ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "power.h"
  29 #include "expairseq.h"
  30 #include "add.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "numeric.h"
  34 #include "constant.h"
  35 #include "operators.h"
  36 #include "inifcns.h" // for log() in power::derivative()
  37 #include "matrix.h"
  38 #include "indexed.h"
  39 #include "symbol.h"
  40 #include "lst.h"
  41 #include "archive.h"
  42 #include "utils.h"
  43 #include "relational.h"
  44 #include "compiler.h"
  45
  46 namespace GiNaC {
  47
  48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  49   print_func<print_dflt>(&power::do_print_dflt).
  50   print_func<print_latex>(&power::do_print_latex).
  51   print_func<print_csrc>(&power::do_print_csrc).
  52   print_func<print_python>(&power::do_print_python).
  53   print_func<print_python_repr>(&power::do_print_python_repr).
  54   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() : inherited(&power::tinfo_static) { }
  63
  64 //////////
  65 // other constructors
  66 //////////
  67
  68 // all inlined
  69
  70 //////////
  71 // archiving
  72 //////////
  73
  74 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  75 {
  76         n.find_ex("basis", basis, sym_lst);
  77         n.find_ex("exponent", exponent, sym_lst);
  78 }
  79
  80 void power::archive(archive_node &n) const
  81 {
  82         inherited::archive(n);
  83         n.add_ex("basis", basis);
  84         n.add_ex("exponent", exponent);
  85 }
  86
  87 DEFAULT_UNARCHIVE(power)
  88
  89 //////////
  90 // functions overriding virtual functions from base classes
  91 //////////
  92
  93 // public
  94
  95 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  96 {
  97         // Ordinary output of powers using '^' or '**'
  98         if (precedence() <= level)
  99                 c.s << openbrace << '(';
 100         basis.print(c, precedence());
 101         c.s << powersymbol;
 102         c.s << openbrace;
 103         exponent.print(c, precedence());
 104         c.s << closebrace;
 105         if (precedence() <= level)
 106                 c.s << ')' << closebrace;
 107 }
 108
 109 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 110 {
 111         if (exponent.is_equal(_ex1_2)) {
 112
 113                 // Square roots are printed in a special way
 114                 c.s << "sqrt(";
 115                 basis.print(c);
 116                 c.s << ')';
 117
 118         } else
 119                 print_power(c, "^", "", "", level);
 120 }
 121
 122 void power::do_print_latex(const print_latex & c, unsigned level) const
 123 {
 124         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 125
 126                 // Powers with negative numeric exponents are printed as fractions
 127                 c.s << "\\frac{1}{";
 128                 power(basis, -exponent).eval().print(c);
 129                 c.s << '}';
 130
 131         } else if (exponent.is_equal(_ex1_2)) {
 132
 133                 // Square roots are printed in a special way
 134                 c.s << "\\sqrt{";
 135                 basis.print(c);
 136                 c.s << '}';
 137
 138         } else
 139                 print_power(c, "^", "{", "}", level);
 140 }
 141
 142 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 143 {
 144         // Optimal output of integer powers of symbols to aid compiler CSE.
 145         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 146         // to learn why such a parenthesation is really necessary.
 147         if (exp == 1) {
 148                 x.print(c);
 149         } else if (exp == 2) {
 150                 x.print(c);
 151                 c.s << "*";
 152                 x.print(c);
 153         } else if (exp & 1) {
 154                 x.print(c);
 155                 c.s << "*";
 156                 print_sym_pow(c, x, exp-1);
 157         } else {
 158                 c.s << "(";
 159                 print_sym_pow(c, x, exp >> 1);
 160                 c.s << ")*(";
 161                 print_sym_pow(c, x, exp >> 1);
 162                 c.s << ")";
 163         }
 164 }
 165
 166 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
 167 {
 168         if (exponent.is_equal(_ex_1)) {
 169                 c.s << "recip(";
 170                 basis.print(c);
 171                 c.s << ')';
 172                 return;
 173         }
 174         c.s << "expt(";
 175         basis.print(c);
 176         c.s << ", ";
 177         exponent.print(c);
 178         c.s << ')';
 179 }
 180
 181 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 182 {
 183         // Integer powers of symbols are printed in a special, optimized way
 184         if (exponent.info(info_flags::integer)
 185          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 186                 int exp = ex_to<numeric>(exponent).to_int();
 187                 if (exp > 0)
 188                         c.s << '(';
 189                 else {
 190                         exp = -exp;
 191                         c.s << "1.0/(";
 192                 }
 193                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 194                 c.s << ')';
 195
 196         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 197         } else if (exponent.is_equal(_ex_1)) {
 198                 c.s << "1.0/(";
 199                 basis.print(c);
 200                 c.s << ')';
 201
 202         // Otherwise, use the pow() function
 203         } else {
 204                 c.s << "pow(";
 205                 basis.print(c);
 206                 c.s << ',';
 207                 exponent.print(c);
 208                 c.s << ')';
 209         }
 210 }
 211
 212 void power::do_print_python(const print_python & c, unsigned level) const
 213 {
 214         print_power(c, "**", "", "", level);
 215 }
 216
 217 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 218 {
 219         c.s << class_name() << '(';
 220         basis.print(c);
 221         c.s << ',';
 222         exponent.print(c);
 223         c.s << ')';
 224 }
 225
 226 bool power::info(unsigned inf) const
 227 {
 228         switch (inf) {
 229                 case info_flags::polynomial:
 230                 case info_flags::integer_polynomial:
 231                 case info_flags::cinteger_polynomial:
 232                 case info_flags::rational_polynomial:
 233                 case info_flags::crational_polynomial:
 234                         return exponent.info(info_flags::nonnegint) &&
 235                                basis.info(inf);
 236                 case info_flags::rational_function:
 237                         return exponent.info(info_flags::integer) &&
 238                                basis.info(inf);
 239                 case info_flags::algebraic:
 240                         return !exponent.info(info_flags::integer) ||
 241                                basis.info(inf);
 242                 case info_flags::expanded:
 243                         return (flags & status_flags::expanded);
 244         }
 245         return inherited::info(inf);
 246 }
 247
 248 size_t power::nops() const
 249 {
 250         return 2;
 251 }
 252
 253 ex power::op(size_t i) const
 254 {
 255         GINAC_ASSERT(i<2);
 256
 257         return i==0 ? basis : exponent;
 258 }
 259
 260 ex power::map(map_function & f) const
 261 {
 262         const ex &mapped_basis = f(basis);
 263         const ex &mapped_exponent = f(exponent);
 264
 265         if (!are_ex_trivially_equal(basis, mapped_basis)
 266          || !are_ex_trivially_equal(exponent, mapped_exponent))
 267                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 268         else
 269                 return *this;
 270 }
 271
 272 bool power::is_polynomial(const ex & var) const
 273 {
 274         if (exponent.has(var))
 275                 return false;
 276         if (!exponent.info(info_flags::nonnegint))
 277                 return false;
 278         return basis.is_polynomial(var);
 279 }
 280
 281 int power::degree(const ex & s) const
 282 {
 283         if (is_equal(ex_to<basic>(s)))
 284                 return 1;
 285         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 286                 if (basis.is_equal(s))
 287                         return ex_to<numeric>(exponent).to_int();
 288                 else
 289                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 290         } else if (basis.has(s))
 291                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 292         else
 293                 return 0;
 294 }
 295
 296 int power::ldegree(const ex & s) const
 297 {
 298         if (is_equal(ex_to<basic>(s)))
 299                 return 1;
 300         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 301                 if (basis.is_equal(s))
 302                         return ex_to<numeric>(exponent).to_int();
 303                 else
 304                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 305         } else if (basis.has(s))
 306                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 307         else
 308                 return 0;
 309 }
 310
 311 ex power::coeff(const ex & s, int n) const
 312 {
 313         if (is_equal(ex_to<basic>(s)))
 314                 return n==1 ? _ex1 : _ex0;
 315         else if (!basis.is_equal(s)) {
 316                 // basis not equal to s
 317                 if (n == 0)
 318                         return *this;
 319                 else
 320                         return _ex0;
 321         } else {
 322                 // basis equal to s
 323                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 324                         // integer exponent
 325                         int int_exp = ex_to<numeric>(exponent).to_int();
 326                         if (n == int_exp)
 327                                 return _ex1;
 328                         else
 329                                 return _ex0;
 330                 } else {
 331                         // non-integer exponents are treated as zero
 332                         if (n == 0)
 333                                 return *this;
 334                         else
 335                                 return _ex0;
 336                 }
 337         }
 338 }
 339
 340 /** Perform automatic term rewriting rules in this class.  In the following
 341  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 342  *  stand for such expressions that contain a plain number.
 343  *  - ^(x,0) -> 1  (also handles ^(0,0))
 344  *  - ^(x,1) -> x
 345  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 346  *  - ^(1,x) -> 1
 347  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 348  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 349  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 350  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 351  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 352  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 353  *
 354  *  @param level cut-off in recursive evaluation */
 355 ex power::eval(int level) const
 356 {
 357         if ((level==1) && (flags & status_flags::evaluated))
 358                 return *this;
 359         else if (level == -max_recursion_level)
 360                 throw(std::runtime_error("max recursion level reached"));
 361
 362         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 363         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 364
 365         bool basis_is_numerical = false;
 366         bool exponent_is_numerical = false;
 367         const numeric *num_basis;
 368         const numeric *num_exponent;
 369
 370         if (is_exactly_a<numeric>(ebasis)) {
 371                 basis_is_numerical = true;
 372                 num_basis = &ex_to<numeric>(ebasis);
 373         }
 374         if (is_exactly_a<numeric>(eexponent)) {
 375                 exponent_is_numerical = true;
 376                 num_exponent = &ex_to<numeric>(eexponent);
 377         }
 378
 379         // ^(x,0) -> 1  (0^0 also handled here)
 380         if (eexponent.is_zero()) {
 381                 if (ebasis.is_zero())
 382                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 383                 else
 384                         return _ex1;
 385         }
 386
 387         // ^(x,1) -> x
 388         if (eexponent.is_equal(_ex1))
 389                 return ebasis;
 390
 391         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 392         if (ebasis.is_zero() && exponent_is_numerical) {
 393                 if ((num_exponent->real()).is_zero())
 394                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 395                 else if ((num_exponent->real()).is_negative())
 396                         throw (pole_error("power::eval(): division by zero",1));
 397                 else
 398                         return _ex0;
 399         }
 400
 401         // ^(1,x) -> 1
 402         if (ebasis.is_equal(_ex1))
 403                 return _ex1;
 404
 405         // power of a function calculated by separate rules defined for this function
 406         if (is_exactly_a<function>(ebasis))
 407                 return ex_to<function>(ebasis).power(eexponent);
 408
 409         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 410         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 411                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 412
 413         if (exponent_is_numerical) {
 414
 415                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 416                 // except if c1,c2 are rational, but c1^c2 is not)
 417                 if (basis_is_numerical) {
 418                         const bool basis_is_crational = num_basis->is_crational();
 419                         const bool exponent_is_crational = num_exponent->is_crational();
 420                         if (!basis_is_crational || !exponent_is_crational) {
 421                                 // return a plain float
 422                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 423                                                                                                status_flags::evaluated |
 424                                                                                                status_flags::expanded);
 425                         }
 426
 427                         const numeric res = num_basis->power(*num_exponent);
 428                         if (res.is_crational()) {
 429                                 return res;
 430                         }
 431                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 432
 433                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 434                         if (basis_is_crational && exponent_is_crational
 435                             && num_exponent->is_real()
 436                             && !num_exponent->is_integer()) {
 437                                 const numeric n = num_exponent->numer();
 438                                 const numeric m = num_exponent->denom();
 439                                 numeric r;
 440                                 numeric q = iquo(n, m, r);
 441                                 if (r.is_negative()) {
 442                                         r += m;
 443                                         --q;
 444                                 }
 445                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 446                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 447                                                 // try it for numerator and denominator separately, in order to
 448                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 449                                                 const numeric bnum = num_basis->numer();
 450                                                 const numeric bden = num_basis->denom();
 451                                                 const numeric res_bnum = bnum.power(*num_exponent);
 452                                                 const numeric res_bden = bden.power(*num_exponent);
 453                                                 if (res_bnum.is_integer())
 454                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 455                                                 if (res_bden.is_integer())
 456                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 457                                         }
 458                                         return this->hold();
 459                                 } else {
 460                                         // assemble resulting product, but allowing for a re-evaluation,
 461                                         // because otherwise we'll end up with something like
 462                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 463                                         // instead of 7/16*7^(1/3).
 464                                         ex prod = power(*num_basis,r.div(m));
 465                                         return prod*power(*num_basis,q);
 466                                 }
 467                         }
 468                 }
 469
 470                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 471                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 472                 // case c1==1 should not happen, see below!)
 473                 if (is_exactly_a<power>(ebasis)) {
 474                         const power & sub_power = ex_to<power>(ebasis);
 475                         const ex & sub_basis = sub_power.basis;
 476                         const ex & sub_exponent = sub_power.exponent;
 477                         if (is_exactly_a<numeric>(sub_exponent)) {
 478                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 479                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 480                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
 481                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 482                                 }
 483                         }
 484                 }
 485
 486                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 487                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 488                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 489                 }
 490
 491                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 492                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 493                         numeric icont = ebasis.integer_content();
 494                         const numeric& lead_coeff =
 495                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div_dyn(icont);
 496
 497                         const bool canonicalizable = lead_coeff.is_integer();
 498                         const bool unit_normal = lead_coeff.is_pos_integer();
 499                         if (canonicalizable && (! unit_normal))
 500                                 icont = icont.mul(*_num_1_p);
 501
 502                         if (canonicalizable && (icont != *_num1_p)) {
 503                                 const add& addref = ex_to<add>(ebasis);
 504                                 add* addp = new add(addref);
 505                                 addp->setflag(status_flags::dynallocated);
 506                                 addp->clearflag(status_flags::hash_calculated);
 507                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 508                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 509                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 510
 511                                 const numeric c = icont.power(*num_exponent);
 512                                 if (likely(c != *_num1_p))
 513                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 514                                 else
 515                                         return power(*addp, *num_exponent);
 516                         }
 517                 }
 518
 519                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 520                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 521                 if (is_exactly_a<mul>(ebasis)) {
 522                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 523                         const mul & mulref = ex_to<mul>(ebasis);
 524                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 525                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 526                                 if (num_coeff.is_real()) {
 527                                         if (num_coeff.is_positive()) {
 528                                                 mul *mulp = new mul(mulref);
 529                                                 mulp->overall_coeff = _ex1;
 530                                                 mulp->clearflag(status_flags::evaluated);
 531                                                 mulp->clearflag(status_flags::hash_calculated);
 532                                                 return (new mul(power(*mulp,exponent),
 533                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 534                                         } else {
 535                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 536                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 537                                                         mul *mulp = new mul(mulref);
 538                                                         mulp->overall_coeff = _ex_1;
 539                                                         mulp->clearflag(status_flags::evaluated);
 540                                                         mulp->clearflag(status_flags::hash_calculated);
 541                                                         return (new mul(power(*mulp,exponent),
 542                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 543                                                 }
 544                                         }
 545                                 }
 546                         }
 547                 }
 548
 549                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 550                 if (num_exponent->is_pos_integer() &&
 551                     ebasis.return_type() != return_types::commutative &&
 552                     !is_a<matrix>(ebasis)) {
 553                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 554                 }
 555         }
 556
 557         if (are_ex_trivially_equal(ebasis,basis) &&
 558             are_ex_trivially_equal(eexponent,exponent)) {
 559                 return this->hold();
 560         }
 561         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 562                                                        status_flags::evaluated);
 563 }
 564
 565 ex power::evalf(int level) const
 566 {
 567         ex ebasis;
 568         ex eexponent;
 569
 570         if (level==1) {
 571                 ebasis = basis;
 572                 eexponent = exponent;
 573         } else if (level == -max_recursion_level) {
 574                 throw(std::runtime_error("max recursion level reached"));
 575         } else {
 576                 ebasis = basis.evalf(level-1);
 577                 if (!is_exactly_a<numeric>(exponent))
 578                         eexponent = exponent.evalf(level-1);
 579                 else
 580                         eexponent = exponent;
 581         }
 582
 583         return power(ebasis,eexponent);
 584 }
 585
 586 ex power::evalm() const
 587 {
 588         const ex ebasis = basis.evalm();
 589         const ex eexponent = exponent.evalm();
 590         if (is_a<matrix>(ebasis)) {
 591                 if (is_exactly_a<numeric>(eexponent)) {
 592                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 593                 }
 594         }
 595         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 596 }
 597
 598 bool power::has(const ex & other, unsigned options) const
 599 {
 600         if (!(options & has_options::algebraic))
 601                 return basic::has(other, options);
 602         if (!is_a<power>(other))
 603                 return basic::has(other, options);
 604         if (!exponent.info(info_flags::integer)
 605                         || !other.op(1).info(info_flags::integer))
 606                 return basic::has(other, options);
 607         if (exponent.info(info_flags::posint)
 608                         && other.op(1).info(info_flags::posint)
 609                         && ex_to<numeric>(exponent).to_int()
 610                                         > ex_to<numeric>(other.op(1)).to_int()
 611                         && basis.match(other.op(0)))
 612                 return true;
 613         if (exponent.info(info_flags::negint)
 614                         && other.op(1).info(info_flags::negint)
 615                         && ex_to<numeric>(exponent).to_int()
 616                                         < ex_to<numeric>(other.op(1)).to_int()
 617                         && basis.match(other.op(0)))
 618                 return true;
 619         return basic::has(other, options);
 620 }
 621
 622 // from mul.cpp
 623 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 624
 625 ex power::subs(const exmap & m, unsigned options) const
 626 {
 627         const ex &subsed_basis = basis.subs(m, options);
 628         const ex &subsed_exponent = exponent.subs(m, options);
 629
 630         if (!are_ex_trivially_equal(basis, subsed_basis)
 631          || !are_ex_trivially_equal(exponent, subsed_exponent))
 632                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 633
 634         if (!(options & subs_options::algebraic))
 635                 return subs_one_level(m, options);
 636
 637         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 638                 int nummatches = std::numeric_limits<int>::max();
 639                 lst repls;
 640                 if (tryfactsubs(*this, it->first, nummatches, repls))
 641                         return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
 642         }
 643
 644         return subs_one_level(m, options);
 645 }
 646
 647 ex power::eval_ncmul(const exvector & v) const
 648 {
 649         return inherited::eval_ncmul(v);
 650 }
 651
 652 ex power::conjugate() const
 653 {
 654         ex newbasis = basis.conjugate();
 655         ex newexponent = exponent.conjugate();
 656         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 657                 return *this;
 658         }
 659         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 660 }
 661
 662 ex power::real_part() const
 663 {
 664         if (exponent.info(info_flags::integer)) {
 665                 ex basis_real = basis.real_part();
 666                 if (basis_real == basis)
 667                         return *this;
 668                 realsymbol a("a"),b("b");
 669                 ex result;
 670                 if (exponent.info(info_flags::posint))
 671                         result = power(a+I*b,exponent);
 672                 else
 673                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 674                 result = result.expand();
 675                 result = result.real_part();
 676                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 677                 return result;
 678         }
 679
 680         ex a = basis.real_part();
 681         ex b = basis.imag_part();
 682         ex c = exponent.real_part();
 683         ex d = exponent.imag_part();
 684         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 685 }
 686
 687 ex power::imag_part() const
 688 {
 689         if (exponent.info(info_flags::integer)) {
 690                 ex basis_real = basis.real_part();
 691                 if (basis_real == basis)
 692                         return 0;
 693                 realsymbol a("a"),b("b");
 694                 ex result;
 695                 if (exponent.info(info_flags::posint))
 696                         result = power(a+I*b,exponent);
 697                 else
 698                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 699                 result = result.expand();
 700                 result = result.imag_part();
 701                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 702                 return result;
 703         }
 704
 705         ex a=basis.real_part();
 706         ex b=basis.imag_part();
 707         ex c=exponent.real_part();
 708         ex d=exponent.imag_part();
 709         return
 710                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 711 }
 712
 713 // protected
 714
 715 // protected
 716
 717 /** Implementation of ex::diff() for a power.
 718  *  @see ex::diff */
 719 ex power::derivative(const symbol & s) const
 720 {
 721         if (is_a<numeric>(exponent)) {
 722                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 723                 epvector newseq;
 724                 newseq.reserve(2);
 725                 newseq.push_back(expair(basis, exponent - _ex1));
 726                 newseq.push_back(expair(basis.diff(s), _ex1));
 727                 return mul(newseq, exponent);
 728         } else {
 729                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 730                 return mul(*this,
 731                            add(mul(exponent.diff(s), log(basis)),
 732                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 733         }
 734 }
 735
 736 int power::compare_same_type(const basic & other) const
 737 {
 738         GINAC_ASSERT(is_exactly_a<power>(other));
 739         const power &o = static_cast<const power &>(other);
 740
 741         int cmpval = basis.compare(o.basis);
 742         if (cmpval)
 743                 return cmpval;
 744         else
 745                 return exponent.compare(o.exponent);
 746 }
 747
 748 unsigned power::return_type() const
 749 {
 750         return basis.return_type();
 751 }
 752
 753 tinfo_t power::return_type_tinfo() const
 754 {
 755         return basis.return_type_tinfo();
 756 }
 757
 758 ex power::expand(unsigned options) const
 759 {
 760         if (options == 0 && (flags & status_flags::expanded))
 761                 return *this;
 762
 763         const ex expanded_basis = basis.expand(options);
 764         const ex expanded_exponent = exponent.expand(options);
 765
 766         // x^(a+b) -> x^a * x^b
 767         if (is_exactly_a<add>(expanded_exponent)) {
 768                 const add &a = ex_to<add>(expanded_exponent);
 769                 exvector distrseq;
 770                 distrseq.reserve(a.seq.size() + 1);
 771                 epvector::const_iterator last = a.seq.end();
 772                 epvector::const_iterator cit = a.seq.begin();
 773                 while (cit!=last) {
 774                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 775                         ++cit;
 776                 }
 777
 778                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 779                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 780                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 781                         int int_exponent = num_exponent.to_int();
 782                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 783                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 784                         else
 785                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 786                 } else
 787                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 788
 789                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 790                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 791                 return r.expand(options);
 792         }
 793
 794         if (!is_exactly_a<numeric>(expanded_exponent) ||
 795                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 796                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 797                         return this->hold();
 798                 } else {
 799                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 800                 }
 801         }
 802
 803         // integer numeric exponent
 804         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 805         int int_exponent = num_exponent.to_int();
 806
 807         // (x+y)^n, n>0
 808         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 809                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 810
 811         // (x*y)^n -> x^n * y^n
 812         if (is_exactly_a<mul>(expanded_basis))
 813                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 814
 815         // cannot expand further
 816         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 817                 return this->hold();
 818         else
 819                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 820 }
 821
 822 //////////
 823 // new virtual functions which can be overridden by derived classes
 824 //////////
 825
 826 // none
 827
 828 //////////
 829 // non-virtual functions in this class
 830 //////////
 831
 832 /** expand a^n where a is an add and n is a positive integer.
 833  *  @see power::expand */
 834 ex power::expand_add(const add & a, int n, unsigned options) const
 835 {
 836         if (n==2)
 837                 return expand_add_2(a, options);
 838
 839         const size_t m = a.nops();
 840         exvector result;
 841         // The number of terms will be the number of combinatorial compositions,
 842         // i.e. the number of unordered arrangements of m nonnegative integers
 843         // which sum up to n.  It is frequently written as C_n(m) and directly
 844         // related with binomial coefficients:
 845         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 846         intvector k(m-1);
 847         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 848         intvector upper_limit(m-1);
 849         int l;
 850
 851         for (size_t l=0; l<m-1; ++l) {
 852                 k[l] = 0;
 853                 k_cum[l] = 0;
 854                 upper_limit[l] = n;
 855         }
 856
 857         while (true) {
 858                 exvector term;
 859                 term.reserve(m+1);
 860                 for (l=0; l<m-1; ++l) {
 861                         const ex & b = a.op(l);
 862                         GINAC_ASSERT(!is_exactly_a<add>(b));
 863                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 864                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 865                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 866                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 867                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 868                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 869                         if (is_exactly_a<mul>(b))
 870                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 871                         else
 872                                 term.push_back(power(b,k[l]));
 873                 }
 874
 875                 const ex & b = a.op(l);
 876                 GINAC_ASSERT(!is_exactly_a<add>(b));
 877                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 878                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 879                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 880                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 881                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 882                              !is_exactly_a<power>(ex_to<power>(b).basis));
 883                 if (is_exactly_a<mul>(b))
 884                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 885                 else
 886                         term.push_back(power(b,n-k_cum[m-2]));
 887
 888                 numeric f = binomial(numeric(n),numeric(k[0]));
 889                 for (l=1; l<m-1; ++l)
 890                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 891
 892                 term.push_back(f);
 893
 894                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 895
 896                 // increment k[]
 897                 l = m-2;
 898                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 899                         k[l] = 0;
 900                         --l;
 901                 }
 902                 if (l<0) break;
 903
 904                 // recalc k_cum[] and upper_limit[]
 905                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 906
 907                 for (size_t i=l+1; i<m-1; ++i)
 908                         k_cum[i] = k_cum[i-1]+k[i];
 909
 910                 for (size_t i=l+1; i<m-1; ++i)
 911                         upper_limit[i] = n-k_cum[i-1];
 912         }
 913
 914         return (new add(result))->setflag(status_flags::dynallocated |
 915                                           status_flags::expanded);
 916 }
 917
 918
 919 /** Special case of power::expand_add. Expands a^2 where a is an add.
 920  *  @see power::expand_add */
 921 ex power::expand_add_2(const add & a, unsigned options) const
 922 {
 923         epvector sum;
 924         size_t a_nops = a.nops();
 925         sum.reserve((a_nops*(a_nops+1))/2);
 926         epvector::const_iterator last = a.seq.end();
 927
 928         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 929         // first part: ignore overall_coeff and expand other terms
 930         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 931                 const ex & r = cit0->rest;
 932                 const ex & c = cit0->coeff;
 933
 934                 GINAC_ASSERT(!is_exactly_a<add>(r));
 935                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 936                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 937                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 938                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 939                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 940                              !is_exactly_a<power>(ex_to<power>(r).basis));
 941
 942                 if (c.is_equal(_ex1)) {
 943                         if (is_exactly_a<mul>(r)) {
 944                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 945                                                      _ex1));
 946                         } else {
 947                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 948                                                      _ex1));
 949                         }
 950                 } else {
 951                         if (is_exactly_a<mul>(r)) {
 952                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 953                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 954                         } else {
 955                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 956                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 957                         }
 958                 }
 959
 960                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 961                         const ex & r1 = cit1->rest;
 962                         const ex & c1 = cit1->coeff;
 963                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 964                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 965                 }
 966         }
 967
 968         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 969
 970         // second part: add terms coming from overall_factor (if != 0)
 971         if (!a.overall_coeff.is_zero()) {
 972                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 973                 while (i != end) {
 974                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 975                         ++i;
 976                 }
 977                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 978         }
 979
 980         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 981
 982         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 983 }
 984
 985 /** Expand factors of m in m^n where m is a mul and n is an integer.
 986  *  @see power::expand */
 987 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 988 {
 989         GINAC_ASSERT(n.is_integer());
 990
 991         if (n.is_zero()) {
 992                 return _ex1;
 993         }
 994
 995         // Leave it to multiplication since dummy indices have to be renamed
 996         if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
 997                 ex result = m;
 998                 exvector va = get_all_dummy_indices(m);
 999                 sort(va.begin(), va.end(), ex_is_less());
1000
1001                 for (int i=1; i < n.to_int(); i++)
1002                         result *= rename_dummy_indices_uniquely(va, m);
1003                 return result;
1004         }
1005
1006         epvector distrseq;
1007         distrseq.reserve(m.seq.size());
1008         bool need_reexpand = false;
1009
1010         epvector::const_iterator last = m.seq.end();
1011         epvector::const_iterator cit = m.seq.begin();
1012         while (cit!=last) {
1013                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1014                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1015                         // this happens when e.g. (a+b)^(1/2) gets squared and
1016                         // the resulting product needs to be reexpanded
1017                         need_reexpand = true;
1018                 }
1019                 distrseq.push_back(p);
1020                 ++cit;
1021         }
1022
1023         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1024         if (need_reexpand)
1025                 return ex(result).expand(options);
1026         if (from_expand)
1027                 return result.setflag(status_flags::expanded);
1028         return result;
1029 }
1030
1031 } // namespace GiNaC