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 #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)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 339  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 340  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 341  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 342  *
 343  *  @param level cut-off in recursive evaluation */
 344 ex power::eval(int level) const
 345 {
 346         if ((level==1) && (flags & status_flags::evaluated))
 347                 return *this;
 348         else if (level == -max_recursion_level)
 349                 throw(std::runtime_error("max recursion level reached"));
 350
 351         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 352         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 353
 354         bool basis_is_numerical = false;
 355         bool exponent_is_numerical = false;
 356         const numeric *num_basis;
 357         const numeric *num_exponent;
 358
 359         if (is_exactly_a<numeric>(ebasis)) {
 360                 basis_is_numerical = true;
 361                 num_basis = &ex_to<numeric>(ebasis);
 362         }
 363         if (is_exactly_a<numeric>(eexponent)) {
 364                 exponent_is_numerical = true;
 365                 num_exponent = &ex_to<numeric>(eexponent);
 366         }
 367
 368         // ^(x,0) -> 1  (0^0 also handled here)
 369         if (eexponent.is_zero()) {
 370                 if (ebasis.is_zero())
 371                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 372                 else
 373                         return _ex1;
 374         }
 375
 376         // ^(x,1) -> x
 377         if (eexponent.is_equal(_ex1))
 378                 return ebasis;
 379
 380         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 381         if (ebasis.is_zero() && exponent_is_numerical) {
 382                 if ((num_exponent->real()).is_zero())
 383                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 384                 else if ((num_exponent->real()).is_negative())
 385                         throw (pole_error("power::eval(): division by zero",1));
 386                 else
 387                         return _ex0;
 388         }
 389
 390         // ^(1,x) -> 1
 391         if (ebasis.is_equal(_ex1))
 392                 return _ex1;
 393
 394         // power of a function calculated by separate rules defined for this function
 395         if (is_exactly_a<function>(ebasis))
 396                 return ex_to<function>(ebasis).power(eexponent);
 397
 398         if (exponent_is_numerical) {
 399
 400                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 401                 // except if c1,c2 are rational, but c1^c2 is not)
 402                 if (basis_is_numerical) {
 403                         const bool basis_is_crational = num_basis->is_crational();
 404                         const bool exponent_is_crational = num_exponent->is_crational();
 405                         if (!basis_is_crational || !exponent_is_crational) {
 406                                 // return a plain float
 407                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 408                                                                                                status_flags::evaluated |
 409                                                                                                status_flags::expanded);
 410                         }
 411
 412                         const numeric res = num_basis->power(*num_exponent);
 413                         if (res.is_crational()) {
 414                                 return res;
 415                         }
 416                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 417
 418                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 419                         if (basis_is_crational && exponent_is_crational
 420                             && num_exponent->is_real()
 421                             && !num_exponent->is_integer()) {
 422                                 const numeric n = num_exponent->numer();
 423                                 const numeric m = num_exponent->denom();
 424                                 numeric r;
 425                                 numeric q = iquo(n, m, r);
 426                                 if (r.is_negative()) {
 427                                         r += m;
 428                                         --q;
 429                                 }
 430                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 431                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 432                                                 // try it for numerator and denominator separately, in order to
 433                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 434                                                 const numeric bnum = num_basis->numer();
 435                                                 const numeric bden = num_basis->denom();
 436                                                 const numeric res_bnum = bnum.power(*num_exponent);
 437                                                 const numeric res_bden = bden.power(*num_exponent);
 438                                                 if (res_bnum.is_integer())
 439                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 440                                                 if (res_bden.is_integer())
 441                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 442                                         }
 443                                         return this->hold();
 444                                 } else {
 445                                         // assemble resulting product, but allowing for a re-evaluation,
 446                                         // because otherwise we'll end up with something like
 447                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 448                                         // instead of 7/16*7^(1/3).
 449                                         ex prod = power(*num_basis,r.div(m));
 450                                         return prod*power(*num_basis,q);
 451                                 }
 452                         }
 453                 }
 454
 455                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 456                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 457                 // case c1==1 should not happen, see below!)
 458                 if (is_exactly_a<power>(ebasis)) {
 459                         const power & sub_power = ex_to<power>(ebasis);
 460                         const ex & sub_basis = sub_power.basis;
 461                         const ex & sub_exponent = sub_power.exponent;
 462                         if (is_exactly_a<numeric>(sub_exponent)) {
 463                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 464                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 465                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative())
 466                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 467                         }
 468                 }
 469
 470                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 471                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 472                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 473                 }
 474
 475                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 476                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 477                 if (is_exactly_a<mul>(ebasis)) {
 478                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 479                         const mul & mulref = ex_to<mul>(ebasis);
 480                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 481                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 482                                 if (num_coeff.is_real()) {
 483                                         if (num_coeff.is_positive()) {
 484                                                 mul *mulp = new mul(mulref);
 485                                                 mulp->overall_coeff = _ex1;
 486                                                 mulp->clearflag(status_flags::evaluated);
 487                                                 mulp->clearflag(status_flags::hash_calculated);
 488                                                 return (new mul(power(*mulp,exponent),
 489                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 490                                         } else {
 491                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 492                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 493                                                         mul *mulp = new mul(mulref);
 494                                                         mulp->overall_coeff = _ex_1;
 495                                                         mulp->clearflag(status_flags::evaluated);
 496                                                         mulp->clearflag(status_flags::hash_calculated);
 497                                                         return (new mul(power(*mulp,exponent),
 498                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 499                                                 }
 500                                         }
 501                                 }
 502                         }
 503                 }
 504
 505                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 506                 if (num_exponent->is_pos_integer() &&
 507                     ebasis.return_type() != return_types::commutative &&
 508                     !is_a<matrix>(ebasis)) {
 509                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 510                 }
 511         }
 512
 513         if (are_ex_trivially_equal(ebasis,basis) &&
 514             are_ex_trivially_equal(eexponent,exponent)) {
 515                 return this->hold();
 516         }
 517         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 518                                                        status_flags::evaluated);
 519 }
 520
 521 ex power::evalf(int level) const
 522 {
 523         ex ebasis;
 524         ex eexponent;
 525
 526         if (level==1) {
 527                 ebasis = basis;
 528                 eexponent = exponent;
 529         } else if (level == -max_recursion_level) {
 530                 throw(std::runtime_error("max recursion level reached"));
 531         } else {
 532                 ebasis = basis.evalf(level-1);
 533                 if (!is_exactly_a<numeric>(exponent))
 534                         eexponent = exponent.evalf(level-1);
 535                 else
 536                         eexponent = exponent;
 537         }
 538
 539         return power(ebasis,eexponent);
 540 }
 541
 542 ex power::evalm() const
 543 {
 544         const ex ebasis = basis.evalm();
 545         const ex eexponent = exponent.evalm();
 546         if (is_a<matrix>(ebasis)) {
 547                 if (is_exactly_a<numeric>(eexponent)) {
 548                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 549                 }
 550         }
 551         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 552 }
 553
 554 bool power::has(const ex & other, unsigned options) const
 555 {
 556         if (!(options & has_options::algebraic))
 557                 return basic::has(other, options);
 558         if (!is_a<power>(other))
 559                 return basic::has(other, options);
 560         if (!exponent.info(info_flags::integer)
 561                         || !other.op(1).info(info_flags::integer))
 562                 return basic::has(other, options);
 563         if (exponent.info(info_flags::posint)
 564                         && other.op(1).info(info_flags::posint)
 565                         && ex_to<numeric>(exponent).to_int()
 566                                         > ex_to<numeric>(other.op(1)).to_int()
 567                         && basis.match(other.op(0)))
 568                 return true;
 569         if (exponent.info(info_flags::negint)
 570                         && other.op(1).info(info_flags::negint)
 571                         && ex_to<numeric>(exponent).to_int()
 572                                         < ex_to<numeric>(other.op(1)).to_int()
 573                         && basis.match(other.op(0)))
 574                 return true;
 575         return basic::has(other, options);
 576 }
 577
 578 // from mul.cpp
 579 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 580
 581 ex power::subs(const exmap & m, unsigned options) const
 582 {
 583         const ex &subsed_basis = basis.subs(m, options);
 584         const ex &subsed_exponent = exponent.subs(m, options);
 585
 586         if (!are_ex_trivially_equal(basis, subsed_basis)
 587          || !are_ex_trivially_equal(exponent, subsed_exponent))
 588                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 589
 590         if (!(options & subs_options::algebraic))
 591                 return subs_one_level(m, options);
 592
 593         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 594                 int nummatches = std::numeric_limits<int>::max();
 595                 lst repls;
 596                 if (tryfactsubs(*this, it->first, nummatches, repls))
 597                         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);
 598         }
 599
 600         return subs_one_level(m, options);
 601 }
 602
 603 ex power::eval_ncmul(const exvector & v) const
 604 {
 605         return inherited::eval_ncmul(v);
 606 }
 607
 608 ex power::conjugate() const
 609 {
 610         ex newbasis = basis.conjugate();
 611         ex newexponent = exponent.conjugate();
 612         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 613                 return *this;
 614         }
 615         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 616 }
 617
 618 ex power::real_part() const
 619 {
 620         if (exponent.info(info_flags::integer)) {
 621                 ex basis_real = basis.real_part();
 622                 if (basis_real == basis)
 623                         return *this;
 624                 realsymbol a("a"),b("b");
 625                 ex result;
 626                 if (exponent.info(info_flags::posint))
 627                         result = power(a+I*b,exponent);
 628                 else
 629                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 630                 result = result.expand();
 631                 result = result.real_part();
 632                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 633                 return result;
 634         }
 635
 636         ex a = basis.real_part();
 637         ex b = basis.imag_part();
 638         ex c = exponent.real_part();
 639         ex d = exponent.imag_part();
 640         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 641 }
 642
 643 ex power::imag_part() const
 644 {
 645         if (exponent.info(info_flags::integer)) {
 646                 ex basis_real = basis.real_part();
 647                 if (basis_real == basis)
 648                         return 0;
 649                 realsymbol a("a"),b("b");
 650                 ex result;
 651                 if (exponent.info(info_flags::posint))
 652                         result = power(a+I*b,exponent);
 653                 else
 654                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 655                 result = result.expand();
 656                 result = result.imag_part();
 657                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 658                 return result;
 659         }
 660
 661         ex a=basis.real_part();
 662         ex b=basis.imag_part();
 663         ex c=exponent.real_part();
 664         ex d=exponent.imag_part();
 665         return
 666                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 667 }
 668
 669 // protected
 670
 671 // protected
 672
 673 /** Implementation of ex::diff() for a power.
 674  *  @see ex::diff */
 675 ex power::derivative(const symbol & s) const
 676 {
 677         if (is_a<numeric>(exponent)) {
 678                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 679                 epvector newseq;
 680                 newseq.reserve(2);
 681                 newseq.push_back(expair(basis, exponent - _ex1));
 682                 newseq.push_back(expair(basis.diff(s), _ex1));
 683                 return mul(newseq, exponent);
 684         } else {
 685                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 686                 return mul(*this,
 687                            add(mul(exponent.diff(s), log(basis)),
 688                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 689         }
 690 }
 691
 692 int power::compare_same_type(const basic & other) const
 693 {
 694         GINAC_ASSERT(is_exactly_a<power>(other));
 695         const power &o = static_cast<const power &>(other);
 696
 697         int cmpval = basis.compare(o.basis);
 698         if (cmpval)
 699                 return cmpval;
 700         else
 701                 return exponent.compare(o.exponent);
 702 }
 703
 704 unsigned power::return_type() const
 705 {
 706         return basis.return_type();
 707 }
 708
 709 tinfo_t power::return_type_tinfo() const
 710 {
 711         return basis.return_type_tinfo();
 712 }
 713
 714 ex power::expand(unsigned options) const
 715 {
 716         if (options == 0 && (flags & status_flags::expanded))
 717                 return *this;
 718
 719         const ex expanded_basis = basis.expand(options);
 720         const ex expanded_exponent = exponent.expand(options);
 721
 722         // x^(a+b) -> x^a * x^b
 723         if (is_exactly_a<add>(expanded_exponent)) {
 724                 const add &a = ex_to<add>(expanded_exponent);
 725                 exvector distrseq;
 726                 distrseq.reserve(a.seq.size() + 1);
 727                 epvector::const_iterator last = a.seq.end();
 728                 epvector::const_iterator cit = a.seq.begin();
 729                 while (cit!=last) {
 730                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 731                         ++cit;
 732                 }
 733
 734                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 735                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 736                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 737                         int int_exponent = num_exponent.to_int();
 738                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 739                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 740                         else
 741                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 742                 } else
 743                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 744
 745                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 746                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 747                 return r.expand(options);
 748         }
 749
 750         if (!is_exactly_a<numeric>(expanded_exponent) ||
 751                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 752                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 753                         return this->hold();
 754                 } else {
 755                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 756                 }
 757         }
 758
 759         // integer numeric exponent
 760         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 761         int int_exponent = num_exponent.to_int();
 762
 763         // (x+y)^n, n>0
 764         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 765                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 766
 767         // (x*y)^n -> x^n * y^n
 768         if (is_exactly_a<mul>(expanded_basis))
 769                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 770
 771         // cannot expand further
 772         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 773                 return this->hold();
 774         else
 775                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 776 }
 777
 778 //////////
 779 // new virtual functions which can be overridden by derived classes
 780 //////////
 781
 782 // none
 783
 784 //////////
 785 // non-virtual functions in this class
 786 //////////
 787
 788 /** expand a^n where a is an add and n is a positive integer.
 789  *  @see power::expand */
 790 ex power::expand_add(const add & a, int n, unsigned options) const
 791 {
 792         if (n==2)
 793                 return expand_add_2(a, options);
 794
 795         const size_t m = a.nops();
 796         exvector result;
 797         // The number of terms will be the number of combinatorial compositions,
 798         // i.e. the number of unordered arrangements of m nonnegative integers
 799         // which sum up to n.  It is frequently written as C_n(m) and directly
 800         // related with binomial coefficients:
 801         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 802         intvector k(m-1);
 803         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 804         intvector upper_limit(m-1);
 805         int l;
 806
 807         for (size_t l=0; l<m-1; ++l) {
 808                 k[l] = 0;
 809                 k_cum[l] = 0;
 810                 upper_limit[l] = n;
 811         }
 812
 813         while (true) {
 814                 exvector term;
 815                 term.reserve(m+1);
 816                 for (l=0; l<m-1; ++l) {
 817                         const ex & b = a.op(l);
 818                         GINAC_ASSERT(!is_exactly_a<add>(b));
 819                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 820                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 821                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 822                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 823                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 824                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 825                         if (is_exactly_a<mul>(b))
 826                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 827                         else
 828                                 term.push_back(power(b,k[l]));
 829                 }
 830
 831                 const ex & b = a.op(l);
 832                 GINAC_ASSERT(!is_exactly_a<add>(b));
 833                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 834                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 835                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 836                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 837                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 838                              !is_exactly_a<power>(ex_to<power>(b).basis));
 839                 if (is_exactly_a<mul>(b))
 840                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 841                 else
 842                         term.push_back(power(b,n-k_cum[m-2]));
 843
 844                 numeric f = binomial(numeric(n),numeric(k[0]));
 845                 for (l=1; l<m-1; ++l)
 846                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 847
 848                 term.push_back(f);
 849
 850                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 851
 852                 // increment k[]
 853                 l = m-2;
 854                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 855                         k[l] = 0;
 856                         --l;
 857                 }
 858                 if (l<0) break;
 859
 860                 // recalc k_cum[] and upper_limit[]
 861                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 862
 863                 for (size_t i=l+1; i<m-1; ++i)
 864                         k_cum[i] = k_cum[i-1]+k[i];
 865
 866                 for (size_t i=l+1; i<m-1; ++i)
 867                         upper_limit[i] = n-k_cum[i-1];
 868         }
 869
 870         return (new add(result))->setflag(status_flags::dynallocated |
 871                                           status_flags::expanded);
 872 }
 873
 874
 875 /** Special case of power::expand_add. Expands a^2 where a is an add.
 876  *  @see power::expand_add */
 877 ex power::expand_add_2(const add & a, unsigned options) const
 878 {
 879         epvector sum;
 880         size_t a_nops = a.nops();
 881         sum.reserve((a_nops*(a_nops+1))/2);
 882         epvector::const_iterator last = a.seq.end();
 883
 884         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 885         // first part: ignore overall_coeff and expand other terms
 886         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 887                 const ex & r = cit0->rest;
 888                 const ex & c = cit0->coeff;
 889
 890                 GINAC_ASSERT(!is_exactly_a<add>(r));
 891                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 892                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 893                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 894                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 895                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 896                              !is_exactly_a<power>(ex_to<power>(r).basis));
 897
 898                 if (c.is_equal(_ex1)) {
 899                         if (is_exactly_a<mul>(r)) {
 900                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 901                                                      _ex1));
 902                         } else {
 903                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 904                                                      _ex1));
 905                         }
 906                 } else {
 907                         if (is_exactly_a<mul>(r)) {
 908                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 909                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 910                         } else {
 911                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 912                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 913                         }
 914                 }
 915
 916                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 917                         const ex & r1 = cit1->rest;
 918                         const ex & c1 = cit1->coeff;
 919                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 920                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 921                 }
 922         }
 923
 924         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 925
 926         // second part: add terms coming from overall_factor (if != 0)
 927         if (!a.overall_coeff.is_zero()) {
 928                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 929                 while (i != end) {
 930                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 931                         ++i;
 932                 }
 933                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 934         }
 935
 936         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 937
 938         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 939 }
 940
 941 /** Expand factors of m in m^n where m is a mul and n is and integer.
 942  *  @see power::expand */
 943 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 944 {
 945         GINAC_ASSERT(n.is_integer());
 946
 947         if (n.is_zero()) {
 948                 return _ex1;
 949         }
 950
 951         // Leave it to multiplication since dummy indices have to be renamed
 952         if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
 953                 ex result = m;
 954                 exvector va = get_all_dummy_indices(m);
 955                 sort(va.begin(), va.end(), ex_is_less());
 956
 957                 for (int i=1; i < n.to_int(); i++)
 958                         result *= rename_dummy_indices_uniquely(va, m);
 959                 return result;
 960         }
 961
 962         epvector distrseq;
 963         distrseq.reserve(m.seq.size());
 964         bool need_reexpand = false;
 965
 966         epvector::const_iterator last = m.seq.end();
 967         epvector::const_iterator cit = m.seq.begin();
 968         while (cit!=last) {
 969                 if (is_exactly_a<numeric>(cit->rest)) {
 970                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
 971                 } else {
 972                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 973                         // since n is an integer
 974                         numeric new_coeff = ex_to<numeric>(cit->coeff).mul(n);
 975                         if (from_expand && is_exactly_a<add>(cit->rest) && new_coeff.is_pos_integer()) {
 976                                 // this happens when e.g. (a+b)^(1/2) gets squared and
 977                                 // the resulting product needs to be reexpanded
 978                                 need_reexpand = true;
 979                         }
 980                         distrseq.push_back(expair(cit->rest, new_coeff));
 981                 }
 982                 ++cit;
 983         }
 984
 985         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
 986         if (need_reexpand)
 987                 return ex(result).expand(options);
 988         if (from_expand)
 989                 return result.setflag(status_flags::expanded);
 990         return result;
 991 }
 992
 993 } // namespace GiNaC