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
  45 namespace GiNaC {
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  48   print_func<print_dflt>(&power::do_print_dflt).
  49   print_func<print_latex>(&power::do_print_latex).
  50   print_func<print_csrc>(&power::do_print_csrc).
  51   print_func<print_python>(&power::do_print_python).
  52   print_func<print_python_repr>(&power::do_print_python_repr))
  53
  54 typedef std::vector<int> intvector;
  55
  56 //////////
  57 // default constructor
  58 //////////
  59
  60 power::power() : inherited(&power::tinfo_static) { }
  61
  62 //////////
  63 // other constructors
  64 //////////
  65
  66 // all inlined
  67
  68 //////////
  69 // archiving
  70 //////////
  71
  72 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  73 {
  74         n.find_ex("basis", basis, sym_lst);
  75         n.find_ex("exponent", exponent, sym_lst);
  76 }
  77
  78 void power::archive(archive_node &n) const
  79 {
  80         inherited::archive(n);
  81         n.add_ex("basis", basis);
  82         n.add_ex("exponent", exponent);
  83 }
  84
  85 DEFAULT_UNARCHIVE(power)
  86
  87 //////////
  88 // functions overriding virtual functions from base classes
  89 //////////
  90
  91 // public
  92
  93 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  94 {
  95         // Ordinary output of powers using '^' or '**'
  96         if (precedence() <= level)
  97                 c.s << openbrace << '(';
  98         basis.print(c, precedence());
  99         c.s << powersymbol;
 100         c.s << openbrace;
 101         exponent.print(c, precedence());
 102         c.s << closebrace;
 103         if (precedence() <= level)
 104                 c.s << ')' << closebrace;
 105 }
 106
 107 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 108 {
 109         if (exponent.is_equal(_ex1_2)) {
 110
 111                 // Square roots are printed in a special way
 112                 c.s << "sqrt(";
 113                 basis.print(c);
 114                 c.s << ')';
 115
 116         } else
 117                 print_power(c, "^", "", "", level);
 118 }
 119
 120 void power::do_print_latex(const print_latex & c, unsigned level) const
 121 {
 122         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 123
 124                 // Powers with negative numeric exponents are printed as fractions
 125                 c.s << "\\frac{1}{";
 126                 power(basis, -exponent).eval().print(c);
 127                 c.s << '}';
 128
 129         } else if (exponent.is_equal(_ex1_2)) {
 130
 131                 // Square roots are printed in a special way
 132                 c.s << "\\sqrt{";
 133                 basis.print(c);
 134                 c.s << '}';
 135
 136         } else
 137                 print_power(c, "^", "{", "}", level);
 138 }
 139
 140 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 141 {
 142         // Optimal output of integer powers of symbols to aid compiler CSE.
 143         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 144         // to learn why such a parenthesation is really necessary.
 145         if (exp == 1) {
 146                 x.print(c);
 147         } else if (exp == 2) {
 148                 x.print(c);
 149                 c.s << "*";
 150                 x.print(c);
 151         } else if (exp & 1) {
 152                 x.print(c);
 153                 c.s << "*";
 154                 print_sym_pow(c, x, exp-1);
 155         } else {
 156                 c.s << "(";
 157                 print_sym_pow(c, x, exp >> 1);
 158                 c.s << ")*(";
 159                 print_sym_pow(c, x, exp >> 1);
 160                 c.s << ")";
 161         }
 162 }
 163
 164 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 165 {
 166         // Integer powers of symbols are printed in a special, optimized way
 167         if (exponent.info(info_flags::integer)
 168          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 169                 int exp = ex_to<numeric>(exponent).to_int();
 170                 if (exp > 0)
 171                         c.s << '(';
 172                 else {
 173                         exp = -exp;
 174                         if (is_a<print_csrc_cl_N>(c))
 175                                 c.s << "recip(";
 176                         else
 177                                 c.s << "1.0/(";
 178                 }
 179                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 180                 c.s << ')';
 181
 182         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 183         } else if (exponent.is_equal(_ex_1)) {
 184                 if (is_a<print_csrc_cl_N>(c))
 185                         c.s << "recip(";
 186                 else
 187                         c.s << "1.0/(";
 188                 basis.print(c);
 189                 c.s << ')';
 190
 191         // Otherwise, use the pow() or expt() (CLN) functions
 192         } else {
 193                 if (is_a<print_csrc_cl_N>(c))
 194                         c.s << "expt(";
 195                 else
 196                         c.s << "pow(";
 197                 basis.print(c);
 198                 c.s << ',';
 199                 exponent.print(c);
 200                 c.s << ')';
 201         }
 202 }
 203
 204 void power::do_print_python(const print_python & c, unsigned level) const
 205 {
 206         print_power(c, "**", "", "", level);
 207 }
 208
 209 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 210 {
 211         c.s << class_name() << '(';
 212         basis.print(c);
 213         c.s << ',';
 214         exponent.print(c);
 215         c.s << ')';
 216 }
 217
 218 bool power::info(unsigned inf) const
 219 {
 220         switch (inf) {
 221                 case info_flags::polynomial:
 222                 case info_flags::integer_polynomial:
 223                 case info_flags::cinteger_polynomial:
 224                 case info_flags::rational_polynomial:
 225                 case info_flags::crational_polynomial:
 226                         return exponent.info(info_flags::nonnegint) &&
 227                                basis.info(inf);
 228                 case info_flags::rational_function:
 229                         return exponent.info(info_flags::integer) &&
 230                                basis.info(inf);
 231                 case info_flags::algebraic:
 232                         return !exponent.info(info_flags::integer) ||
 233                                basis.info(inf);
 234         }
 235         return inherited::info(inf);
 236 }
 237
 238 size_t power::nops() const
 239 {
 240         return 2;
 241 }
 242
 243 ex power::op(size_t i) const
 244 {
 245         GINAC_ASSERT(i<2);
 246
 247         return i==0 ? basis : exponent;
 248 }
 249
 250 ex power::map(map_function & f) const
 251 {
 252         const ex &mapped_basis = f(basis);
 253         const ex &mapped_exponent = f(exponent);
 254
 255         if (!are_ex_trivially_equal(basis, mapped_basis)
 256          || !are_ex_trivially_equal(exponent, mapped_exponent))
 257                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 258         else
 259                 return *this;
 260 }
 261
 262 bool power::is_polynomial(const ex & var) const
 263 {
 264         if (exponent.has(var))
 265                 return false;
 266         if (!exponent.info(info_flags::nonnegint))
 267                 return false;
 268         return basis.is_polynomial(var);
 269 }
 270
 271 int power::degree(const ex & s) const
 272 {
 273         if (is_equal(ex_to<basic>(s)))
 274                 return 1;
 275         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 276                 if (basis.is_equal(s))
 277                         return ex_to<numeric>(exponent).to_int();
 278                 else
 279                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 280         } else if (basis.has(s))
 281                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 282         else
 283                 return 0;
 284 }
 285
 286 int power::ldegree(const ex & s) const
 287 {
 288         if (is_equal(ex_to<basic>(s)))
 289                 return 1;
 290         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 291                 if (basis.is_equal(s))
 292                         return ex_to<numeric>(exponent).to_int();
 293                 else
 294                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 295         } else if (basis.has(s))
 296                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 297         else
 298                 return 0;
 299 }
 300
 301 ex power::coeff(const ex & s, int n) const
 302 {
 303         if (is_equal(ex_to<basic>(s)))
 304                 return n==1 ? _ex1 : _ex0;
 305         else if (!basis.is_equal(s)) {
 306                 // basis not equal to s
 307                 if (n == 0)
 308                         return *this;
 309                 else
 310                         return _ex0;
 311         } else {
 312                 // basis equal to s
 313                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 314                         // integer exponent
 315                         int int_exp = ex_to<numeric>(exponent).to_int();
 316                         if (n == int_exp)
 317                                 return _ex1;
 318                         else
 319                                 return _ex0;
 320                 } else {
 321                         // non-integer exponents are treated as zero
 322                         if (n == 0)
 323                                 return *this;
 324                         else
 325                                 return _ex0;
 326                 }
 327         }
 328 }
 329
 330 /** Perform automatic term rewriting rules in this class.  In the following
 331  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 332  *  stand for such expressions that contain a plain number.
 333  *  - ^(x,0) -> 1  (also handles ^(0,0))
 334  *  - ^(x,1) -> x
 335  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 336  *  - ^(1,x) -> 1
 337  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 338  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 339  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 340  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 341  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 342  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 343  *
 344  *  @param level cut-off in recursive evaluation */
 345 ex power::eval(int level) const
 346 {
 347         if ((level==1) && (flags & status_flags::evaluated))
 348                 return *this;
 349         else if (level == -max_recursion_level)
 350                 throw(std::runtime_error("max recursion level reached"));
 351
 352         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 353         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 354
 355         bool basis_is_numerical = false;
 356         bool exponent_is_numerical = false;
 357         const numeric *num_basis;
 358         const numeric *num_exponent;
 359
 360         if (is_exactly_a<numeric>(ebasis)) {
 361                 basis_is_numerical = true;
 362                 num_basis = &ex_to<numeric>(ebasis);
 363         }
 364         if (is_exactly_a<numeric>(eexponent)) {
 365                 exponent_is_numerical = true;
 366                 num_exponent = &ex_to<numeric>(eexponent);
 367         }
 368
 369         // ^(x,0) -> 1  (0^0 also handled here)
 370         if (eexponent.is_zero()) {
 371                 if (ebasis.is_zero())
 372                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 373                 else
 374                         return _ex1;
 375         }
 376
 377         // ^(x,1) -> x
 378         if (eexponent.is_equal(_ex1))
 379                 return ebasis;
 380
 381         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 382         if (ebasis.is_zero() && exponent_is_numerical) {
 383                 if ((num_exponent->real()).is_zero())
 384                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 385                 else if ((num_exponent->real()).is_negative())
 386                         throw (pole_error("power::eval(): division by zero",1));
 387                 else
 388                         return _ex0;
 389         }
 390
 391         // ^(1,x) -> 1
 392         if (ebasis.is_equal(_ex1))
 393                 return _ex1;
 394
 395         // power of a function calculated by separate rules defined for this function
 396         if (is_exactly_a<function>(ebasis))
 397                 return ex_to<function>(ebasis).power(eexponent);
 398
 399         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 400         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 401                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 402
 403         if (exponent_is_numerical) {
 404
 405                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 406                 // except if c1,c2 are rational, but c1^c2 is not)
 407                 if (basis_is_numerical) {
 408                         const bool basis_is_crational = num_basis->is_crational();
 409                         const bool exponent_is_crational = num_exponent->is_crational();
 410                         if (!basis_is_crational || !exponent_is_crational) {
 411                                 // return a plain float
 412                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 413                                                                                                status_flags::evaluated |
 414                                                                                                status_flags::expanded);
 415                         }
 416
 417                         const numeric res = num_basis->power(*num_exponent);
 418                         if (res.is_crational()) {
 419                                 return res;
 420                         }
 421                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 422
 423                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 424                         if (basis_is_crational && exponent_is_crational
 425                             && num_exponent->is_real()
 426                             && !num_exponent->is_integer()) {
 427                                 const numeric n = num_exponent->numer();
 428                                 const numeric m = num_exponent->denom();
 429                                 numeric r;
 430                                 numeric q = iquo(n, m, r);
 431                                 if (r.is_negative()) {
 432                                         r += m;
 433                                         --q;
 434                                 }
 435                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 436                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 437                                                 // try it for numerator and denominator separately, in order to
 438                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 439                                                 const numeric bnum = num_basis->numer();
 440                                                 const numeric bden = num_basis->denom();
 441                                                 const numeric res_bnum = bnum.power(*num_exponent);
 442                                                 const numeric res_bden = bden.power(*num_exponent);
 443                                                 if (res_bnum.is_integer())
 444                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 445                                                 if (res_bden.is_integer())
 446                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 447                                         }
 448                                         return this->hold();
 449                                 } else {
 450                                         // assemble resulting product, but allowing for a re-evaluation,
 451                                         // because otherwise we'll end up with something like
 452                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 453                                         // instead of 7/16*7^(1/3).
 454                                         ex prod = power(*num_basis,r.div(m));
 455                                         return prod*power(*num_basis,q);
 456                                 }
 457                         }
 458                 }
 459
 460                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 461                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 462                 // case c1==1 should not happen, see below!)
 463                 if (is_exactly_a<power>(ebasis)) {
 464                         const power & sub_power = ex_to<power>(ebasis);
 465                         const ex & sub_basis = sub_power.basis;
 466                         const ex & sub_exponent = sub_power.exponent;
 467                         if (is_exactly_a<numeric>(sub_exponent)) {
 468                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 469                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 470                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
 471                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 472                                 }
 473                         }
 474                 }
 475
 476                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 477                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 478                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 479                 }
 480
 481                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 482                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 483                         const numeric icont = ebasis.integer_content();
 484                         const numeric& lead_coeff =
 485                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div_dyn(icont);
 486
 487                         const bool canonicalizable = lead_coeff.is_integer();
 488                         const bool unit_normal = lead_coeff.is_pos_integer();
 489
 490                         if (icont != *_num1_p) {
 491                                 return (new mul(power(ebasis/icont, *num_exponent), power(icont, *num_exponent))
 492                                        )->setflag(status_flags::dynallocated);
 493                         }
 494
 495                         if (canonicalizable && (! unit_normal)) {
 496                                 if (num_exponent->is_even()) {
 497                                         return power(-ebasis, *num_exponent);
 498                                 } else {
 499                                         return (new mul(power(-ebasis, *num_exponent), *_num_1_p)
 500                                                )->setflag(status_flags::dynallocated);
 501                                 }
 502                         }
 503                 }
 504
 505                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 506                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 507                 if (is_exactly_a<mul>(ebasis)) {
 508                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 509                         const mul & mulref = ex_to<mul>(ebasis);
 510                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 511                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 512                                 if (num_coeff.is_real()) {
 513                                         if (num_coeff.is_positive()) {
 514                                                 mul *mulp = new mul(mulref);
 515                                                 mulp->overall_coeff = _ex1;
 516                                                 mulp->clearflag(status_flags::evaluated);
 517                                                 mulp->clearflag(status_flags::hash_calculated);
 518                                                 return (new mul(power(*mulp,exponent),
 519                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 520                                         } else {
 521                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 522                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 523                                                         mul *mulp = new mul(mulref);
 524                                                         mulp->overall_coeff = _ex_1;
 525                                                         mulp->clearflag(status_flags::evaluated);
 526                                                         mulp->clearflag(status_flags::hash_calculated);
 527                                                         return (new mul(power(*mulp,exponent),
 528                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 529                                                 }
 530                                         }
 531                                 }
 532                         }
 533                 }
 534
 535                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 536                 if (num_exponent->is_pos_integer() &&
 537                     ebasis.return_type() != return_types::commutative &&
 538                     !is_a<matrix>(ebasis)) {
 539                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 540                 }
 541         }
 542
 543         if (are_ex_trivially_equal(ebasis,basis) &&
 544             are_ex_trivially_equal(eexponent,exponent)) {
 545                 return this->hold();
 546         }
 547         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 548                                                        status_flags::evaluated);
 549 }
 550
 551 ex power::evalf(int level) const
 552 {
 553         ex ebasis;
 554         ex eexponent;
 555
 556         if (level==1) {
 557                 ebasis = basis;
 558                 eexponent = exponent;
 559         } else if (level == -max_recursion_level) {
 560                 throw(std::runtime_error("max recursion level reached"));
 561         } else {
 562                 ebasis = basis.evalf(level-1);
 563                 if (!is_exactly_a<numeric>(exponent))
 564                         eexponent = exponent.evalf(level-1);
 565                 else
 566                         eexponent = exponent;
 567         }
 568
 569         return power(ebasis,eexponent);
 570 }
 571
 572 ex power::evalm() const
 573 {
 574         const ex ebasis = basis.evalm();
 575         const ex eexponent = exponent.evalm();
 576         if (is_a<matrix>(ebasis)) {
 577                 if (is_exactly_a<numeric>(eexponent)) {
 578                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 579                 }
 580         }
 581         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 582 }
 583
 584 bool power::has(const ex & other, unsigned options) const
 585 {
 586         if (!(options & has_options::algebraic))
 587                 return basic::has(other, options);
 588         if (!is_a<power>(other))
 589                 return basic::has(other, options);
 590         if (!exponent.info(info_flags::integer)
 591                         || !other.op(1).info(info_flags::integer))
 592                 return basic::has(other, options);
 593         if (exponent.info(info_flags::posint)
 594                         && other.op(1).info(info_flags::posint)
 595                         && ex_to<numeric>(exponent).to_int()
 596                                         > ex_to<numeric>(other.op(1)).to_int()
 597                         && basis.match(other.op(0)))
 598                 return true;
 599         if (exponent.info(info_flags::negint)
 600                         && other.op(1).info(info_flags::negint)
 601                         && ex_to<numeric>(exponent).to_int()
 602                                         < ex_to<numeric>(other.op(1)).to_int()
 603                         && basis.match(other.op(0)))
 604                 return true;
 605         return basic::has(other, options);
 606 }
 607
 608 // from mul.cpp
 609 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 610
 611 ex power::subs(const exmap & m, unsigned options) const
 612 {
 613         const ex &subsed_basis = basis.subs(m, options);
 614         const ex &subsed_exponent = exponent.subs(m, options);
 615
 616         if (!are_ex_trivially_equal(basis, subsed_basis)
 617          || !are_ex_trivially_equal(exponent, subsed_exponent))
 618                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 619
 620         if (!(options & subs_options::algebraic))
 621                 return subs_one_level(m, options);
 622
 623         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 624                 int nummatches = std::numeric_limits<int>::max();
 625                 lst repls;
 626                 if (tryfactsubs(*this, it->first, nummatches, repls))
 627                         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);
 628         }
 629
 630         return subs_one_level(m, options);
 631 }
 632
 633 ex power::eval_ncmul(const exvector & v) const
 634 {
 635         return inherited::eval_ncmul(v);
 636 }
 637
 638 ex power::conjugate() const
 639 {
 640         ex newbasis = basis.conjugate();
 641         ex newexponent = exponent.conjugate();
 642         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 643                 return *this;
 644         }
 645         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 646 }
 647
 648 ex power::real_part() const
 649 {
 650         if (exponent.info(info_flags::integer)) {
 651                 ex basis_real = basis.real_part();
 652                 if (basis_real == basis)
 653                         return *this;
 654                 realsymbol a("a"),b("b");
 655                 ex result;
 656                 if (exponent.info(info_flags::posint))
 657                         result = power(a+I*b,exponent);
 658                 else
 659                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 660                 result = result.expand();
 661                 result = result.real_part();
 662                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 663                 return result;
 664         }
 665
 666         ex a = basis.real_part();
 667         ex b = basis.imag_part();
 668         ex c = exponent.real_part();
 669         ex d = exponent.imag_part();
 670         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 671 }
 672
 673 ex power::imag_part() const
 674 {
 675         if (exponent.info(info_flags::integer)) {
 676                 ex basis_real = basis.real_part();
 677                 if (basis_real == basis)
 678                         return 0;
 679                 realsymbol a("a"),b("b");
 680                 ex result;
 681                 if (exponent.info(info_flags::posint))
 682                         result = power(a+I*b,exponent);
 683                 else
 684                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 685                 result = result.expand();
 686                 result = result.imag_part();
 687                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 688                 return result;
 689         }
 690
 691         ex a=basis.real_part();
 692         ex b=basis.imag_part();
 693         ex c=exponent.real_part();
 694         ex d=exponent.imag_part();
 695         return
 696                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 697 }
 698
 699 // protected
 700
 701 // protected
 702
 703 /** Implementation of ex::diff() for a power.
 704  *  @see ex::diff */
 705 ex power::derivative(const symbol & s) const
 706 {
 707         if (is_a<numeric>(exponent)) {
 708                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 709                 epvector newseq;
 710                 newseq.reserve(2);
 711                 newseq.push_back(expair(basis, exponent - _ex1));
 712                 newseq.push_back(expair(basis.diff(s), _ex1));
 713                 return mul(newseq, exponent);
 714         } else {
 715                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 716                 return mul(*this,
 717                            add(mul(exponent.diff(s), log(basis)),
 718                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 719         }
 720 }
 721
 722 int power::compare_same_type(const basic & other) const
 723 {
 724         GINAC_ASSERT(is_exactly_a<power>(other));
 725         const power &o = static_cast<const power &>(other);
 726
 727         int cmpval = basis.compare(o.basis);
 728         if (cmpval)
 729                 return cmpval;
 730         else
 731                 return exponent.compare(o.exponent);
 732 }
 733
 734 unsigned power::return_type() const
 735 {
 736         return basis.return_type();
 737 }
 738
 739 tinfo_t power::return_type_tinfo() const
 740 {
 741         return basis.return_type_tinfo();
 742 }
 743
 744 ex power::expand(unsigned options) const
 745 {
 746         if (options == 0 && (flags & status_flags::expanded))
 747                 return *this;
 748
 749         const ex expanded_basis = basis.expand(options);
 750         const ex expanded_exponent = exponent.expand(options);
 751
 752         // x^(a+b) -> x^a * x^b
 753         if (is_exactly_a<add>(expanded_exponent)) {
 754                 const add &a = ex_to<add>(expanded_exponent);
 755                 exvector distrseq;
 756                 distrseq.reserve(a.seq.size() + 1);
 757                 epvector::const_iterator last = a.seq.end();
 758                 epvector::const_iterator cit = a.seq.begin();
 759                 while (cit!=last) {
 760                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 761                         ++cit;
 762                 }
 763
 764                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 765                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 766                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 767                         int int_exponent = num_exponent.to_int();
 768                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 769                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 770                         else
 771                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 772                 } else
 773                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 774
 775                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 776                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 777                 return r.expand(options);
 778         }
 779
 780         if (!is_exactly_a<numeric>(expanded_exponent) ||
 781                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 782                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 783                         return this->hold();
 784                 } else {
 785                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 786                 }
 787         }
 788
 789         // integer numeric exponent
 790         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 791         int int_exponent = num_exponent.to_int();
 792
 793         // (x+y)^n, n>0
 794         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 795                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 796
 797         // (x*y)^n -> x^n * y^n
 798         if (is_exactly_a<mul>(expanded_basis))
 799                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 800
 801         // cannot expand further
 802         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 803                 return this->hold();
 804         else
 805                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 806 }
 807
 808 //////////
 809 // new virtual functions which can be overridden by derived classes
 810 //////////
 811
 812 // none
 813
 814 //////////
 815 // non-virtual functions in this class
 816 //////////
 817
 818 /** expand a^n where a is an add and n is a positive integer.
 819  *  @see power::expand */
 820 ex power::expand_add(const add & a, int n, unsigned options) const
 821 {
 822         if (n==2)
 823                 return expand_add_2(a, options);
 824
 825         const size_t m = a.nops();
 826         exvector result;
 827         // The number of terms will be the number of combinatorial compositions,
 828         // i.e. the number of unordered arrangements of m nonnegative integers
 829         // which sum up to n.  It is frequently written as C_n(m) and directly
 830         // related with binomial coefficients:
 831         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 832         intvector k(m-1);
 833         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 834         intvector upper_limit(m-1);
 835         int l;
 836
 837         for (size_t l=0; l<m-1; ++l) {
 838                 k[l] = 0;
 839                 k_cum[l] = 0;
 840                 upper_limit[l] = n;
 841         }
 842
 843         while (true) {
 844                 exvector term;
 845                 term.reserve(m+1);
 846                 for (l=0; l<m-1; ++l) {
 847                         const ex & b = a.op(l);
 848                         GINAC_ASSERT(!is_exactly_a<add>(b));
 849                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 850                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 851                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 852                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 853                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 854                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 855                         if (is_exactly_a<mul>(b))
 856                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 857                         else
 858                                 term.push_back(power(b,k[l]));
 859                 }
 860
 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(n-k_cum[m-2]), options, true));
 871                 else
 872                         term.push_back(power(b,n-k_cum[m-2]));
 873
 874                 numeric f = binomial(numeric(n),numeric(k[0]));
 875                 for (l=1; l<m-1; ++l)
 876                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 877
 878                 term.push_back(f);
 879
 880                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 881
 882                 // increment k[]
 883                 l = m-2;
 884                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 885                         k[l] = 0;
 886                         --l;
 887                 }
 888                 if (l<0) break;
 889
 890                 // recalc k_cum[] and upper_limit[]
 891                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 892
 893                 for (size_t i=l+1; i<m-1; ++i)
 894                         k_cum[i] = k_cum[i-1]+k[i];
 895
 896                 for (size_t i=l+1; i<m-1; ++i)
 897                         upper_limit[i] = n-k_cum[i-1];
 898         }
 899
 900         return (new add(result))->setflag(status_flags::dynallocated |
 901                                           status_flags::expanded);
 902 }
 903
 904
 905 /** Special case of power::expand_add. Expands a^2 where a is an add.
 906  *  @see power::expand_add */
 907 ex power::expand_add_2(const add & a, unsigned options) const
 908 {
 909         epvector sum;
 910         size_t a_nops = a.nops();
 911         sum.reserve((a_nops*(a_nops+1))/2);
 912         epvector::const_iterator last = a.seq.end();
 913
 914         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 915         // first part: ignore overall_coeff and expand other terms
 916         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 917                 const ex & r = cit0->rest;
 918                 const ex & c = cit0->coeff;
 919
 920                 GINAC_ASSERT(!is_exactly_a<add>(r));
 921                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 922                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 923                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 924                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 925                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 926                              !is_exactly_a<power>(ex_to<power>(r).basis));
 927
 928                 if (c.is_equal(_ex1)) {
 929                         if (is_exactly_a<mul>(r)) {
 930                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 931                                                      _ex1));
 932                         } else {
 933                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 934                                                      _ex1));
 935                         }
 936                 } else {
 937                         if (is_exactly_a<mul>(r)) {
 938                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 939                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 940                         } else {
 941                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 942                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 943                         }
 944                 }
 945
 946                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 947                         const ex & r1 = cit1->rest;
 948                         const ex & c1 = cit1->coeff;
 949                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 950                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 951                 }
 952         }
 953
 954         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 955
 956         // second part: add terms coming from overall_factor (if != 0)
 957         if (!a.overall_coeff.is_zero()) {
 958                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 959                 while (i != end) {
 960                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 961                         ++i;
 962                 }
 963                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 964         }
 965
 966         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 967
 968         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 969 }
 970
 971 /** Expand factors of m in m^n where m is a mul and n is an integer.
 972  *  @see power::expand */
 973 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 974 {
 975         GINAC_ASSERT(n.is_integer());
 976
 977         if (n.is_zero()) {
 978                 return _ex1;
 979         }
 980
 981         // Leave it to multiplication since dummy indices have to be renamed
 982         if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
 983                 ex result = m;
 984                 exvector va = get_all_dummy_indices(m);
 985                 sort(va.begin(), va.end(), ex_is_less());
 986
 987                 for (int i=1; i < n.to_int(); i++)
 988                         result *= rename_dummy_indices_uniquely(va, m);
 989                 return result;
 990         }
 991
 992         epvector distrseq;
 993         distrseq.reserve(m.seq.size());
 994         bool need_reexpand = false;
 995
 996         epvector::const_iterator last = m.seq.end();
 997         epvector::const_iterator cit = m.seq.begin();
 998         while (cit!=last) {
 999                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1000                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1001                         // this happens when e.g. (a+b)^(1/2) gets squared and
1002                         // the resulting product needs to be reexpanded
1003                         need_reexpand = true;
1004                 }
1005                 distrseq.push_back(p);
1006                 ++cit;
1007         }
1008
1009         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1010         if (need_reexpand)
1011                 return ex(result).expand(options);
1012         if (from_expand)
1013                 return result.setflag(status_flags::expanded);
1014         return result;
1015 }
1016
1017 } // namespace GiNaC