status_flags::dynallocated);
 521                                 addp->clearflag(status_flags::hash_calculated);
 522                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 523                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 524                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 525
 526                                 const numeric c = icont.power(*num_exponent);
 527                                 if (likely(c != *_num1_p))
 528                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 529                                 else
 530                                         return power(*addp, *num_exponent);
 531                         }
 532                 }
 533
 534                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 535                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 536                 if (is_exactly_a<mul>(ebasis)) {
 537                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 538                         const mul & mulref = ex_to<mul>(ebasis);
 539                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 540                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 541                                 if (num_coeff.is_real()) {
 542                                         if (num_coeff.is_positive()) {
 543                                                 mul *mulp = new mul(mulref);
 544                                                 mulp->overall_coeff = _ex1;
 545                                                 mulp->setflag(status_flags::dynallocated);
 546                                                 mulp->clearflag(status_flags::evaluated);
 547                                                 mulp->clearflag(status_flags::hash_calculated);
 548                                                 return (new mul(power(*mulp,exponent),
 549                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 550                                         } else {
 551                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 552                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 553                                                         mul *mulp = new mul(mulref);
 554                                                         mulp->overall_coeff = _ex_1;
 555                                                         mulp->setflag(status_flags::dynallocated);
 556                                                         mulp->clearflag(status_flags::evaluated);
 557                                                         mulp->clearflag(status_flags::hash_calculated);
 558                                                         return (new mul(power(*mulp,exponent),
 559                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 560                                                 }
 561                                         }
 562                                 }
 563                         }
 564                 }
 565
 566                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 567                 if (num_exponent->is_pos_integer() &&
 568                     ebasis.return_type() != return_types::commutative &&
 569                     !is_a<matrix>(ebasis)) {
 570                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 571                 }
 572         }
 573
 574         if (are_ex_trivially_equal(ebasis,basis) &&
 575             are_ex_trivially_equal(eexponent,exponent)) {
 576                 return this->hold();
 577         }
 578         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 579                                                        status_flags::evaluated);
 580 }
 581
 582 ex power::evalf(int level) const
 583 {
 584         ex ebasis;
 585         ex eexponent;
 586
 587         if (level==1) {
 588                 ebasis = basis;
 589                 eexponent = exponent;
 590         } else if (level == -max_recursion_level) {
 591                 throw(std::runtime_error("max recursion level reached"));
 592         } else {
 593                 ebasis = basis.evalf(level-1);
 594                 if (!is_exactly_a<numeric>(exponent))
 595                         eexponent = exponent.evalf(level-1);
 596                 else
 597                         eexponent = exponent;
 598         }
 599
 600         return power(ebasis,eexponent);
 601 }
 602
 603 ex power::evalm() const
 604 {
 605         const ex ebasis = basis.evalm();
 606         const ex eexponent = exponent.evalm();
 607         if (is_a<matrix>(ebasis)) {
 608                 if (is_exactly_a<numeric>(eexponent)) {
 609                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 610                 }
 611         }
 612         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 613 }
 614
 615 bool power::has(const ex & other, unsigned options) const
 616 {
 617         if (!(options & has_options::algebraic))
 618                 return basic::has(other, options);
 619         if (!is_a<power>(other))
 620                 return basic::has(other, options);
 621         if (!exponent.info(info_flags::integer)
 622                         || !other.op(1).info(info_flags::integer))
 623                 return basic::has(other, options);
 624         if (exponent.info(info_flags::posint)
 625                         && other.op(1).info(info_flags::posint)
 626                         && ex_to<numeric>(exponent).to_int()
 627                                         > ex_to<numeric>(other.op(1)).to_int()
 628                         && basis.match(other.op(0)))
 629                 return true;
 630         if (exponent.info(info_flags::negint)
 631                         && other.op(1).info(info_flags::negint)
 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         return basic::has(other, options);
 637 }
 638
 639 // from mul.cpp
 640 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 641
 642 ex power::subs(const exmap & m, unsigned options) const
 643 {
 644         const ex &subsed_basis = basis.subs(m, options);
 645         const ex &subsed_exponent = exponent.subs(m, options);
 646
 647         if (!are_ex_trivially_equal(basis, subsed_basis)
 648          || !are_ex_trivially_equal(exponent, subsed_exponent))
 649                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 650
 651         if (!(options & subs_options::algebraic))
 652                 return subs_one_level(m, options);
 653
 654         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 655                 int nummatches = std::numeric_limits<int>::max();
 656                 exmap repls;
 657                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 658                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 659                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 660                         ex result = (*this)*power(anum/aden, nummatches);
 661                         return (ex_to<basic>(result)).subs_one_level(m, options);
 662                 }
 663         }
 664
 665         return subs_one_level(m, options);
 666 }
 667
 668 ex power::eval_ncmul(const exvector & v) const
 669 {
 670         return inherited::eval_ncmul(v);
 671 }
 672
 673 ex power::conjugate() const
 674 {
 675         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 676         // branch cut which runs along the negative real axis.
 677         if (basis.info(info_flags::positive)) {
 678                 ex newexponent = exponent.conjugate();
 679                 if (are_ex_trivially_equal(exponent, newexponent)) {
 680                         return *this;
 681                 }
 682                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
 683         }
 684         if (exponent.info(info_flags::integer)) {
 685                 ex newbasis = basis.conjugate();
 686                 if (are_ex_trivially_equal(basis, newbasis)) {
 687                         return *this;
 688                 }
 689                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
 690         }
 691         return conjugate_function(*this).hold();
 692 }
 693
 694 ex power::real_part() const
 695 {
 696         if (exponent.info(info_flags::integer)) {
 697                 ex basis_real = basis.real_part();
 698                 if (basis_real == basis)
 699                         return *this;
 700                 realsymbol a("a"),b("b");
 701                 ex result;
 702                 if (exponent.info(info_flags::posint))
 703                         result = power(a+I*b,exponent);
 704                 else
 705                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 706                 result = result.expand();
 707                 result = result.real_part();
 708                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 709                 return result;
 710         }
 711
 712         ex a = basis.real_part();
 713         ex b = basis.imag_part();
 714         ex c = exponent.real_part();
 715         ex d = exponent.imag_part();
 716         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 717 }
 718
 719 ex power::imag_part() const
 720 {
 721         if (exponent.info(info_flags::integer)) {
 722                 ex basis_real = basis.real_part();
 723                 if (basis_real == basis)
 724                         return 0;
 725                 realsymbol a("a"),b("b");
 726                 ex result;
 727                 if (exponent.info(info_flags::posint))
 728                         result = power(a+I*b,exponent);
 729                 else
 730                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 731                 result = result.expand();
 732                 result = result.imag_part();
 733                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 734                 return result;
 735         }
 736
 737         ex a=basis.real_part();
 738         ex b=basis.imag_part();
 739         ex c=exponent.real_part();
 740         ex d=exponent.imag_part();
 741         return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 742 }
 743
 744 // protected
 745
 746 // protected
 747
 748 /** Implementation of ex::diff() for a power.
 749  *  @see ex::diff */
 750 ex power::derivative(const symbol & s) const
 751 {
 752         if (is_a<numeric>(exponent)) {
 753                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 754                 epvector newseq;
 755                 newseq.reserve(2);
 756                 newseq.push_back(expair(basis, exponent - _ex1));
 757                 newseq.push_back(expair(basis.diff(s), _ex1));
 758                 return mul(newseq, exponent);
 759         } else {
 760                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 761                 return mul(*this,
 762                            add(mul(exponent.diff(s), log(basis)),
 763                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 764         }
 765 }
 766
 767 int power::compare_same_type(const basic & other) const
 768 {
 769         GINAC_ASSERT(is_exactly_a<power>(other));
 770         const power &o = static_cast<const power &>(other);
 771
 772         int cmpval = basis.compare(o.basis);
 773         if (cmpval)
 774                 return cmpval;
 775         else
 776                 return exponent.compare(o.exponent);
 777 }
 778
 779 unsigned power::return_type() const
 780 {
 781         return basis.return_type();
 782 }
 783
 784 return_type_t power::return_type_tinfo() const
 785 {
 786         return basis.return_type_tinfo();
 787 }
 788
 789 ex power::expand(unsigned options) const
 790 {
 791         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 792                 // A special case worth optimizing.
 793                 setflag(status_flags::expanded);
 794                 return *this;
 795         }
 796
 797         const ex expanded_basis = basis.expand(options);
 798         const ex expanded_exponent = exponent.expand(options);
 799
 800         // x^(a+b) -> x^a * x^b
 801         if (is_exactly_a<add>(expanded_exponent)) {
 802                 const add &a = ex_to<add>(expanded_exponent);
 803                 exvector distrseq;
 804                 distrseq.reserve(a.seq.size() + 1);
 805                 epvector::const_iterator last = a.seq.end();
 806                 epvector::const_iterator cit = a.seq.begin();
 807                 while (cit!=last) {
 808                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 809                         ++cit;
 810                 }
 811
 812                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 813                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 814                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 815                         int int_exponent = num_exponent.to_int();
 816                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 817                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 818                         else
 819                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 820                 } else
 821                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 822
 823                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 824                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 825                 return r.expand(options);
 826         }
 827
 828         if (!is_exactly_a<numeric>(expanded_exponent) ||
 829                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 830                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 831                         return this->hold();
 832                 } else {
 833                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 834                 }
 835         }
 836
 837         // integer numeric exponent
 838         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 839         int int_exponent = num_exponent.to_int();
 840
 841         // (x+y)^n, n>0
 842         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 843                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 844
 845         // (x*y)^n -> x^n * y^n
 846         if (is_exactly_a<mul>(expanded_basis))
 847                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 848
 849         // cannot expand further
 850         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 851                 return this->hold();
 852         else
 853                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 854 }
 855
 856 //////////
 857 // new virtual functions which can be overridden by derived classes
 858 //////////
 859
 860 // none
 861
 862 //////////
 863 // non-virtual functions in this class
 864 //////////
 865
 866 /** expand a^n where a is an add and n is a positive integer.
 867  *  @see power::expand */
 868 ex power::expand_add(const add & a, int n, unsigned options) const
 869 {
 870         if (n==2)
 871                 return expand_add_2(a, options);
 872
 873         const size_t m = a.nops();
 874         exvector result;
 875         // The number of terms will be the number of combinatorial compositions,
 876         // i.e. the number of unordered arrangements of m nonnegative integers
 877         // which sum up to n.  It is frequently written as C_n(m) and directly
 878         // related with binomial coefficients:
 879         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 880         intvector k(m-1);
 881         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 882         intvector upper_limit(m-1);
 883
 884         for (size_t l=0; l<m-1; ++l) {
 885                 k[l] = 0;
 886                 k_cum[l] = 0;
 887                 upper_limit[l] = n;
 888         }
 889
 890         while (true) {
 891                 exvector term;
 892                 term.reserve(m+1);
 893                 for (std::size_t l = 0; l < m - 1; ++l) {
 894                         const ex & b = a.op(l);
 895                         GINAC_ASSERT(!is_exactly_a<add>(b));
 896                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 897                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 898                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 899                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 900                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 901                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 902                         if (is_exactly_a<mul>(b))
 903                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 904                         else
 905                                 term.push_back(power(b,k[l]));
 906                 }
 907
 908                 const ex & b = a.op(m - 1);
 909                 GINAC_ASSERT(!is_exactly_a<add>(b));
 910                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 911                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 912                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 913                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 914                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 915                              !is_exactly_a<power>(ex_to<power>(b).basis));
 916                 if (is_exactly_a<mul>(b))
 917                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 918                 else
 919                         term.push_back(power(b,n-k_cum[m-2]));
 920
 921                 numeric f = binomial(numeric(n),numeric(k[0]));
 922                 for (std::size_t l = 1; l < m - 1; ++l)
 923                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 924
 925                 term.push_back(f);
 926
 927                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 928
 929                 // increment k[]
 930                 bool done = false;
 931                 std::size_t l = m - 2;
 932                 while ((++k[l]) > upper_limit[l]) {
 933                         k[l] = 0;
 934                         if (l != 0)
 935                                 --l;
 936                         else {
 937                                 done = true;
 938                                 break;
 939                         }
 940                 }
 941                 if (done)
 942                         break;
 943
 944                 // recalc k_cum[] and upper_limit[]
 945                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 946
 947                 for (size_t i=l+1; i<m-1; ++i)
 948                         k_cum[i] = k_cum[i-1]+k[i];
 949
 950                 for (size_t i=l+1; i<m-1; ++i)
 951                         upper_limit[i] = n-k_cum[i-1];
 952         }
 953
 954         return (new add(result))->setflag(status_flags::dynallocated |
 955                                           status_flags::expanded);
 956 }
 957
 958
 959 /** Special case of power::expand_add. Expands a^2 where a is an add.
 960  *  @see power::expand_add */
 961 ex power::expand_add_2(const add & a, unsigned options) const
 962 {
 963         epvector sum;
 964         size_t a_nops = a.nops();
 965         sum.reserve((a_nops*(a_nops+1))/2);
 966         epvector::const_iterator last = a.seq.end();
 967
 968         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 969         // first part: ignore overall_coeff and expand other terms
 970         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 971                 const ex & r = cit0->rest;
 972                 const ex & c = cit0->coeff;
 973
 974                 GINAC_ASSERT(!is_exactly_a<add>(r));
 975                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 976                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 977                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 978                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 979                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 980                              !is_exactly_a<power>(ex_to<power>(r).basis));
 981
 982                 if (c.is_equal(_ex1)) {
 983                         if (is_exactly_a<mul>(r)) {
 984                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 985                                                      _ex1));
 986                         } else {
 987                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 988                                                      _ex1));
 989                         }
 990                 } else {
 991                         if (is_exactly_a<mul>(r)) {
 992                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 993                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 994                         } else {
 995                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 996                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 997                         }
 998                 }
 999
1000                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1001                         const ex & r1 = cit1->rest;
1002                         const ex & c1 = cit1->coeff;
1003                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
1004                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1005                 }
1006         }
1007
1008         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1009
1010         // second part: add terms coming from overall_factor (if != 0)
1011         if (!a.overall_coeff.is_zero()) {
1012                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1013                 while (i != end) {
1014                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1015                         ++i;
1016                 }
1017                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1018         }
1019
1020         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1021
1022         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1023 }
1024
1025 /** Expand factors of m in m^n where m is a mul and n is an integer.
1026  *  @see power::expand */
1027 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1028 {
1029         GINAC_ASSERT(n.is_integer());
1030
1031         if (n.is_zero()) {
1032                 return _ex1;
1033         }
1034
1035         // do not bother to rename indices if there are no any.
1036         if ((!(options & expand_options::expand_rename_idx))
1037                         && m.info(info_flags::has_indices))
1038                 options |= expand_options::expand_rename_idx;
1039         // Leave it to multiplication since dummy indices have to be renamed
1040         if ((options & expand_options::expand_rename_idx) &&
1041                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1042                 ex result = m;
1043                 exvector va = get_all_dummy_indices(m);
1044                 sort(va.begin(), va.end(), ex_is_less());
1045
1046                 for (int i=1; i < n.to_int(); i++)
1047                         result *= rename_dummy_indices_uniquely(va, m);
1048                 return result;
1049         }
1050
1051         epvector distrseq;
1052         distrseq.reserve(m.seq.size());
1053         bool need_reexpand = false;
1054
1055         epvector::const_iterator last = m.seq.end();
1056         epvector::const_iterator cit = m.seq.begin();
1057         while (cit!=last) {
1058                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1059                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1060                         // this happens when e.g. (a+b)^(1/2) gets squared and
1061                         // the resulting product needs to be reexpanded
1062                         need_reexpand = true;
1063                 }
1064                 distrseq.push_back(p);
1065                 ++cit;
1066         }
1067
1068         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1069         if (need_reexpand)
1070                 return ex(result).expand(options);
1071         if (from_expand)
1072                 return result.setflag(status_flags::expanded);
1073         return result;
1074 }
1075
1076 GINAC_BIND_UNARCHIVER(power);
1077
1078 } // namespace GiNaC