ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2006 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
  44 namespace GiNaC {
  45
  46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  47   print_func<print_dflt>(&power::do_print_dflt).
  48   print_func<print_latex>(&power::do_print_latex).
  49   print_func<print_csrc>(&power::do_print_csrc).
  50   print_func<print_python>(&power::do_print_python).
  51   print_func<print_python_repr>(&power::do_print_python_repr))
  52
  53 typedef std::vector<int> intvector;
  54
  55 //////////
  56 // default constructor
  57 //////////
  58
  59 power::power() : inherited(&power::tinfo_static) { }
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 // all inlined
  66
  67 //////////
  68 // archiving
  69 //////////
  70
  71 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  72 {
  73         n.find_ex("basis", basis, sym_lst);
  74         n.find_ex("exponent", exponent, sym_lst);
  75 }
  76
  77 void power::archive(archive_node &n) const
  78 {
  79         inherited::archive(n);
  80         n.add_ex("basis", basis);
  81         n.add_ex("exponent", exponent);
  82 }
  83
  84 DEFAULT_UNARCHIVE(power)
  85
  86 //////////
  87 // functions overriding virtual functions from base classes
  88 //////////
  89
  90 // public
  91
  92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  93 {
  94         // Ordinary output of powers using '^' or '**'
  95         if (precedence() <= level)
  96                 c.s << openbrace << '(';
  97         basis.print(c, precedence());
  98         c.s << powersymbol;
  99         c.s << openbrace;
 100         exponent.print(c, precedence());
 101         c.s << closebrace;
 102         if (precedence() <= level)
 103                 c.s << ')' << closebrace;
 104 }
 105
 106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 107 {
 108         if (exponent.is_equal(_ex1_2)) {
 109
 110                 // Square roots are printed in a special way
 111                 c.s << "sqrt(";
 112                 basis.print(c);
 113                 c.s << ')';
 114
 115         } else
 116                 print_power(c, "^", "", "", level);
 117 }
 118
 119 void power::do_print_latex(const print_latex & c, unsigned level) const
 120 {
 121         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 122
 123                 // Powers with negative numeric exponents are printed as fractions
 124                 c.s << "\\frac{1}{";
 125                 power(basis, -exponent).eval().print(c);
 126                 c.s << '}';
 127
 128         } else if (exponent.is_equal(_ex1_2)) {
 129
 130                 // Square roots are printed in a special way
 131                 c.s << "\\sqrt{";
 132                 basis.print(c);
 133                 c.s << '}';
 134
 135         } else
 136                 print_power(c, "^", "{", "}", level);
 137 }
 138
 139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 140 {
 141         // Optimal output of integer powers of symbols to aid compiler CSE.
 142         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 143         // to learn why such a parenthesation is really necessary.
 144         if (exp == 1) {
 145                 x.print(c);
 146         } else if (exp == 2) {
 147                 x.print(c);
 148                 c.s << "*";
 149                 x.print(c);
 150         } else if (exp & 1) {
 151                 x.print(c);
 152                 c.s << "*";
 153                 print_sym_pow(c, x, exp-1);
 154         } else {
 155                 c.s << "(";
 156                 print_sym_pow(c, x, exp >> 1);
 157                 c.s << ")*(";
 158                 print_sym_pow(c, x, exp >> 1);
 159                 c.s << ")";
 160         }
 161 }
 162
 163 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 164 {
 165         // Integer powers of symbols are printed in a special, optimized way
 166         if (exponent.info(info_flags::integer)
 167          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 168                 int exp = ex_to<numeric>(exponent).to_int();
 169                 if (exp > 0)
 170                         c.s << '(';
 171                 else {
 172                         exp = -exp;
 173                         if (is_a<print_csrc_cl_N>(c))
 174                                 c.s << "recip(";
 175                         else
 176                                 c.s << "1.0/(";
 177                 }
 178                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 179                 c.s << ')';
 180
 181         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 182         } else if (exponent.is_equal(_ex_1)) {
 183                 if (is_a<print_csrc_cl_N>(c))
 184                         c.s << "recip(";
 185                 else
 186                         c.s << "1.0/(";
 187                 basis.print(c);
 188                 c.s << ')';
 189
 190         // Otherwise, use the pow() or expt() (CLN) functions
 191         } else {
 192                 if (is_a<print_csrc_cl_N>(c))
 193                         c.s << "expt(";
 194                 else
 195                         c.s << "pow(";
 196                 basis.print(c);
 197                 c.s << ',';
 198                 exponent.print(c);
 199                 c.s << ')';
 200         }
 201 }
 202
 203 void power::do_print_python(const print_python & c, unsigned level) const
 204 {
 205         print_power(c, "**", "", "", level);
 206 }
 207
 208 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 209 {
 210         c.s << class_name() << '(';
 211         basis.print(c);
 212         c.s << ',';
 213         exponent.print(c);
 214         c.s << ')';
 215 }
 216
 217 bool power::info(unsigned inf) const
 218 {
 219         switch (inf) {
 220                 case info_flags::polynomial:
 221                 case info_flags::integer_polynomial:
 222                 case info_flags::cinteger_polynomial:
 223                 case info_flags::rational_polynomial:
 224                 case info_flags::crational_polynomial:
 225                         return exponent.info(info_flags::nonnegint) &&
 226                                basis.info(inf);
 227                 case info_flags::rational_function:
 228                         return exponent.info(info_flags::integer) &&
 229                                basis.info(inf);
 230                 case info_flags::algebraic:
 231                         return !exponent.info(info_flags::integer) ||
 232                                basis.info(inf);
 233         }
 234         return inherited::info(inf);
 235 }
 236
 237 size_t power::nops() const
 238 {
 239         return 2;
 240 }
 241
 242 ex power::op(size_t i) const
 243 {
 244         GINAC_ASSERT(i<2);
 245
 246         return i==0 ? basis : exponent;
 247 }
 248
 249 ex power::map(map_function & f) const
 250 {
 251         const ex &mapped_basis = f(basis);
 252         const ex &mapped_exponent = f(exponent);
 253
 254         if (!are_ex_trivially_equal(basis, mapped_basis)
 255          || !are_ex_trivially_equal(exponent, mapped_exponent))
 256                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 257         else
 258                 return *this;
 259 }
 260
 261 bool power::is_polynomial(const ex & var) const
 262 {
 263         if (exponent.has(var))
 264                 return false;
 265         if (!exponent.info(info_flags::nonnegint))
 266                 return false;
 267         return basis.is_polynomial(var);
 268 }
 269
 270 int power::degree(const ex & s) const
 271 {
 272         if (is_equal(ex_to<basic>(s)))
 273                 return 1;
 274         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 275                 if (basis.is_equal(s))
 276                         return ex_to<numeric>(exponent).to_int();
 277                 else
 278                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 279         } else if (basis.has(s))
 280                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 281         else
 282                 return 0;
 283 }
 284
 285 int power::ldegree(const ex & s) const
 286 {
 287         if (is_equal(ex_to<basic>(s)))
 288                 return 1;
 289         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 290                 if (basis.is_equal(s))
 291                         return ex_to<numeric>(exponent).to_int();
 292                 else
 293                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 294         } else if (basis.has(s))
 295                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 296         else
 297                 return 0;
 298 }
 299
 300 ex power::coeff(const ex & s, int n) const
 301 {
 302         if (is_equal(ex_to<basic>(s)))
 303                 return n==1 ? _ex1 : _ex0;
 304         else if (!basis.is_equal(s)) {
 305                 // basis not equal to s
 306                 if (n == 0)
 307                         return *this;
 308                 else
 309                         return _ex0;
 310         } else {
 311                 // basis equal to s
 312                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 313                         // integer exponent
 314                         int int_exp = ex_to<numeric>(exponent).to_int();
 315                         if (n == int_exp)
 316                                 return _ex1;
 317                         else
 318                                 return _ex0;
 319                 } else {
 320                         // non-integer exponents are treated as zero
 321                         if (n == 0)
 322                                 return *this;
 323                         else
 324                                 return _ex0;
 325                 }
 326         }
 327 }
 328
 329 /** Perform automatic term rewriting rules in this class.  In the following
 330  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 331  *  stand for such expressions that contain a plain number.
 332  *  - ^(x,0) -> 1  (also handles ^(0,0))
 333  *  - ^(x,1) -> x
 334  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 335  *  - ^(1,x) -> 1
 336  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 337  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 338  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 339  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 340  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 341  *
 342  *  @param level cut-off in recursive evaluation */
 343 ex power::eval(int level) const
 344 {
 345         if ((level==1) && (flags & status_flags::evaluated))
 346                 return *this;
 347         else if (level == -max_recursion_level)
 348                 throw(std::runtime_error("max recursion level reached"));
 349
 350         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 351         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 352
 353         bool basis_is_numerical = false;
 354         bool exponent_is_numerical = false;
 355         const numeric *num_basis;
 356         const numeric *num_exponent;
 357
 358         if (is_exactly_a<numeric>(ebasis)) {
 359                 basis_is_numerical = true;
 360                 num_basis = &ex_to<numeric>(ebasis);
 361         }
 362         if (is_exactly_a<numeric>(eexponent)) {
 363                 exponent_is_numerical = true;
 364                 num_exponent = &ex_to<numeric>(eexponent);
 365         }
 366
 367         // ^(x,0) -> 1  (0^0 also handled here)
 368         if (eexponent.is_zero()) {
 369                 if (ebasis.is_zero())
 370                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 371                 else
 372                         return _ex1;
 373         }
 374
 375         // ^(x,1) -> x
 376         if (eexponent.is_equal(_ex1))
 377                 return ebasis;
 378
 379         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 380         if (ebasis.is_zero() && exponent_is_numerical) {
 381                 if ((num_exponent->real()).is_zero())
 382                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 383                 else if ((num_exponent->real()).is_negative())
 384                         throw (pole_error("power::eval(): division by zero",1));
 385                 else
 386                         return _ex0;
 387         }
 388
 389         // ^(1,x) -> 1
 390         if (ebasis.is_equal(_ex1))
 391                 return _ex1;
 392
 393         // power of a function calculated by separate rules defined for this function
 394         if (is_exactly_a<function>(ebasis))
 395                 return ex_to<function>(ebasis).power(eexponent);
 396
 397         if (exponent_is_numerical) {
 398
 399                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 400                 // except if c1,c2 are rational, but c1^c2 is not)
 401                 if (basis_is_numerical) {
 402                         const bool basis_is_crational = num_basis->is_crational();
 403                         const bool exponent_is_crational = num_exponent->is_crational();
 404                         if (!basis_is_crational || !exponent_is_crational) {
 405                                 // return a plain float
 406                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 407                                                                                                status_flags::evaluated |
 408                                                                                                status_flags::expanded);
 409                         }
 410
 411                         const numeric res = num_basis->power(*num_exponent);
 412                         if (res.is_crational()) {
 413                                 return res;
 414                         }
 415                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 416
 417                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 418                         if (basis_is_crational && exponent_is_crational
 419                             && num_exponent->is_real()
 420                             && !num_exponent->is_integer()) {
 421                                 const numeric n = num_exponent->numer();
 422                                 const numeric m = num_exponent->denom();
 423                                 numeric r;
 424                                 numeric q = iquo(n, m, r);
 425                                 if (r.is_negative()) {
 426                                         r += m;
 427                                         --q;
 428                                 }
 429                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 430                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 431                                                 // try it for numerator and denominator separately, in order to
 432                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 433                                                 const numeric bnum = num_basis->numer();
 434                                                 const numeric bden = num_basis->denom();
 435                                                 const numeric res_bnum = bnum.power(*num_exponent);
 436                                                 const numeric res_bden = bden.power(*num_exponent);
 437                                                 if (res_bnum.is_integer())
 438                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 439                                                 if (res_bden.is_integer())
 440                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 441                                         }
 442                                         return this->hold();
 443                                 } else {
 444                                         // assemble resulting product, but allowing for a re-evaluation,
 445                                         // because otherwise we'll end up with something like
 446                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 447                                         // instead of 7/16*7^(1/3).
 448                                         ex prod = power(*num_basis,r.div(m));
 449                                         return prod*power(*num_basis,q);
 450                                 }
 451                         }
 452                 }
 453
 454                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 455                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 456                 // case c1==1 should not happen, see below!)
 457                 if (is_exactly_a<power>(ebasis)) {
 458                         const power & sub_power = ex_to<power>(ebasis);
 459                         const ex & sub_basis = sub_power.basis;
 460                         const ex & sub_exponent = sub_power.exponent;
 461                         if (is_exactly_a<numeric>(sub_exponent)) {
 462                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 463                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 464                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative())
 465                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 466                         }
 467                 }
 468
 469                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 470                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 471                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 472                 }
 473
 474                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 475                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 476                 if (is_exactly_a<mul>(ebasis)) {
 477                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 478                         const mul & mulref = ex_to<mul>(ebasis);
 479                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 480                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 481                                 if (num_coeff.is_real()) {
 482                                         if (num_coeff.is_positive()) {
 483                                                 mul *mulp = new mul(mulref);
 484                                                 mulp->overall_coeff = _ex1;
 485                                                 mulp->clearflag(status_flags::evaluated);
 486                                                 mulp->clearflag(status_flags::hash_calculated);
 487                                                 return (new mul(power(*mulp,exponent),
 488                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 489                                         } else {
 490                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 491                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 492                                                         mul *mulp = new mul(mulref);
 493                                                         mulp->overall_coeff = _ex_1;
 494                                                         mulp->clearflag(status_flags::evaluated);
 495                                                         mulp->clearflag(status_flags::hash_calculated);
 496                                                         return (new mul(power(*mulp,exponent),
 497                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 498                                                 }
 499                                         }
 500                                 }
 501                         }
 502                 }
 503
 504                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 505                 if (num_exponent->is_pos_integer() &&
 506                     ebasis.return_type() != return_types::commutative &&
 507                     !is_a<matrix>(ebasis)) {
 508                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 509                 }
 510         }
 511
 512         if (are_ex_trivially_equal(ebasis,basis) &&
 513             are_ex_trivially_equal(eexponent,exponent)) {
 514                 return this->hold();
 515         }
 516         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 517                                                        status_flags::evaluated);
 518 }
 519
 520 ex power::evalf(int level) const
 521 {
 522         ex ebasis;
 523         ex eexponent;
 524
 525         if (level==1) {
 526                 ebasis = basis;
 527                 eexponent = exponent;
 528         } else if (level == -max_recursion_level) {
 529                 throw(std::runtime_error("max recursion level reached"));
 530         } else {
 531                 ebasis = basis.evalf(level-1);
 532                 if (!is_exactly_a<numeric>(exponent))
 533                         eexponent = exponent.evalf(level-1);
 534                 else
 535                         eexponent = exponent;
 536         }
 537
 538         return power(ebasis,eexponent);
 539 }
 540
 541 ex power::evalm() const
 542 {
 543         const ex ebasis = basis.evalm();
 544         const ex eexponent = exponent.evalm();
 545         if (is_a<matrix>(ebasis)) {
 546                 if (is_exactly_a<numeric>(eexponent)) {
 547                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 548                 }
 549         }
 550         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 551 }
 552
 553 bool power::has(const ex & other, unsigned options) const
 554 {
 555         if (!(options & has_options::algebraic))
 556                 return basic::has(other, options);
 557         if (!is_a<power>(other))
 558                 return basic::has(other, options);
 559         if (!exponent.info(info_flags::integer)
 560                         || !other.op(1).info(info_flags::integer))
 561                 return basic::has(other, options);
 562         if (exponent.info(info_flags::posint)
 563                         && other.op(1).info(info_flags::posint)
 564                         && ex_to<numeric>(exponent).to_int()
 565                                         > ex_to<numeric>(other.op(1)).to_int()
 566                         && basis.match(other.op(0)))
 567                 return true;
 568         if (exponent.info(info_flags::negint)
 569                         && other.op(1).info(info_flags::negint)
 570                         && ex_to<numeric>(exponent).to_int()
 571                                         < ex_to<numeric>(other.op(1)).to_int()
 572                         && basis.match(other.op(0)))
 573                 return true;
 574         return basic::has(other, options);
 575 }
 576
 577 // from mul.cpp
 578 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 579
 580 ex power::subs(const exmap & m, unsigned options) const
 581 {
 582         const ex &subsed_basis = basis.subs(m, options);
 583         const ex &subsed_exponent = exponent.subs(m, options);
 584
 585         if (!are_ex_trivially_equal(basis, subsed_basis)
 586          || !are_ex_trivially_equal(exponent, subsed_exponent))
 587                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 588
 589         if (!(options & subs_options::algebraic))
 590                 return subs_one_level(m, options);
 591
 592         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 593                 int nummatches = std::numeric_limits<int>::max();
 594                 lst repls;
 595                 if (tryfactsubs(*this, it->first, nummatches, repls))
 596                         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);
 597         }
 598
 599         return subs_one_level(m, options);
 600 }
 601
 602 ex power::eval_ncmul(const exvector & v) const
 603 {
 604         return inherited::eval_ncmul(v);
 605 }
 606
 607 ex power::conjugate() const
 608 {
 609         ex newbasis = basis.conjugate();
 610         ex newexponent = exponent.conjugate();
 611         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 612                 return *this;
 613         }
 614         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 615 }
 616
 617 // protected
 618
 619 /** Implementation of ex::diff() for a power.
 620  *  @see ex::diff */
 621 ex power::derivative(const symbol & s) const
 622 {
 623         if (exponent.info(info_flags::real)) {
 624                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 625                 epvector newseq;
 626                 newseq.reserve(2);
 627                 newseq.push_back(expair(basis, exponent - _ex1));
 628                 newseq.push_back(expair(basis.diff(s), _ex1));
 629                 return mul(newseq, exponent);
 630         } else {
 631                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 632                 return mul(*this,
 633                            add(mul(exponent.diff(s), log(basis)),
 634                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 635         }
 636 }
 637
 638 int power::compare_same_type(const basic & other) const
 639 {
 640         GINAC_ASSERT(is_exactly_a<power>(other));
 641         const power &o = static_cast<const power &>(other);
 642
 643         int cmpval = basis.compare(o.basis);
 644         if (cmpval)
 645                 return cmpval;
 646         else
 647                 return exponent.compare(o.exponent);
 648 }
 649
 650 unsigned power::return_type() const
 651 {
 652         return basis.return_type();
 653 }
 654
 655 tinfo_t power::return_type_tinfo() const
 656 {
 657         return basis.return_type_tinfo();
 658 }
 659
 660 ex power::expand(unsigned options) const
 661 {
 662         if (options == 0 && (flags & status_flags::expanded))
 663                 return *this;
 664
 665         const ex expanded_basis = basis.expand(options);
 666         const ex expanded_exponent = exponent.expand(options);
 667
 668         // x^(a+b) -> x^a * x^b
 669         if (is_exactly_a<add>(expanded_exponent)) {
 670                 const add &a = ex_to<add>(expanded_exponent);
 671                 exvector distrseq;
 672                 distrseq.reserve(a.seq.size() + 1);
 673                 epvector::const_iterator last = a.seq.end();
 674                 epvector::const_iterator cit = a.seq.begin();
 675                 while (cit!=last) {
 676                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 677                         ++cit;
 678                 }
 679
 680                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 681                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 682                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 683                         int int_exponent = num_exponent.to_int();
 684                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 685                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 686                         else
 687                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 688                 } else
 689                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 690
 691                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 692                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 693                 return r.expand(options);
 694         }
 695
 696         if (!is_exactly_a<numeric>(expanded_exponent) ||
 697                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 698                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 699                         return this->hold();
 700                 } else {
 701                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 702                 }
 703         }
 704
 705         // integer numeric exponent
 706         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 707         int int_exponent = num_exponent.to_int();
 708
 709         // (x+y)^n, n>0
 710         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 711                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 712
 713         // (x*y)^n -> x^n * y^n
 714         if (is_exactly_a<mul>(expanded_basis))
 715                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 716
 717         // cannot expand further
 718         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 719                 return this->hold();
 720         else
 721                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 722 }
 723
 724 //////////
 725 // new virtual functions which can be overridden by derived classes
 726 //////////
 727
 728 // none
 729
 730 //////////
 731 // non-virtual functions in this class
 732 //////////
 733
 734 /** expand a^n where a is an add and n is a positive integer.
 735  *  @see power::expand */
 736 ex power::expand_add(const add & a, int n, unsigned options) const
 737 {
 738         if (n==2)
 739                 return expand_add_2(a, options);
 740
 741         const size_t m = a.nops();
 742         exvector result;
 743         // The number of terms will be the number of combinatorial compositions,
 744         // i.e. the number of unordered arrangements of m nonnegative integers
 745         // which sum up to n.  It is frequently written as C_n(m) and directly
 746         // related with binomial coefficients:
 747         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 748         intvector k(m-1);
 749         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 750         intvector upper_limit(m-1);
 751         int l;
 752
 753         for (size_t l=0; l<m-1; ++l) {
 754                 k[l] = 0;
 755                 k_cum[l] = 0;
 756                 upper_limit[l] = n;
 757         }
 758
 759         while (true) {
 760                 exvector term;
 761                 term.reserve(m+1);
 762                 for (l=0; l<m-1; ++l) {
 763                         const ex & b = a.op(l);
 764                         GINAC_ASSERT(!is_exactly_a<add>(b));
 765                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 766                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 767                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 768                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 769                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 770                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 771                         if (is_exactly_a<mul>(b))
 772                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 773                         else
 774                                 term.push_back(power(b,k[l]));
 775                 }
 776
 777                 const ex & b = a.op(l);
 778                 GINAC_ASSERT(!is_exactly_a<add>(b));
 779                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 780                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 781                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 782                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 783                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 784                              !is_exactly_a<power>(ex_to<power>(b).basis));
 785                 if (is_exactly_a<mul>(b))
 786                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 787                 else
 788                         term.push_back(power(b,n-k_cum[m-2]));
 789
 790                 numeric f = binomial(numeric(n),numeric(k[0]));
 791                 for (l=1; l<m-1; ++l)
 792                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 793
 794                 term.push_back(f);
 795
 796                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 797
 798                 // increment k[]
 799                 l = m-2;
 800                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 801                         k[l] = 0;
 802                         --l;
 803                 }
 804                 if (l<0) break;
 805
 806                 // recalc k_cum[] and upper_limit[]
 807                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 808
 809                 for (size_t i=l+1; i<m-1; ++i)
 810                         k_cum[i] = k_cum[i-1]+k[i];
 811
 812                 for (size_t i=l+1; i<m-1; ++i)
 813                         upper_limit[i] = n-k_cum[i-1];
 814         }
 815
 816         return (new add(result))->setflag(status_flags::dynallocated |
 817                                           status_flags::expanded);
 818 }
 819
 820
 821 /** Special case of power::expand_add. Expands a^2 where a is an add.
 822  *  @see power::expand_add */
 823 ex power::expand_add_2(const add & a, unsigned options) const
 824 {
 825         epvector sum;
 826         size_t a_nops = a.nops();
 827         sum.reserve((a_nops*(a_nops+1))/2);
 828         epvector::const_iterator last = a.seq.end();
 829
 830         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 831         // first part: ignore overall_coeff and expand other terms
 832         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 833                 const ex & r = cit0->rest;
 834                 const ex & c = cit0->coeff;
 835
 836                 GINAC_ASSERT(!is_exactly_a<add>(r));
 837                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 838                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 839                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 840                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 841                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 842                              !is_exactly_a<power>(ex_to<power>(r).basis));
 843
 844                 if (c.is_equal(_ex1)) {
 845                         if (is_exactly_a<mul>(r)) {
 846                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 847                                                      _ex1));
 848                         } else {
 849                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 850                                                      _ex1));
 851                         }
 852                 } else {
 853                         if (is_exactly_a<mul>(r)) {
 854                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 855                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 856                         } else {
 857                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 858                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 859                         }
 860                 }
 861
 862                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 863                         const ex & r1 = cit1->rest;
 864                         const ex & c1 = cit1->coeff;
 865                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 866                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 867                 }
 868         }
 869
 870         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 871
 872         // second part: add terms coming from overall_factor (if != 0)
 873         if (!a.overall_coeff.is_zero()) {
 874                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 875                 while (i != end) {
 876                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 877                         ++i;
 878                 }
 879                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 880         }
 881
 882         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 883
 884         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 885 }
 886
 887 /** Expand factors of m in m^n where m is a mul and n is and integer.
 888  *  @see power::expand */
 889 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 890 {
 891         GINAC_ASSERT(n.is_integer());
 892
 893         if (n.is_zero()) {
 894                 return _ex1;
 895         }
 896
 897         // Leave it to multiplication since dummy indices have to be renamed
 898         if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
 899                 ex result = m;
 900                 exvector va = get_all_dummy_indices(m);
 901                 sort(va.begin(), va.end(), ex_is_less());
 902
 903                 for (int i=1; i < n.to_int(); i++)
 904                         result *= rename_dummy_indices_uniquely(va, m);
 905                 return result;
 906         }
 907
 908         epvector distrseq;
 909         distrseq.reserve(m.seq.size());
 910         bool need_reexpand = false;
 911
 912         epvector::const_iterator last = m.seq.end();
 913         epvector::const_iterator cit = m.seq.begin();
 914         while (cit!=last) {
 915                 if (is_exactly_a<numeric>(cit->rest)) {
 916                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
 917                 } else {
 918                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 919                         // since n is an integer
 920                         numeric new_coeff = ex_to<numeric>(cit->coeff).mul(n);
 921                         if (from_expand && is_exactly_a<add>(cit->rest) && new_coeff.is_pos_integer()) {
 922                                 // this happens when e.g. (a+b)^(1/2) gets squared and
 923                                 // the resulting product needs to be reexpanded
 924                                 need_reexpand = true;
 925                         }
 926                         distrseq.push_back(expair(cit->rest, new_coeff));
 927                 }
 928                 ++cit;
 929         }
 930
 931         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
 932         if (need_reexpand)
 933                 return ex(result).expand(options);
 934         if (from_expand)
 935                 return result.setflag(status_flags::expanded);
 936         return result;
 937 }
 938
 939 } // namespace GiNaC