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 int power::degree(const ex & s) const
 262 {
 263         if (is_equal(ex_to<basic>(s)))
 264                 return 1;
 265         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 266                 if (basis.is_equal(s))
 267                         return ex_to<numeric>(exponent).to_int();
 268                 else
 269                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 270         } else if (basis.has(s))
 271                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 272         else
 273                 return 0;
 274 }
 275
 276 int power::ldegree(const ex & s) const
 277 {
 278         if (is_equal(ex_to<basic>(s)))
 279                 return 1;
 280         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 281                 if (basis.is_equal(s))
 282                         return ex_to<numeric>(exponent).to_int();
 283                 else
 284                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 285         } else if (basis.has(s))
 286                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 287         else
 288                 return 0;
 289 }
 290
 291 ex power::coeff(const ex & s, int n) const
 292 {
 293         if (is_equal(ex_to<basic>(s)))
 294                 return n==1 ? _ex1 : _ex0;
 295         else if (!basis.is_equal(s)) {
 296                 // basis not equal to s
 297                 if (n == 0)
 298                         return *this;
 299                 else
 300                         return _ex0;
 301         } else {
 302                 // basis equal to s
 303                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 304                         // integer exponent
 305                         int int_exp = ex_to<numeric>(exponent).to_int();
 306                         if (n == int_exp)
 307                                 return _ex1;
 308                         else
 309                                 return _ex0;
 310                 } else {
 311                         // non-integer exponents are treated as zero
 312                         if (n == 0)
 313                                 return *this;
 314                         else
 315                                 return _ex0;
 316                 }
 317         }
 318 }
 319
 320 /** Perform automatic term rewriting rules in this class.  In the following
 321  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 322  *  stand for such expressions that contain a plain number.
 323  *  - ^(x,0) -> 1  (also handles ^(0,0))
 324  *  - ^(x,1) -> x
 325  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 326  *  - ^(1,x) -> 1
 327  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 328  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 329  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 330  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 331  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 332  *
 333  *  @param level cut-off in recursive evaluation */
 334 ex power::eval(int level) const
 335 {
 336         if ((level==1) && (flags & status_flags::evaluated))
 337                 return *this;
 338         else if (level == -max_recursion_level)
 339                 throw(std::runtime_error("max recursion level reached"));
 340
 341         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 342         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 343
 344         bool basis_is_numerical = false;
 345         bool exponent_is_numerical = false;
 346         const numeric *num_basis;
 347         const numeric *num_exponent;
 348
 349         if (is_exactly_a<numeric>(ebasis)) {
 350                 basis_is_numerical = true;
 351                 num_basis = &ex_to<numeric>(ebasis);
 352         }
 353         if (is_exactly_a<numeric>(eexponent)) {
 354                 exponent_is_numerical = true;
 355                 num_exponent = &ex_to<numeric>(eexponent);
 356         }
 357
 358         // ^(x,0) -> 1  (0^0 also handled here)
 359         if (eexponent.is_zero()) {
 360                 if (ebasis.is_zero())
 361                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 362                 else
 363                         return _ex1;
 364         }
 365
 366         // ^(x,1) -> x
 367         if (eexponent.is_equal(_ex1))
 368                 return ebasis;
 369
 370         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 371         if (ebasis.is_zero() && exponent_is_numerical) {
 372                 if ((num_exponent->real()).is_zero())
 373                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 374                 else if ((num_exponent->real()).is_negative())
 375                         throw (pole_error("power::eval(): division by zero",1));
 376                 else
 377                         return _ex0;
 378         }
 379
 380         // ^(1,x) -> 1
 381         if (ebasis.is_equal(_ex1))
 382                 return _ex1;
 383
 384         // power of a function calculated by separate rules defined for this function
 385         if (is_exactly_a<function>(ebasis))
 386                 return ex_to<function>(ebasis).power(eexponent);
 387
 388         if (exponent_is_numerical) {
 389
 390                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 391                 // except if c1,c2 are rational, but c1^c2 is not)
 392                 if (basis_is_numerical) {
 393                         const bool basis_is_crational = num_basis->is_crational();
 394                         const bool exponent_is_crational = num_exponent->is_crational();
 395                         if (!basis_is_crational || !exponent_is_crational) {
 396                                 // return a plain float
 397                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 398                                                                                                status_flags::evaluated |
 399                                                                                                status_flags::expanded);
 400                         }
 401
 402                         const numeric res = num_basis->power(*num_exponent);
 403                         if (res.is_crational()) {
 404                                 return res;
 405                         }
 406                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 407
 408                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 409                         if (basis_is_crational && exponent_is_crational
 410                             && num_exponent->is_real()
 411                             && !num_exponent->is_integer()) {
 412                                 const numeric n = num_exponent->numer();
 413                                 const numeric m = num_exponent->denom();
 414                                 numeric r;
 415                                 numeric q = iquo(n, m, r);
 416                                 if (r.is_negative()) {
 417                                         r += m;
 418                                         --q;
 419                                 }
 420                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 421                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 422                                                 // try it for numerator and denominator separately, in order to
 423                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 424                                                 const numeric bnum = num_basis->numer();
 425                                                 const numeric bden = num_basis->denom();
 426                                                 const numeric res_bnum = bnum.power(*num_exponent);
 427                                                 const numeric res_bden = bden.power(*num_exponent);
 428                                                 if (res_bnum.is_integer())
 429                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 430                                                 if (res_bden.is_integer())
 431                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 432                                         }
 433                                         return this->hold();
 434                                 } else {
 435                                         // assemble resulting product, but allowing for a re-evaluation,
 436                                         // because otherwise we'll end up with something like
 437                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 438                                         // instead of 7/16*7^(1/3).
 439                                         ex prod = power(*num_basis,r.div(m));
 440                                         return prod*power(*num_basis,q);
 441                                 }
 442                         }
 443                 }
 444
 445                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 446                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 447                 // case c1==1 should not happen, see below!)
 448                 if (is_exactly_a<power>(ebasis)) {
 449                         const power & sub_power = ex_to<power>(ebasis);
 450                         const ex & sub_basis = sub_power.basis;
 451                         const ex & sub_exponent = sub_power.exponent;
 452                         if (is_exactly_a<numeric>(sub_exponent)) {
 453                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 454                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 455                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative())
 456                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 457                         }
 458                 }
 459
 460                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 461                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 462                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 463                 }
 464
 465                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 466                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 467                 if (is_exactly_a<mul>(ebasis)) {
 468                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 469                         const mul & mulref = ex_to<mul>(ebasis);
 470                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 471                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 472                                 if (num_coeff.is_real()) {
 473                                         if (num_coeff.is_positive()) {
 474                                                 mul *mulp = new mul(mulref);
 475                                                 mulp->overall_coeff = _ex1;
 476                                                 mulp->clearflag(status_flags::evaluated);
 477                                                 mulp->clearflag(status_flags::hash_calculated);
 478                                                 return (new mul(power(*mulp,exponent),
 479                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 480                                         } else {
 481                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 482                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 483                                                         mul *mulp = new mul(mulref);
 484                                                         mulp->overall_coeff = _ex_1;
 485                                                         mulp->clearflag(status_flags::evaluated);
 486                                                         mulp->clearflag(status_flags::hash_calculated);
 487                                                         return (new mul(power(*mulp,exponent),
 488                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 489                                                 }
 490                                         }
 491                                 }
 492                         }
 493                 }
 494
 495                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 496                 if (num_exponent->is_pos_integer() &&
 497                     ebasis.return_type() != return_types::commutative &&
 498                     !is_a<matrix>(ebasis)) {
 499                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 500                 }
 501         }
 502
 503         if (are_ex_trivially_equal(ebasis,basis) &&
 504             are_ex_trivially_equal(eexponent,exponent)) {
 505                 return this->hold();
 506         }
 507         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 508                                                        status_flags::evaluated);
 509 }
 510
 511 ex power::evalf(int level) const
 512 {
 513         ex ebasis;
 514         ex eexponent;
 515
 516         if (level==1) {
 517                 ebasis = basis;
 518                 eexponent = exponent;
 519         } else if (level == -max_recursion_level) {
 520                 throw(std::runtime_error("max recursion level reached"));
 521         } else {
 522                 ebasis = basis.evalf(level-1);
 523                 if (!is_exactly_a<numeric>(exponent))
 524                         eexponent = exponent.evalf(level-1);
 525                 else
 526                         eexponent = exponent;
 527         }
 528
 529         return power(ebasis,eexponent);
 530 }
 531
 532 ex power::evalm() const
 533 {
 534         const ex ebasis = basis.evalm();
 535         const ex eexponent = exponent.evalm();
 536         if (is_a<matrix>(ebasis)) {
 537                 if (is_exactly_a<numeric>(eexponent)) {
 538                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 539                 }
 540         }
 541         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 542 }
 543
 544 bool power::has(const ex & other, unsigned options) const
 545 {
 546         if (!(options & has_options::algebraic))
 547                 return basic::has(other, options);
 548         if (!is_a<power>(other))
 549                 return basic::has(other, options);
 550         if (!exponent.info(info_flags::integer)
 551                         || !other.op(1).info(info_flags::integer))
 552                 return basic::has(other, options);
 553         if (exponent.info(info_flags::posint)
 554                         && other.op(1).info(info_flags::posint)
 555                         && ex_to<numeric>(exponent).to_int()
 556                                         > ex_to<numeric>(other.op(1)).to_int()
 557                         && basis.match(other.op(0)))
 558                 return true;
 559         if (exponent.info(info_flags::negint)
 560                         && other.op(1).info(info_flags::negint)
 561                         && ex_to<numeric>(exponent).to_int()
 562                                         < ex_to<numeric>(other.op(1)).to_int()
 563                         && basis.match(other.op(0)))
 564                 return true;
 565         return basic::has(other, options);
 566 }
 567
 568 // from mul.cpp
 569 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 570
 571 ex power::subs(const exmap & m, unsigned options) const
 572 {
 573         const ex &subsed_basis = basis.subs(m, options);
 574         const ex &subsed_exponent = exponent.subs(m, options);
 575
 576         if (!are_ex_trivially_equal(basis, subsed_basis)
 577          || !are_ex_trivially_equal(exponent, subsed_exponent))
 578                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 579
 580         if (!(options & subs_options::algebraic))
 581                 return subs_one_level(m, options);
 582
 583         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 584                 int nummatches = std::numeric_limits<int>::max();
 585                 lst repls;
 586                 if (tryfactsubs(*this, it->first, nummatches, repls))
 587                         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);
 588         }
 589
 590         return subs_one_level(m, options);
 591 }
 592
 593 ex power::eval_ncmul(const exvector & v) const
 594 {
 595         return inherited::eval_ncmul(v);
 596 }
 597
 598 ex power::conjugate() const
 599 {
 600         ex newbasis = basis.conjugate();
 601         ex newexponent = exponent.conjugate();
 602         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 603                 return *this;
 604         }
 605         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 606 }
 607
 608 // protected
 609
 610 /** Implementation of ex::diff() for a power.
 611  *  @see ex::diff */
 612 ex power::derivative(const symbol & s) const
 613 {
 614         if (exponent.info(info_flags::real)) {
 615                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 616                 epvector newseq;
 617                 newseq.reserve(2);
 618                 newseq.push_back(expair(basis, exponent - _ex1));
 619                 newseq.push_back(expair(basis.diff(s), _ex1));
 620                 return mul(newseq, exponent);
 621         } else {
 622                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 623                 return mul(*this,
 624                            add(mul(exponent.diff(s), log(basis)),
 625                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 626         }
 627 }
 628
 629 int power::compare_same_type(const basic & other) const
 630 {
 631         GINAC_ASSERT(is_exactly_a<power>(other));
 632         const power &o = static_cast<const power &>(other);
 633
 634         int cmpval = basis.compare(o.basis);
 635         if (cmpval)
 636                 return cmpval;
 637         else
 638                 return exponent.compare(o.exponent);
 639 }
 640
 641 unsigned power::return_type() const
 642 {
 643         return basis.return_type();
 644 }
 645
 646 tinfo_t power::return_type_tinfo() const
 647 {
 648         return basis.return_type_tinfo();
 649 }
 650
 651 ex power::expand(unsigned options) const
 652 {
 653         if (options == 0 && (flags & status_flags::expanded))
 654                 return *this;
 655
 656         const ex expanded_basis = basis.expand(options);
 657         const ex expanded_exponent = exponent.expand(options);
 658
 659         // x^(a+b) -> x^a * x^b
 660         if (is_exactly_a<add>(expanded_exponent)) {
 661                 const add &a = ex_to<add>(expanded_exponent);
 662                 exvector distrseq;
 663                 distrseq.reserve(a.seq.size() + 1);
 664                 epvector::const_iterator last = a.seq.end();
 665                 epvector::const_iterator cit = a.seq.begin();
 666                 while (cit!=last) {
 667                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 668                         ++cit;
 669                 }
 670
 671                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 672                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 673                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 674                         int int_exponent = num_exponent.to_int();
 675                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 676                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 677                         else
 678                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 679                 } else
 680                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 681
 682                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 683                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 684                 return r.expand(options);
 685         }
 686
 687         if (!is_exactly_a<numeric>(expanded_exponent) ||
 688                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 689                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 690                         return this->hold();
 691                 } else {
 692                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 693                 }
 694         }
 695
 696         // integer numeric exponent
 697         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 698         int int_exponent = num_exponent.to_int();
 699
 700         // (x+y)^n, n>0
 701         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 702                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 703
 704         // (x*y)^n -> x^n * y^n
 705         if (is_exactly_a<mul>(expanded_basis))
 706                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 707
 708         // cannot expand further
 709         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 710                 return this->hold();
 711         else
 712                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 713 }
 714
 715 //////////
 716 // new virtual functions which can be overridden by derived classes
 717 //////////
 718
 719 // none
 720
 721 //////////
 722 // non-virtual functions in this class
 723 //////////
 724
 725 /** expand a^n where a is an add and n is a positive integer.
 726  *  @see power::expand */
 727 ex power::expand_add(const add & a, int n, unsigned options) const
 728 {
 729         if (n==2)
 730                 return expand_add_2(a, options);
 731
 732         const size_t m = a.nops();
 733         exvector result;
 734         // The number of terms will be the number of combinatorial compositions,
 735         // i.e. the number of unordered arrangements of m nonnegative integers
 736         // which sum up to n.  It is frequently written as C_n(m) and directly
 737         // related with binomial coefficients:
 738         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 739         intvector k(m-1);
 740         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 741         intvector upper_limit(m-1);
 742         int l;
 743
 744         for (size_t l=0; l<m-1; ++l) {
 745                 k[l] = 0;
 746                 k_cum[l] = 0;
 747                 upper_limit[l] = n;
 748         }
 749
 750         while (true) {
 751                 exvector term;
 752                 term.reserve(m+1);
 753                 for (l=0; l<m-1; ++l) {
 754                         const ex & b = a.op(l);
 755                         GINAC_ASSERT(!is_exactly_a<add>(b));
 756                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 757                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 758                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 759                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 760                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 761                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 762                         if (is_exactly_a<mul>(b))
 763                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 764                         else
 765                                 term.push_back(power(b,k[l]));
 766                 }
 767
 768                 const ex & b = a.op(l);
 769                 GINAC_ASSERT(!is_exactly_a<add>(b));
 770                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 771                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 772                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 773                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 774                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 775                              !is_exactly_a<power>(ex_to<power>(b).basis));
 776                 if (is_exactly_a<mul>(b))
 777                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 778                 else
 779                         term.push_back(power(b,n-k_cum[m-2]));
 780
 781                 numeric f = binomial(numeric(n),numeric(k[0]));
 782                 for (l=1; l<m-1; ++l)
 783                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 784
 785                 term.push_back(f);
 786
 787                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 788
 789                 // increment k[]
 790                 l = m-2;
 791                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 792                         k[l] = 0;
 793                         --l;
 794                 }
 795                 if (l<0) break;
 796
 797                 // recalc k_cum[] and upper_limit[]
 798                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 799
 800                 for (size_t i=l+1; i<m-1; ++i)
 801                         k_cum[i] = k_cum[i-1]+k[i];
 802
 803                 for (size_t i=l+1; i<m-1; ++i)
 804                         upper_limit[i] = n-k_cum[i-1];
 805         }
 806
 807         return (new add(result))->setflag(status_flags::dynallocated |
 808                                           status_flags::expanded);
 809 }
 810
 811
 812 /** Special case of power::expand_add. Expands a^2 where a is an add.
 813  *  @see power::expand_add */
 814 ex power::expand_add_2(const add & a, unsigned options) const
 815 {
 816         epvector sum;
 817         size_t a_nops = a.nops();
 818         sum.reserve((a_nops*(a_nops+1))/2);
 819         epvector::const_iterator last = a.seq.end();
 820
 821         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 822         // first part: ignore overall_coeff and expand other terms
 823         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 824                 const ex & r = cit0->rest;
 825                 const ex & c = cit0->coeff;
 826
 827                 GINAC_ASSERT(!is_exactly_a<add>(r));
 828                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 829                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 830                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 831                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 832                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 833                              !is_exactly_a<power>(ex_to<power>(r).basis));
 834
 835                 if (c.is_equal(_ex1)) {
 836                         if (is_exactly_a<mul>(r)) {
 837                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 838                                                      _ex1));
 839                         } else {
 840                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 841                                                      _ex1));
 842                         }
 843                 } else {
 844                         if (is_exactly_a<mul>(r)) {
 845                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 846                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 847                         } else {
 848                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 849                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 850                         }
 851                 }
 852
 853                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 854                         const ex & r1 = cit1->rest;
 855                         const ex & c1 = cit1->coeff;
 856                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 857                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 858                 }
 859         }
 860
 861         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 862
 863         // second part: add terms coming from overall_factor (if != 0)
 864         if (!a.overall_coeff.is_zero()) {
 865                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 866                 while (i != end) {
 867                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 868                         ++i;
 869                 }
 870                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 871         }
 872
 873         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 874
 875         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 876 }
 877
 878 /** Expand factors of m in m^n where m is a mul and n is and integer.
 879  *  @see power::expand */
 880 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 881 {
 882         GINAC_ASSERT(n.is_integer());
 883
 884         if (n.is_zero()) {
 885                 return _ex1;
 886         }
 887
 888         // Leave it to multiplication since dummy indices have to be renamed
 889         if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
 890                 ex result = m;
 891                 exvector va = get_all_dummy_indices(m);
 892                 sort(va.begin(), va.end(), ex_is_less());
 893
 894                 for (int i=1; i < n.to_int(); i++)
 895                         result *= rename_dummy_indices_uniquely(va, m);
 896                 return result;
 897         }
 898
 899         epvector distrseq;
 900         distrseq.reserve(m.seq.size());
 901         bool need_reexpand = false;
 902
 903         epvector::const_iterator last = m.seq.end();
 904         epvector::const_iterator cit = m.seq.begin();
 905         while (cit!=last) {
 906                 if (is_exactly_a<numeric>(cit->rest)) {
 907                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
 908                 } else {
 909                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 910                         // since n is an integer
 911                         numeric new_coeff = ex_to<numeric>(cit->coeff).mul(n);
 912                         if (from_expand && is_exactly_a<add>(cit->rest) && new_coeff.is_pos_integer()) {
 913                                 // this happens when e.g. (a+b)^(1/2) gets squared and
 914                                 // the resulting product needs to be reexpanded
 915                                 need_reexpand = true;
 916                         }
 917                         distrseq.push_back(expair(cit->rest, new_coeff));
 918                 }
 919                 ++cit;
 920         }
 921
 922         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
 923         if (need_reexpand)
 924                 return ex(result).expand(options);
 925         if (from_expand)
 926                 return result.setflag(status_flags::expanded);
 927         return result;
 928 }
 929
 930 } // namespace GiNaC