ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "power.h"
  29 #include "expairseq.h"
  30 #include "add.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "numeric.h"
  34 #include "constant.h"
  35 #include "operators.h"
  36 #include "inifcns.h" // for log() in power::derivative()
  37 #include "matrix.h"
  38 #include "indexed.h"
  39 #include "symbol.h"
  40 #include "lst.h"
  41 #include "archive.h"
  42 #include "utils.h"
  43 #include "relational.h"
  44 #include "compiler.h"
  45
  46 namespace GiNaC {
  47
  48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  49   print_func<print_dflt>(&power::do_print_dflt).
  50   print_func<print_latex>(&power::do_print_latex).
  51   print_func<print_csrc>(&power::do_print_csrc).
  52   print_func<print_python>(&power::do_print_python).
  53   print_func<print_python_repr>(&power::do_print_python_repr).
  54   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::has_indices: {
 244                         if (flags & status_flags::has_indices)
 245                                 return true;
 246                         else if (flags & status_flags::has_no_indices)
 247                                 return false;
 248                         else if (basis.info(info_flags::has_indices)) {
 249                                 setflag(status_flags::has_indices);
 250                                 clearflag(status_flags::has_no_indices);
 251                                 return true;
 252                         } else {
 253                                 clearflag(status_flags::has_indices);
 254                                 setflag(status_flags::has_no_indices);
 255                                 return false;
 256                         }
 257                 }
 258         }
 259         return inherited::info(inf);
 260 }
 261
 262 size_t power::nops() const
 263 {
 264         return 2;
 265 }
 266
 267 ex power::op(size_t i) const
 268 {
 269         GINAC_ASSERT(i<2);
 270
 271         return i==0 ? basis : exponent;
 272 }
 273
 274 ex power::map(map_function & f) const
 275 {
 276         const ex &mapped_basis = f(basis);
 277         const ex &mapped_exponent = f(exponent);
 278
 279         if (!are_ex_trivially_equal(basis, mapped_basis)
 280          || !are_ex_trivially_equal(exponent, mapped_exponent))
 281                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 282         else
 283                 return *this;
 284 }
 285
 286 bool power::is_polynomial(const ex & var) const
 287 {
 288         if (exponent.has(var))
 289                 return false;
 290         if (!exponent.info(info_flags::nonnegint))
 291                 return false;
 292         return basis.is_polynomial(var);
 293 }
 294
 295 int power::degree(const ex & s) const
 296 {
 297         if (is_equal(ex_to<basic>(s)))
 298                 return 1;
 299         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 300                 if (basis.is_equal(s))
 301                         return ex_to<numeric>(exponent).to_int();
 302                 else
 303                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 304         } else if (basis.has(s))
 305                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 306         else
 307                 return 0;
 308 }
 309
 310 int power::ldegree(const ex & s) const
 311 {
 312         if (is_equal(ex_to<basic>(s)))
 313                 return 1;
 314         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 315                 if (basis.is_equal(s))
 316                         return ex_to<numeric>(exponent).to_int();
 317                 else
 318                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 319         } else if (basis.has(s))
 320                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 321         else
 322                 return 0;
 323 }
 324
 325 ex power::coeff(const ex & s, int n) const
 326 {
 327         if (is_equal(ex_to<basic>(s)))
 328                 return n==1 ? _ex1 : _ex0;
 329         else if (!basis.is_equal(s)) {
 330                 // basis not equal to s
 331                 if (n == 0)
 332                         return *this;
 333                 else
 334                         return _ex0;
 335         } else {
 336                 // basis equal to s
 337                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 338                         // integer exponent
 339                         int int_exp = ex_to<numeric>(exponent).to_int();
 340                         if (n == int_exp)
 341                                 return _ex1;
 342                         else
 343                                 return _ex0;
 344                 } else {
 345                         // non-integer exponents are treated as zero
 346                         if (n == 0)
 347                                 return *this;
 348                         else
 349                                 return _ex0;
 350                 }
 351         }
 352 }
 353
 354 /** Perform automatic term rewriting rules in this class.  In the following
 355  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 356  *  stand for such expressions that contain a plain number.
 357  *  - ^(x,0) -> 1  (also handles ^(0,0))
 358  *  - ^(x,1) -> x
 359  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 360  *  - ^(1,x) -> 1
 361  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 362  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 363  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 364  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 365  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 366  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 367  *
 368  *  @param level cut-off in recursive evaluation */
 369 ex power::eval(int level) const
 370 {
 371         if ((level==1) && (flags & status_flags::evaluated))
 372                 return *this;
 373         else if (level == -max_recursion_level)
 374                 throw(std::runtime_error("max recursion level reached"));
 375
 376         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 377         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 378
 379         bool basis_is_numerical = false;
 380         bool exponent_is_numerical = false;
 381         const numeric *num_basis;
 382         const numeric *num_exponent;
 383
 384         if (is_exactly_a<numeric>(ebasis)) {
 385                 basis_is_numerical = true;
 386                 num_basis = &ex_to<numeric>(ebasis);
 387         }
 388         if (is_exactly_a<numeric>(eexponent)) {
 389                 exponent_is_numerical = true;
 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() && exponent_is_numerical) {
 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 (exponent_is_numerical) {
 428
 429                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 430                 // except if c1,c2 are rational, but c1^c2 is not)
 431                 if (basis_is_numerical) {
 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,
 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                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 496                                 }
 497                         }
 498                 }
 499
 500                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 501                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 502                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 503                 }
 504
 505                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 506                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 507                         numeric icont = ebasis.integer_content();
 508                         const numeric lead_coeff =
 509                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 510
 511                         const bool canonicalizable = lead_coeff.is_integer();
 512                         const bool unit_normal = lead_coeff.is_pos_integer();
 513                         if (canonicalizable && (! unit_normal))
 514                                 icont = icont.mul(*_num_1_p);
 515
 516                         if (canonicalizable && (icont != *_num1_p)) {
 517                                 const add& addref = ex_to<add>(ebasis);
 518                                 add* addp = new add(addref);
 519                                 addp->setflag(status_flags::dynallocated);
 520                                 addp->clearflag(status_flags::hash_calculated);
 521                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 522                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 523                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 524
 525                                 const numeric c = icont.power(*num_exponent);
 526                                 if (likely(c != *_num1_p))
 527                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 528                                 else
 529                                         return power(*addp, *num_exponent);
 530                         }
 531                 }
 532
 533                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 534                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 535                 if (is_exactly_a<mul>(ebasis)) {
 536                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 537                         const mul & mulref = ex_to<mul>(ebasis);
 538                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 539                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 540                                 if (num_coeff.is_real()) {
 541                                         if (num_coeff.is_positive()) {
 542                                                 mul *mulp = new mul(mulref);
 543                                                 mulp->overall_coeff = _ex1;
 544                                                 mulp->clearflag(status_flags::evaluated);
 545                                                 mulp->clearflag(status_flags::hash_calculated);
 546                                                 return (new mul(power(*mulp,exponent),
 547                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 548                                         } else {
 549                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 550                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 551                                                         mul *mulp = new mul(mulref);
 552                                                         mulp->overall_coeff = _ex_1;
 553                                                         mulp->clearflag(status_flags::evaluated);
 554                                                         mulp->clearflag(status_flags::hash_calculated);
 555                                                         return (new mul(power(*mulp,exponent),
 556                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 557                                                 }
 558                                         }
 559                                 }
 560                         }
 561                 }
 562
 563                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 564                 if (num_exponent->is_pos_integer() &&
 565                     ebasis.return_type() != return_types::commutative &&
 566                     !is_a<matrix>(ebasis)) {
 567                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 568                 }
 569         }
 570
 571         if (are_ex_trivially_equal(ebasis,basis) &&
 572             are_ex_trivially_equal(eexponent,exponent)) {
 573                 return this->hold();
 574         }
 575         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 576                                                        status_flags::evaluated);
 577 }
 578
 579 ex power::evalf(int level) const
 580 {
 581         ex ebasis;
 582         ex eexponent;
 583
 584         if (level==1) {
 585                 ebasis = basis;
 586                 eexponent = exponent;
 587         } else if (level == -max_recursion_level) {
 588                 throw(std::runtime_error("max recursion level reached"));
 589         } else {
 590                 ebasis = basis.evalf(level-1);
 591                 if (!is_exactly_a<numeric>(exponent))
 592                         eexponent = exponent.evalf(level-1);
 593                 else
 594                         eexponent = exponent;
 595         }
 596
 597         return power(ebasis,eexponent);
 598 }
 599
 600 ex power::evalm() const
 601 {
 602         const ex ebasis = basis.evalm();
 603         const ex eexponent = exponent.evalm();
 604         if (is_a<matrix>(ebasis)) {
 605                 if (is_exactly_a<numeric>(eexponent)) {
 606                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 607                 }
 608         }
 609         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 610 }
 611
 612 bool power::has(const ex & other, unsigned options) const
 613 {
 614         if (!(options & has_options::algebraic))
 615                 return basic::has(other, options);
 616         if (!is_a<power>(other))
 617                 return basic::has(other, options);
 618         if (!exponent.info(info_flags::integer)
 619                         || !other.op(1).info(info_flags::integer))
 620                 return basic::has(other, options);
 621         if (exponent.info(info_flags::posint)
 622                         && other.op(1).info(info_flags::posint)
 623                         && ex_to<numeric>(exponent).to_int()
 624                                         > ex_to<numeric>(other.op(1)).to_int()
 625                         && basis.match(other.op(0)))
 626                 return true;
 627         if (exponent.info(info_flags::negint)
 628                         && other.op(1).info(info_flags::negint)
 629                         && ex_to<numeric>(exponent).to_int()
 630                                         < ex_to<numeric>(other.op(1)).to_int()
 631                         && basis.match(other.op(0)))
 632                 return true;
 633         return basic::has(other, options);
 634 }
 635
 636 // from mul.cpp
 637 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 638
 639 ex power::subs(const exmap & m, unsigned options) const
 640 {
 641         const ex &subsed_basis = basis.subs(m, options);
 642         const ex &subsed_exponent = exponent.subs(m, options);
 643
 644         if (!are_ex_trivially_equal(basis, subsed_basis)
 645          || !are_ex_trivially_equal(exponent, subsed_exponent))
 646                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 647
 648         if (!(options & subs_options::algebraic))
 649                 return subs_one_level(m, options);
 650
 651         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 652                 int nummatches = std::numeric_limits<int>::max();
 653                 exmap repls;
 654                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 655                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 656                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 657                         ex result = (*this)*power(anum/aden, nummatches);
 658                         return (ex_to<basic>(result)).subs_one_level(m, options);
 659                 }
 660         }
 661
 662         return subs_one_level(m, options);
 663 }
 664
 665 ex power::eval_ncmul(const exvector & v) const
 666 {
 667         return inherited::eval_ncmul(v);
 668 }
 669
 670 ex power::conjugate() const
 671 {
 672         ex newbasis = basis.conjugate();
 673         ex newexponent = exponent.conjugate();
 674         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 675                 return *this;
 676         }
 677         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 678 }
 679
 680 ex power::real_part() const
 681 {
 682         if (exponent.info(info_flags::integer)) {
 683                 ex basis_real = basis.real_part();
 684                 if (basis_real == basis)
 685                         return *this;
 686                 realsymbol a("a"),b("b");
 687                 ex result;
 688                 if (exponent.info(info_flags::posint))
 689                         result = power(a+I*b,exponent);
 690                 else
 691                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 692                 result = result.expand();
 693                 result = result.real_part();
 694                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 695                 return result;
 696         }
 697
 698         ex a = basis.real_part();
 699         ex b = basis.imag_part();
 700         ex c = exponent.real_part();
 701         ex d = exponent.imag_part();
 702         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 703 }
 704
 705 ex power::imag_part() const
 706 {
 707         if (exponent.info(info_flags::integer)) {
 708                 ex basis_real = basis.real_part();
 709                 if (basis_real == basis)
 710                         return 0;
 711                 realsymbol a("a"),b("b");
 712                 ex result;
 713                 if (exponent.info(info_flags::posint))
 714                         result = power(a+I*b,exponent);
 715                 else
 716                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 717                 result = result.expand();
 718                 result = result.imag_part();
 719                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 720                 return result;
 721         }
 722
 723         ex a=basis.real_part();
 724         ex b=basis.imag_part();
 725         ex c=exponent.real_part();
 726         ex d=exponent.imag_part();
 727         return
 728                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 729 }
 730
 731 // protected
 732
 733 // protected
 734
 735 /** Implementation of ex::diff() for a power.
 736  *  @see ex::diff */
 737 ex power::derivative(const symbol & s) const
 738 {
 739         if (is_a<numeric>(exponent)) {
 740                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 741                 epvector newseq;
 742                 newseq.reserve(2);
 743                 newseq.push_back(expair(basis, exponent - _ex1));
 744                 newseq.push_back(expair(basis.diff(s), _ex1));
 745                 return mul(newseq, exponent);
 746         } else {
 747                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 748                 return mul(*this,
 749                            add(mul(exponent.diff(s), log(basis)),
 750                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 751         }
 752 }
 753
 754 int power::compare_same_type(const basic & other) const
 755 {
 756         GINAC_ASSERT(is_exactly_a<power>(other));
 757         const power &o = static_cast<const power &>(other);
 758
 759         int cmpval = basis.compare(o.basis);
 760         if (cmpval)
 761                 return cmpval;
 762         else
 763                 return exponent.compare(o.exponent);
 764 }
 765
 766 unsigned power::return_type() const
 767 {
 768         return basis.return_type();
 769 }
 770
 771 return_type_t power::return_type_tinfo() const
 772 {
 773         return basis.return_type_tinfo();
 774 }
 775
 776 ex power::expand(unsigned options) const
 777 {
 778         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 779                 // A special case worth optimizing.
 780                 setflag(status_flags::expanded);
 781                 return *this;
 782         }
 783
 784         const ex expanded_basis = basis.expand(options);
 785         const ex expanded_exponent = exponent.expand(options);
 786
 787         // x^(a+b) -> x^a * x^b
 788         if (is_exactly_a<add>(expanded_exponent)) {
 789                 const add &a = ex_to<add>(expanded_exponent);
 790                 exvector distrseq;
 791                 distrseq.reserve(a.seq.size() + 1);
 792                 epvector::const_iterator last = a.seq.end();
 793                 epvector::const_iterator cit = a.seq.begin();
 794                 while (cit!=last) {
 795                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 796                         ++cit;
 797                 }
 798
 799                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 800                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 801                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 802                         int int_exponent = num_exponent.to_int();
 803                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 804                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 805                         else
 806                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 807                 } else
 808                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 809
 810                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 811                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 812                 return r.expand(options);
 813         }
 814
 815         if (!is_exactly_a<numeric>(expanded_exponent) ||
 816                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 817                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 818                         return this->hold();
 819                 } else {
 820                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 821                 }
 822         }
 823
 824         // integer numeric exponent
 825         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 826         int int_exponent = num_exponent.to_int();
 827
 828         // (x+y)^n, n>0
 829         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 830                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 831
 832         // (x*y)^n -> x^n * y^n
 833         if (is_exactly_a<mul>(expanded_basis))
 834                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 835
 836         // cannot expand further
 837         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 838                 return this->hold();
 839         else
 840                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 841 }
 842
 843 //////////
 844 // new virtual functions which can be overridden by derived classes
 845 //////////
 846
 847 // none
 848
 849 //////////
 850 // non-virtual functions in this class
 851 //////////
 852
 853 /** expand a^n where a is an add and n is a positive integer.
 854  *  @see power::expand */
 855 ex power::expand_add(const add & a, int n, unsigned options) const
 856 {
 857         if (n==2)
 858                 return expand_add_2(a, options);
 859
 860         const size_t m = a.nops();
 861         exvector result;
 862         // The number of terms will be the number of combinatorial compositions,
 863         // i.e. the number of unordered arrangements of m nonnegative integers
 864         // which sum up to n.  It is frequently written as C_n(m) and directly
 865         // related with binomial coefficients:
 866         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 867         intvector k(m-1);
 868         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 869         intvector upper_limit(m-1);
 870
 871         for (size_t l=0; l<m-1; ++l) {
 872                 k[l] = 0;
 873                 k_cum[l] = 0;
 874                 upper_limit[l] = n;
 875         }
 876
 877         while (true) {
 878                 exvector term;
 879                 term.reserve(m+1);
 880                 for (std::size_t l = 0; l < m - 1; ++l) {
 881                         const ex & b = a.op(l);
 882                         GINAC_ASSERT(!is_exactly_a<add>(b));
 883                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 884                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 885                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 886                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 887                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 888                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 889                         if (is_exactly_a<mul>(b))
 890                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 891                         else
 892                                 term.push_back(power(b,k[l]));
 893                 }
 894
 895                 const ex & b = a.op(m - 1);
 896                 GINAC_ASSERT(!is_exactly_a<add>(b));
 897                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 898                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 899                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 900                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 901                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 902                              !is_exactly_a<power>(ex_to<power>(b).basis));
 903                 if (is_exactly_a<mul>(b))
 904                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 905                 else
 906                         term.push_back(power(b,n-k_cum[m-2]));
 907
 908                 numeric f = binomial(numeric(n),numeric(k[0]));
 909                 for (std::size_t l = 1; l < m - 1; ++l)
 910                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 911
 912                 term.push_back(f);
 913
 914                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 915
 916                 // increment k[]
 917                 bool done = false;
 918                 std::size_t l = m - 2;
 919                 while ((++k[l]) > upper_limit[l]) {
 920                         k[l] = 0;
 921                         if (l != 0)
 922                                 --l;
 923                         else {
 924                                 done = true;
 925                                 break;
 926                         }
 927                 }
 928                 if (done)
 929                         break;
 930
 931                 // recalc k_cum[] and upper_limit[]
 932                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 933
 934                 for (size_t i=l+1; i<m-1; ++i)
 935                         k_cum[i] = k_cum[i-1]+k[i];
 936
 937                 for (size_t i=l+1; i<m-1; ++i)
 938                         upper_limit[i] = n-k_cum[i-1];
 939         }
 940
 941         return (new add(result))->setflag(status_flags::dynallocated |
 942                                           status_flags::expanded);
 943 }
 944
 945
 946 /** Special case of power::expand_add. Expands a^2 where a is an add.
 947  *  @see power::expand_add */
 948 ex power::expand_add_2(const add & a, unsigned options) const
 949 {
 950         epvector sum;
 951         size_t a_nops = a.nops();
 952         sum.reserve((a_nops*(a_nops+1))/2);
 953         epvector::const_iterator last = a.seq.end();
 954
 955         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 956         // first part: ignore overall_coeff and expand other terms
 957         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 958                 const ex & r = cit0->rest;
 959                 const ex & c = cit0->coeff;
 960
 961                 GINAC_ASSERT(!is_exactly_a<add>(r));
 962                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 963                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 964                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 965                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 966                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 967                              !is_exactly_a<power>(ex_to<power>(r).basis));
 968
 969                 if (c.is_equal(_ex1)) {
 970                         if (is_exactly_a<mul>(r)) {
 971                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 972                                                      _ex1));
 973                         } else {
 974                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 975                                                      _ex1));
 976                         }
 977                 } else {
 978                         if (is_exactly_a<mul>(r)) {
 979                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 980                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 981                         } else {
 982                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 983                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 984                         }
 985                 }
 986
 987                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 988                         const ex & r1 = cit1->rest;
 989                         const ex & c1 = cit1->coeff;
 990                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 991                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 992                 }
 993         }
 994
 995         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 996
 997         // second part: add terms coming from overall_factor (if != 0)
 998         if (!a.overall_coeff.is_zero()) {
 999                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1000                 while (i != end) {
1001                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1002                         ++i;
1003                 }
1004                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1005         }
1006
1007         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1008
1009         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1010 }
1011
1012 /** Expand factors of m in m^n where m is a mul and n is an integer.
1013  *  @see power::expand */
1014 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1015 {
1016         GINAC_ASSERT(n.is_integer());
1017
1018         if (n.is_zero()) {
1019                 return _ex1;
1020         }
1021
1022         // do not bother to rename indices if there are no any.
1023         if ((!(options & expand_options::expand_rename_idx))
1024                         && m.info(info_flags::has_indices))
1025                 options |= expand_options::expand_rename_idx;
1026         // Leave it to multiplication since dummy indices have to be renamed
1027         if ((options & expand_options::expand_rename_idx) &&
1028                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1029                 ex result = m;
1030                 exvector va = get_all_dummy_indices(m);
1031                 sort(va.begin(), va.end(), ex_is_less());
1032
1033                 for (int i=1; i < n.to_int(); i++)
1034                         result *= rename_dummy_indices_uniquely(va, m);
1035                 return result;
1036         }
1037
1038         epvector distrseq;
1039         distrseq.reserve(m.seq.size());
1040         bool need_reexpand = false;
1041
1042         epvector::const_iterator last = m.seq.end();
1043         epvector::const_iterator cit = m.seq.begin();
1044         while (cit!=last) {
1045                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1046                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1047                         // this happens when e.g. (a+b)^(1/2) gets squared and
1048                         // the resulting product needs to be reexpanded
1049                         need_reexpand = true;
1050                 }
1051                 distrseq.push_back(p);
1052                 ++cit;
1053         }
1054
1055         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1056         if (need_reexpand)
1057                 return ex(result).expand(options);
1058         if (from_expand)
1059                 return result.setflag(status_flags::expanded);
1060         return result;
1061 }
1062
1063 GINAC_BIND_UNARCHIVER(power);
1064
1065 } // namespace GiNaC