ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "power.h"
  29 #include "expairseq.h"
  30 #include "add.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "numeric.h"
  34 #include "constant.h"
  35 #include "operators.h"
  36 #include "inifcns.h" // for log() in power::derivative()
  37 #include "matrix.h"
  38 #include "indexed.h"
  39 #include "symbol.h"
  40 #include "lst.h"
  41 #include "archive.h"
  42 #include "utils.h"
  43 #include "relational.h"
  44
  45 namespace GiNaC {
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  48   print_func<print_dflt>(&power::do_print_dflt).
  49   print_func<print_latex>(&power::do_print_latex).
  50   print_func<print_csrc>(&power::do_print_csrc).
  51   print_func<print_python>(&power::do_print_python).
  52   print_func<print_python_repr>(&power::do_print_python_repr))
  53
  54 typedef std::vector<int> intvector;
  55
  56 //////////
  57 // default constructor
  58 //////////
  59
  60 power::power() : inherited(&power::tinfo_static) { }
  61
  62 //////////
  63 // other constructors
  64 //////////
  65
  66 // all inlined
  67
  68 //////////
  69 // archiving
  70 //////////
  71
  72 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  73 {
  74         n.find_ex("basis", basis, sym_lst);
  75         n.find_ex("exponent", exponent, sym_lst);
  76 }
  77
  78 void power::archive(archive_node &n) const
  79 {
  80         inherited::archive(n);
  81         n.add_ex("basis", basis);
  82         n.add_ex("exponent", exponent);
  83 }
  84
  85 DEFAULT_UNARCHIVE(power)
  86
  87 //////////
  88 // functions overriding virtual functions from base classes
  89 //////////
  90
  91 // public
  92
  93 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  94 {
  95         // Ordinary output of powers using '^' or '**'
  96         if (precedence() <= level)
  97                 c.s << openbrace << '(';
  98         basis.print(c, precedence());
  99         c.s << powersymbol;
 100         c.s << openbrace;
 101         exponent.print(c, precedence());
 102         c.s << closebrace;
 103         if (precedence() <= level)
 104                 c.s << ')' << closebrace;
 105 }
 106
 107 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 108 {
 109         if (exponent.is_equal(_ex1_2)) {
 110
 111                 // Square roots are printed in a special way
 112                 c.s << "sqrt(";
 113                 basis.print(c);
 114                 c.s << ')';
 115
 116         } else
 117                 print_power(c, "^", "", "", level);
 118 }
 119
 120 void power::do_print_latex(const print_latex & c, unsigned level) const
 121 {
 122         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 123
 124                 // Powers with negative numeric exponents are printed as fractions
 125                 c.s << "\\frac{1}{";
 126                 power(basis, -exponent).eval().print(c);
 127                 c.s << '}';
 128
 129         } else if (exponent.is_equal(_ex1_2)) {
 130
 131                 // Square roots are printed in a special way
 132                 c.s << "\\sqrt{";
 133                 basis.print(c);
 134                 c.s << '}';
 135
 136         } else
 137                 print_power(c, "^", "{", "}", level);
 138 }
 139
 140 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 141 {
 142         // Optimal output of integer powers of symbols to aid compiler CSE.
 143         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 144         // to learn why such a parenthesation is really necessary.
 145         if (exp == 1) {
 146                 x.print(c);
 147         } else if (exp == 2) {
 148                 x.print(c);
 149                 c.s << "*";
 150                 x.print(c);
 151         } else if (exp & 1) {
 152                 x.print(c);
 153                 c.s << "*";
 154                 print_sym_pow(c, x, exp-1);
 155         } else {
 156                 c.s << "(";
 157                 print_sym_pow(c, x, exp >> 1);
 158                 c.s << ")*(";
 159                 print_sym_pow(c, x, exp >> 1);
 160                 c.s << ")";
 161         }
 162 }
 163
 164 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 165 {
 166         // Integer powers of symbols are printed in a special, optimized way
 167         if (exponent.info(info_flags::integer)
 168          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 169                 int exp = ex_to<numeric>(exponent).to_int();
 170                 if (exp > 0)
 171                         c.s << '(';
 172                 else {
 173                         exp = -exp;
 174                         if (is_a<print_csrc_cl_N>(c))
 175                                 c.s << "recip(";
 176                         else
 177                                 c.s << "1.0/(";
 178                 }
 179                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 180                 c.s << ')';
 181
 182         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 183         } else if (exponent.is_equal(_ex_1)) {
 184                 if (is_a<print_csrc_cl_N>(c))
 185                         c.s << "recip(";
 186                 else
 187                         c.s << "1.0/(";
 188                 basis.print(c);
 189                 c.s << ')';
 190
 191         // Otherwise, use the pow() or expt() (CLN) functions
 192         } else {
 193                 if (is_a<print_csrc_cl_N>(c))
 194                         c.s << "expt(";
 195                 else
 196                         c.s << "pow(";
 197                 basis.print(c);
 198                 c.s << ',';
 199                 exponent.print(c);
 200                 c.s << ')';
 201         }
 202 }
 203
 204 void power::do_print_python(const print_python & c, unsigned level) const
 205 {
 206         print_power(c, "**", "", "", level);
 207 }
 208
 209 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 210 {
 211         c.s << class_name() << '(';
 212         basis.print(c);
 213         c.s << ',';
 214         exponent.print(c);
 215         c.s << ')';
 216 }
 217
 218 bool power::info(unsigned inf) const
 219 {
 220         switch (inf) {
 221                 case info_flags::polynomial:
 222                 case info_flags::integer_polynomial:
 223                 case info_flags::cinteger_polynomial:
 224                 case info_flags::rational_polynomial:
 225                 case info_flags::crational_polynomial:
 226                         return exponent.info(info_flags::nonnegint) &&
 227                                basis.info(inf);
 228                 case info_flags::rational_function:
 229                         return exponent.info(info_flags::integer) &&
 230                                basis.info(inf);
 231                 case info_flags::algebraic:
 232                         return !exponent.info(info_flags::integer) ||
 233                                basis.info(inf);
 234         }
 235         return inherited::info(inf);
 236 }
 237
 238 size_t power::nops() const
 239 {
 240         return 2;
 241 }
 242
 243 ex power::op(size_t i) const
 244 {
 245         GINAC_ASSERT(i<2);
 246
 247         return i==0 ? basis : exponent;
 248 }
 249
 250 ex power::map(map_function & f) const
 251 {
 252         const ex &mapped_basis = f(basis);
 253         const ex &mapped_exponent = f(exponent);
 254
 255         if (!are_ex_trivially_equal(basis, mapped_basis)
 256          || !are_ex_trivially_equal(exponent, mapped_exponent))
 257                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 258         else
 259                 return *this;
 260 }
 261
 262 bool power::is_polynomial(const ex & var) const
 263 {
 264         if (exponent.has(var))
 265                 return false;
 266         if (!exponent.info(info_flags::nonnegint))
 267                 return false;
 268         return basis.is_polynomial(var);
 269 }
 270
 271 int power::degree(const ex & s) const
 272 {
 273         if (is_equal(ex_to<basic>(s)))
 274                 return 1;
 275         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 276                 if (basis.is_equal(s))
 277                         return ex_to<numeric>(exponent).to_int();
 278                 else
 279                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 280         } else if (basis.has(s))
 281                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 282         else
 283                 return 0;
 284 }
 285
 286 int power::ldegree(const ex & s) const
 287 {
 288         if (is_equal(ex_to<basic>(s)))
 289                 return 1;
 290         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 291                 if (basis.is_equal(s))
 292                         return ex_to<numeric>(exponent).to_int();
 293                 else
 294                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 295         } else if (basis.has(s))
 296                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 297         else
 298                 return 0;
 299 }
 300
 301 ex power::coeff(const ex & s, int n) const
 302 {
 303         if (is_equal(ex_to<basic>(s)))
 304                 return n==1 ? _ex1 : _ex0;
 305         else if (!basis.is_equal(s)) {
 306                 // basis not equal to s
 307                 if (n == 0)
 308                         return *this;
 309                 else
 310                         return _ex0;
 311         } else {
 312                 // basis equal to s
 313                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 314                         // integer exponent
 315                         int int_exp = ex_to<numeric>(exponent).to_int();
 316                         if (n == int_exp)
 317                                 return _ex1;
 318                         else
 319                                 return _ex0;
 320                 } else {
 321                         // non-integer exponents are treated as zero
 322                         if (n == 0)
 323                                 return *this;
 324                         else
 325                                 return _ex0;
 326                 }
 327         }
 328 }
 329
 330 /** Perform automatic term rewriting rules in this class.  In the following
 331  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 332  *  stand for such expressions that contain a plain number.
 333  *  - ^(x,0) -> 1  (also handles ^(0,0))
 334  *  - ^(x,1) -> x
 335  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 336  *  - ^(1,x) -> 1
 337  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 338  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 339  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 340  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 341  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 342  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 343  *
 344  *  @param level cut-off in recursive evaluation */
 345 ex power::eval(int level) const
 346 {
 347         if ((level==1) && (flags & status_flags::evaluated))
 348                 return *this;
 349         else if (level == -max_recursion_level)
 350                 throw(std::runtime_error("max recursion level reached"));
 351
 352         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 353         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 354
 355         bool basis_is_numerical = false;
 356         bool exponent_is_numerical = false;
 357         const numeric *num_basis;
 358         const numeric *num_exponent;
 359
 360         if (is_exactly_a<numeric>(ebasis)) {
 361                 basis_is_numerical = true;
 362                 num_basis = &ex_to<numeric>(ebasis);
 363         }
 364         if (is_exactly_a<numeric>(eexponent)) {
 365                 exponent_is_numerical = true;
 366                 num_exponent = &ex_to<numeric>(eexponent);
 367         }
 368
 369         // ^(x,0) -> 1  (0^0 also handled here)
 370         if (eexponent.is_zero()) {
 371                 if (ebasis.is_zero())
 372                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 373                 else
 374                         return _ex1;
 375         }
 376
 377         // ^(x,1) -> x
 378         if (eexponent.is_equal(_ex1))
 379                 return ebasis;
 380
 381         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 382         if (ebasis.is_zero() && exponent_is_numerical) {
 383                 if ((num_exponent->real()).is_zero())
 384                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 385                 else if ((num_exponent->real()).is_negative())
 386                         throw (pole_error("power::eval(): division by zero",1));
 387                 else
 388                         return _ex0;
 389         }
 390
 391         // ^(1,x) -> 1
 392         if (ebasis.is_equal(_ex1))
 393                 return _ex1;
 394
 395         // power of a function calculated by separate rules defined for this function
 396         if (is_exactly_a<function>(ebasis))
 397                 return ex_to<function>(ebasis).power(eexponent);
 398
 399         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 400         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 401                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 402
 403         if (exponent_is_numerical) {
 404
 405                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 406                 // except if c1,c2 are rational, but c1^c2 is not)
 407                 if (basis_is_numerical) {
 408                         const bool basis_is_crational = num_basis->is_crational();
 409                         const bool exponent_is_crational = num_exponent->is_crational();
 410                         if (!basis_is_crational || !exponent_is_crational) {
 411                                 // return a plain float
 412                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 413                                                                                                status_flags::evaluated |
 414                                                                                                status_flags::expanded);
 415                         }
 416
 417                         const numeric res = num_basis->power(*num_exponent);
 418                         if (res.is_crational()) {
 419                                 return res;
 420                         }
 421                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 422
 423                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 424                         if (basis_is_crational && exponent_is_crational
 425                             && num_exponent->is_real()
 426                             && !num_exponent->is_integer()) {
 427                                 const numeric n = num_exponent->numer();
 428                                 const numeric m = num_exponent->denom();
 429                                 numeric r;
 430                                 numeric q = iquo(n, m, r);
 431                                 if (r.is_negative()) {
 432                                         r += m;
 433                                         --q;
 434                                 }
 435                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 436                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 437                                                 // try it for numerator and denominator separately, in order to
 438                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 439                                                 const numeric bnum = num_basis->numer();
 440                                                 const numeric bden = num_basis->denom();
 441                                                 const numeric res_bnum = bnum.power(*num_exponent);
 442                                                 const numeric res_bden = bden.power(*num_exponent);
 443                                                 if (res_bnum.is_integer())
 444                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 445                                                 if (res_bden.is_integer())
 446                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 447                                         }
 448                                         return this->hold();
 449                                 } else {
 450                                         // assemble resulting product, but allowing for a re-evaluation,
 451                                         // because otherwise we'll end up with something like
 452                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 453                                         // instead of 7/16*7^(1/3).
 454                                         ex prod = power(*num_basis,r.div(m));
 455                                         return prod*power(*num_basis,q);
 456                                 }
 457                         }
 458                 }
 459
 460                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 461                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 462                 // case c1==1 should not happen, see below!)
 463                 if (is_exactly_a<power>(ebasis)) {
 464                         const power & sub_power = ex_to<power>(ebasis);
 465                         const ex & sub_basis = sub_power.basis;
 466                         const ex & sub_exponent = sub_power.exponent;
 467                         if (is_exactly_a<numeric>(sub_exponent)) {
 468                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 469                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 470                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative())
 471                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 472                         }
 473                 }
 474
 475                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 476                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 477                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 478                 }
 479
 480                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 481                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 482                 if (is_exactly_a<mul>(ebasis)) {
 483                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 484                         const mul & mulref = ex_to<mul>(ebasis);
 485                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 486                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 487                                 if (num_coeff.is_real()) {
 488                                         if (num_coeff.is_positive()) {
 489                                                 mul *mulp = new mul(mulref);
 490                                                 mulp->overall_coeff = _ex1;
 491                                                 mulp->clearflag(status_flags::evaluated);
 492                                                 mulp->clearflag(status_flags::hash_calculated);
 493                                                 return (new mul(power(*mulp,exponent),
 494                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 495                                         } else {
 496                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 497                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 498                                                         mul *mulp = new mul(mulref);
 499                                                         mulp->overall_coeff = _ex_1;
 500                                                         mulp->clearflag(status_flags::evaluated);
 501                                                         mulp->clearflag(status_flags::hash_calculated);
 502                                                         return (new mul(power(*mulp,exponent),
 503                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 504                                                 }
 505                                         }
 506                                 }
 507                         }
 508                 }
 509
 510                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 511                 if (num_exponent->is_pos_integer() &&
 512                     ebasis.return_type() != return_types::commutative &&
 513                     !is_a<matrix>(ebasis)) {
 514                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 515                 }
 516         }
 517
 518         if (are_ex_trivially_equal(ebasis,basis) &&
 519             are_ex_trivially_equal(eexponent,exponent)) {
 520                 return this->hold();
 521         }
 522         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 523                                                        status_flags::evaluated);
 524 }
 525
 526 ex power::evalf(int level) const
 527 {
 528         ex ebasis;
 529         ex eexponent;
 530
 531         if (level==1) {
 532                 ebasis = basis;
 533                 eexponent = exponent;
 534         } else if (level == -max_recursion_level) {
 535                 throw(std::runtime_error("max recursion level reached"));
 536         } else {
 537                 ebasis = basis.evalf(level-1);
 538                 if (!is_exactly_a<numeric>(exponent))
 539                         eexponent = exponent.evalf(level-1);
 540                 else
 541                         eexponent = exponent;
 542         }
 543
 544         return power(ebasis,eexponent);
 545 }
 546
 547 ex power::evalm() const
 548 {
 549         const ex ebasis = basis.evalm();
 550         const ex eexponent = exponent.evalm();
 551         if (is_a<matrix>(ebasis)) {
 552                 if (is_exactly_a<numeric>(eexponent)) {
 553                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 554                 }
 555         }
 556         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 557 }
 558
 559 bool power::has(const ex & other, unsigned options) const
 560 {
 561         if (!(options & has_options::algebraic))
 562                 return basic::has(other, options);
 563         if (!is_a<power>(other))
 564                 return basic::has(other, options);
 565         if (!exponent.info(info_flags::integer)
 566                         || !other.op(1).info(info_flags::integer))
 567                 return basic::has(other, options);
 568         if (exponent.info(info_flags::posint)
 569                         && other.op(1).info(info_flags::posint)
 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         if (exponent.info(info_flags::negint)
 575                         && other.op(1).info(info_flags::negint)
 576                         && ex_to<numeric>(exponent).to_int()
 577                                         < ex_to<numeric>(other.op(1)).to_int()
 578                         && basis.match(other.op(0)))
 579                 return true;
 580         return basic::has(other, options);
 581 }
 582
 583 // from mul.cpp
 584 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 585
 586 ex power::subs(const exmap & m, unsigned options) const
 587 {
 588         const ex &subsed_basis = basis.subs(m, options);
 589         const ex &subsed_exponent = exponent.subs(m, options);
 590
 591         if (!are_ex_trivially_equal(basis, subsed_basis)
 592          || !are_ex_trivially_equal(exponent, subsed_exponent))
 593                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 594
 595         if (!(options & subs_options::algebraic))
 596                 return subs_one_level(m, options);
 597
 598         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 599                 int nummatches = std::numeric_limits<int>::max();
 600                 lst repls;
 601                 if (tryfactsubs(*this, it->first, nummatches, repls))
 602                         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);
 603         }
 604
 605         return subs_one_level(m, options);
 606 }
 607
 608 ex power::eval_ncmul(const exvector & v) const
 609 {
 610         return inherited::eval_ncmul(v);
 611 }
 612
 613 ex power::conjugate() const
 614 {
 615         ex newbasis = basis.conjugate();
 616         ex newexponent = exponent.conjugate();
 617         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 618                 return *this;
 619         }
 620         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 621 }
 622
 623 ex power::real_part() const
 624 {
 625         if (exponent.info(info_flags::integer)) {
 626                 ex basis_real = basis.real_part();
 627                 if (basis_real == basis)
 628                         return *this;
 629                 realsymbol a("a"),b("b");
 630                 ex result;
 631                 if (exponent.info(info_flags::posint))
 632                         result = power(a+I*b,exponent);
 633                 else
 634                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 635                 result = result.expand();
 636                 result = result.real_part();
 637                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 638                 return result;
 639         }
 640
 641         ex a = basis.real_part();
 642         ex b = basis.imag_part();
 643         ex c = exponent.real_part();
 644         ex d = exponent.imag_part();
 645         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 646 }
 647
 648 ex power::imag_part() const
 649 {
 650         if (exponent.info(info_flags::integer)) {
 651                 ex basis_real = basis.real_part();
 652                 if (basis_real == basis)
 653                         return 0;
 654                 realsymbol a("a"),b("b");
 655                 ex result;
 656                 if (exponent.info(info_flags::posint))
 657                         result = power(a+I*b,exponent);
 658                 else
 659                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 660                 result = result.expand();
 661                 result = result.imag_part();
 662                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 663                 return result;
 664         }
 665
 666         ex a=basis.real_part();
 667         ex b=basis.imag_part();
 668         ex c=exponent.real_part();
 669         ex d=exponent.imag_part();
 670         return
 671                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 672 }
 673
 674 // protected
 675
 676 // protected
 677
 678 /** Implementation of ex::diff() for a power.
 679  *  @see ex::diff */
 680 ex power::derivative(const symbol & s) const
 681 {
 682         if (is_a<numeric>(exponent)) {
 683                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 684                 epvector newseq;
 685                 newseq.reserve(2);
 686                 newseq.push_back(expair(basis, exponent - _ex1));
 687                 newseq.push_back(expair(basis.diff(s), _ex1));
 688                 return mul(newseq, exponent);
 689         } else {
 690                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 691                 return mul(*this,
 692                            add(mul(exponent.diff(s), log(basis)),
 693                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 694         }
 695 }
 696
 697 int power::compare_same_type(const basic & other) const
 698 {
 699         GINAC_ASSERT(is_exactly_a<power>(other));
 700         const power &o = static_cast<const power &>(other);
 701
 702         int cmpval = basis.compare(o.basis);
 703         if (cmpval)
 704                 return cmpval;
 705         else
 706                 return exponent.compare(o.exponent);
 707 }
 708
 709 unsigned power::return_type() const
 710 {
 711         return basis.return_type();
 712 }
 713
 714 tinfo_t power::return_type_tinfo() const
 715 {
 716         return basis.return_type_tinfo();
 717 }
 718
 719 ex power::expand(unsigned options) const
 720 {
 721         if (options == 0 && (flags & status_flags::expanded))
 722                 return *this;
 723
 724         const ex expanded_basis = basis.expand(options);
 725         const ex expanded_exponent = exponent.expand(options);
 726
 727         // x^(a+b) -> x^a * x^b
 728         if (is_exactly_a<add>(expanded_exponent)) {
 729                 const add &a = ex_to<add>(expanded_exponent);
 730                 exvector distrseq;
 731                 distrseq.reserve(a.seq.size() + 1);
 732                 epvector::const_iterator last = a.seq.end();
 733                 epvector::const_iterator cit = a.seq.begin();
 734                 while (cit!=last) {
 735                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 736                         ++cit;
 737                 }
 738
 739                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 740                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 741                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 742                         int int_exponent = num_exponent.to_int();
 743                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 744                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 745                         else
 746                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 747                 } else
 748                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 749
 750                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 751                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 752                 return r.expand(options);
 753         }
 754
 755         if (!is_exactly_a<numeric>(expanded_exponent) ||
 756                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 757                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 758                         return this->hold();
 759                 } else {
 760                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 761                 }
 762         }
 763
 764         // integer numeric exponent
 765         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 766         int int_exponent = num_exponent.to_int();
 767
 768         // (x+y)^n, n>0
 769         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 770                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 771
 772         // (x*y)^n -> x^n * y^n
 773         if (is_exactly_a<mul>(expanded_basis))
 774                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 775
 776         // cannot expand further
 777         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 778                 return this->hold();
 779         else
 780                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 781 }
 782
 783 //////////
 784 // new virtual functions which can be overridden by derived classes
 785 //////////
 786
 787 // none
 788
 789 //////////
 790 // non-virtual functions in this class
 791 //////////
 792
 793 /** expand a^n where a is an add and n is a positive integer.
 794  *  @see power::expand */
 795 ex power::expand_add(const add & a, int n, unsigned options) const
 796 {
 797         if (n==2)
 798                 return expand_add_2(a, options);
 799
 800         const size_t m = a.nops();
 801         exvector result;
 802         // The number of terms will be the number of combinatorial compositions,
 803         // i.e. the number of unordered arrangements of m nonnegative integers
 804         // which sum up to n.  It is frequently written as C_n(m) and directly
 805         // related with binomial coefficients:
 806         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 807         intvector k(m-1);
 808         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 809         intvector upper_limit(m-1);
 810         int l;
 811
 812         for (size_t l=0; l<m-1; ++l) {
 813                 k[l] = 0;
 814                 k_cum[l] = 0;
 815                 upper_limit[l] = n;
 816         }
 817
 818         while (true) {
 819                 exvector term;
 820                 term.reserve(m+1);
 821                 for (l=0; l<m-1; ++l) {
 822                         const ex & b = a.op(l);
 823                         GINAC_ASSERT(!is_exactly_a<add>(b));
 824                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 825                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 826                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 827                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 828                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 829                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 830                         if (is_exactly_a<mul>(b))
 831                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 832                         else
 833                                 term.push_back(power(b,k[l]));
 834                 }
 835
 836                 const ex & b = a.op(l);
 837                 GINAC_ASSERT(!is_exactly_a<add>(b));
 838                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 839                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 840                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 841                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 842                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 843                              !is_exactly_a<power>(ex_to<power>(b).basis));
 844                 if (is_exactly_a<mul>(b))
 845                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 846                 else
 847                         term.push_back(power(b,n-k_cum[m-2]));
 848
 849                 numeric f = binomial(numeric(n),numeric(k[0]));
 850                 for (l=1; l<m-1; ++l)
 851                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 852
 853                 term.push_back(f);
 854
 855                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 856
 857                 // increment k[]
 858                 l = m-2;
 859                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 860                         k[l] = 0;
 861                         --l;
 862                 }
 863                 if (l<0) break;
 864
 865                 // recalc k_cum[] and upper_limit[]
 866                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 867
 868                 for (size_t i=l+1; i<m-1; ++i)
 869                         k_cum[i] = k_cum[i-1]+k[i];
 870
 871                 for (size_t i=l+1; i<m-1; ++i)
 872                         upper_limit[i] = n-k_cum[i-1];
 873         }
 874
 875         return (new add(result))->setflag(status_flags::dynallocated |
 876                                           status_flags::expanded);
 877 }
 878
 879
 880 /** Special case of power::expand_add. Expands a^2 where a is an add.
 881  *  @see power::expand_add */
 882 ex power::expand_add_2(const add & a, unsigned options) const
 883 {
 884         epvector sum;
 885         size_t a_nops = a.nops();
 886         sum.reserve((a_nops*(a_nops+1))/2);
 887         epvector::const_iterator last = a.seq.end();
 888
 889         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 890         // first part: ignore overall_coeff and expand other terms
 891         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 892                 const ex & r = cit0->rest;
 893                 const ex & c = cit0->coeff;
 894
 895                 GINAC_ASSERT(!is_exactly_a<add>(r));
 896                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 897                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 898                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 899                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 900                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 901                              !is_exactly_a<power>(ex_to<power>(r).basis));
 902
 903                 if (c.is_equal(_ex1)) {
 904                         if (is_exactly_a<mul>(r)) {
 905                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 906                                                      _ex1));
 907                         } else {
 908                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 909                                                      _ex1));
 910                         }
 911                 } else {
 912                         if (is_exactly_a<mul>(r)) {
 913                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 914                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 915                         } else {
 916                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 917                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 918                         }
 919                 }
 920
 921                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 922                         const ex & r1 = cit1->rest;
 923                         const ex & c1 = cit1->coeff;
 924                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 925                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 926                 }
 927         }
 928
 929         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 930
 931         // second part: add terms coming from overall_factor (if != 0)
 932         if (!a.overall_coeff.is_zero()) {
 933                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 934                 while (i != end) {
 935                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
 936                         ++i;
 937                 }
 938                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
 939         }
 940
 941         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 942
 943         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 944 }
 945
 946 /** Expand factors of m in m^n where m is a mul and n is an integer.
 947  *  @see power::expand */
 948 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
 949 {
 950         GINAC_ASSERT(n.is_integer());
 951
 952         if (n.is_zero()) {
 953                 return _ex1;
 954         }
 955
 956         // Leave it to multiplication since dummy indices have to be renamed
 957         if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
 958                 ex result = m;
 959                 exvector va = get_all_dummy_indices(m);
 960                 sort(va.begin(), va.end(), ex_is_less());
 961
 962                 for (int i=1; i < n.to_int(); i++)
 963                         result *= rename_dummy_indices_uniquely(va, m);
 964                 return result;
 965         }
 966
 967         epvector distrseq;
 968         distrseq.reserve(m.seq.size());
 969         bool need_reexpand = false;
 970
 971         epvector::const_iterator last = m.seq.end();
 972         epvector::const_iterator cit = m.seq.begin();
 973         while (cit!=last) {
 974                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
 975                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.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(p);
 981                 ++cit;
 982         }
 983
 984         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
 985         if (need_reexpand)
 986                 return ex(result).expand(options);
 987         if (from_expand)
 988                 return result.setflag(status_flags::expanded);
 989         return result;
 990 }
 991
 992 } // namespace GiNaC