ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2008 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 #include "compiler.h"
  45
  46 namespace GiNaC {
  47
  48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  49   print_func<print_dflt>(&power::do_print_dflt).
  50   print_func<print_latex>(&power::do_print_latex).
  51   print_func<print_csrc>(&power::do_print_csrc).
  52   print_func<print_python>(&power::do_print_python).
  53   print_func<print_python_repr>(&power::do_print_python_repr).
  54   print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
  55
  56 typedef std::vector<int> intvector;
  57
  58 //////////
  59 // default constructor
  60 //////////
  61
  62 power::power() : inherited(&power::tinfo_static) { }
  63
  64 //////////
  65 // other constructors
  66 //////////
  67
  68 // all inlined
  69
  70 //////////
  71 // archiving
  72 //////////
  73
  74 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  75 {
  76         n.find_ex("basis", basis, sym_lst);
  77         n.find_ex("exponent", exponent, sym_lst);
  78 }
  79
  80 void power::archive(archive_node &n) const
  81 {
  82         inherited::archive(n);
  83         n.add_ex("basis", basis);
  84         n.add_ex("exponent", exponent);
  85 }
  86
  87 DEFAULT_UNARCHIVE(power)
  88
  89 //////////
  90 // functions overriding virtual functions from base classes
  91 //////////
  92
  93 // public
  94
  95 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  96 {
  97         // Ordinary output of powers using '^' or '**'
  98         if (precedence() <= level)
  99                 c.s << openbrace << '(';
 100         basis.print(c, precedence());
 101         c.s << powersymbol;
 102         c.s << openbrace;
 103         exponent.print(c, precedence());
 104         c.s << closebrace;
 105         if (precedence() <= level)
 106                 c.s << ')' << closebrace;
 107 }
 108
 109 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 110 {
 111         if (exponent.is_equal(_ex1_2)) {
 112
 113                 // Square roots are printed in a special way
 114                 c.s << "sqrt(";
 115                 basis.print(c);
 116                 c.s << ')';
 117
 118         } else
 119                 print_power(c, "^", "", "", level);
 120 }
 121
 122 void power::do_print_latex(const print_latex & c, unsigned level) const
 123 {
 124         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 125
 126                 // Powers with negative numeric exponents are printed as fractions
 127                 c.s << "\\frac{1}{";
 128                 power(basis, -exponent).eval().print(c);
 129                 c.s << '}';
 130
 131         } else if (exponent.is_equal(_ex1_2)) {
 132
 133                 // Square roots are printed in a special way
 134                 c.s << "\\sqrt{";
 135                 basis.print(c);
 136                 c.s << '}';
 137
 138         } else
 139                 print_power(c, "^", "{", "}", level);
 140 }
 141
 142 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 143 {
 144         // Optimal output of integer powers of symbols to aid compiler CSE.
 145         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 146         // to learn why such a parenthesation is really necessary.
 147         if (exp == 1) {
 148                 x.print(c);
 149         } else if (exp == 2) {
 150                 x.print(c);
 151                 c.s << "*";
 152                 x.print(c);
 153         } else if (exp & 1) {
 154                 x.print(c);
 155                 c.s << "*";
 156                 print_sym_pow(c, x, exp-1);
 157         } else {
 158                 c.s << "(";
 159                 print_sym_pow(c, x, exp >> 1);
 160                 c.s << ")*(";
 161                 print_sym_pow(c, x, exp >> 1);
 162                 c.s << ")";
 163         }
 164 }
 165
 166 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
 167 {
 168         if (exponent.is_equal(_ex_1)) {
 169                 c.s << "recip(";
 170                 basis.print(c);
 171                 c.s << ')';
 172                 return;
 173         }
 174         c.s << "expt(";
 175         basis.print(c);
 176         c.s << ", ";
 177         exponent.print(c);
 178         c.s << ')';
 179 }
 180
 181 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 182 {
 183         // Integer powers of symbols are printed in a special, optimized way
 184         if (exponent.info(info_flags::integer)
 185          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 186                 int exp = ex_to<numeric>(exponent).to_int();
 187                 if (exp > 0)
 188                         c.s << '(';
 189                 else {
 190                         exp = -exp;
 191                         c.s << "1.0/(";
 192                 }
 193                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 194                 c.s << ')';
 195
 196         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 197         } else if (exponent.is_equal(_ex_1)) {
 198                 c.s << "1.0/(";
 199                 basis.print(c);
 200                 c.s << ')';
 201
 202         // Otherwise, use the pow() function
 203         } else {
 204                 c.s << "pow(";
 205                 basis.print(c);
 206                 c.s << ',';
 207                 exponent.print(c);
 208                 c.s << ')';
 209         }
 210 }
 211
 212 void power::do_print_python(const print_python & c, unsigned level) const
 213 {
 214         print_power(c, "**", "", "", level);
 215 }
 216
 217 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 218 {
 219         c.s << class_name() << '(';
 220         basis.print(c);
 221         c.s << ',';
 222         exponent.print(c);
 223         c.s << ')';
 224 }
 225
 226 bool power::info(unsigned inf) const
 227 {
 228         switch (inf) {
 229                 case info_flags::polynomial:
 230                 case info_flags::integer_polynomial:
 231                 case info_flags::cinteger_polynomial:
 232                 case info_flags::rational_polynomial:
 233                 case info_flags::crational_polynomial:
 234                         return exponent.info(info_flags::nonnegint) &&
 235                                basis.info(inf);
 236                 case info_flags::rational_function:
 237                         return exponent.info(info_flags::integer) &&
 238                                basis.info(inf);
 239                 case info_flags::algebraic:
 240                         return !exponent.info(info_flags::integer) ||
 241                                basis.info(inf);
 242                 case info_flags::expanded:
 243                         return (flags & status_flags::expanded);
 244                 case info_flags::has_indices: {
 245                         if (flags & status_flags::has_indices)
 246                                 return true;
 247                         else if (flags & status_flags::has_no_indices)
 248                                 return false;
 249                         else if (basis.info(info_flags::has_indices)) {
 250                                 setflag(status_flags::has_indices);
 251                                 clearflag(status_flags::has_no_indices);
 252                                 return true;
 253                         } else {
 254                                 clearflag(status_flags::has_indices);
 255                                 setflag(status_flags::has_no_indices);
 256                                 return false;
 257                         }
 258                 }
 259         }
 260         return inherited::info(inf);
 261 }
 262
 263 size_t power::nops() const
 264 {
 265         return 2;
 266 }
 267
 268 ex power::op(size_t i) const
 269 {
 270         GINAC_ASSERT(i<2);
 271
 272         return i==0 ? basis : exponent;
 273 }
 274
 275 ex power::map(map_function & f) const
 276 {
 277         const ex &mapped_basis = f(basis);
 278         const ex &mapped_exponent = f(exponent);
 279
 280         if (!are_ex_trivially_equal(basis, mapped_basis)
 281          || !are_ex_trivially_equal(exponent, mapped_exponent))
 282                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 283         else
 284                 return *this;
 285 }
 286
 287 bool power::is_polynomial(const ex & var) const
 288 {
 289         if (exponent.has(var))
 290                 return false;
 291         if (!exponent.info(info_flags::nonnegint))
 292                 return false;
 293         return basis.is_polynomial(var);
 294 }
 295
 296 int power::degree(const ex & s) const
 297 {
 298         if (is_equal(ex_to<basic>(s)))
 299                 return 1;
 300         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 301                 if (basis.is_equal(s))
 302                         return ex_to<numeric>(exponent).to_int();
 303                 else
 304                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 305         } else if (basis.has(s))
 306                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 307         else
 308                 return 0;
 309 }
 310
 311 int power::ldegree(const ex & s) const
 312 {
 313         if (is_equal(ex_to<basic>(s)))
 314                 return 1;
 315         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 316                 if (basis.is_equal(s))
 317                         return ex_to<numeric>(exponent).to_int();
 318                 else
 319                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 320         } else if (basis.has(s))
 321                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 322         else
 323                 return 0;
 324 }
 325
 326 ex power::coeff(const ex & s, int n) const
 327 {
 328         if (is_equal(ex_to<basic>(s)))
 329                 return n==1 ? _ex1 : _ex0;
 330         else if (!basis.is_equal(s)) {
 331                 // basis not equal to s
 332                 if (n == 0)
 333                         return *this;
 334                 else
 335                         return _ex0;
 336         } else {
 337                 // basis equal to s
 338                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 339                         // integer exponent
 340                         int int_exp = ex_to<numeric>(exponent).to_int();
 341                         if (n == int_exp)
 342                                 return _ex1;
 343                         else
 344                                 return _ex0;
 345                 } else {
 346                         // non-integer exponents are treated as zero
 347                         if (n == 0)
 348                                 return *this;
 349                         else
 350                                 return _ex0;
 351                 }
 352         }
 353 }
 354
 355 /** Perform automatic term rewriting rules in this class.  In the following
 356  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 357  *  stand for such expressions that contain a plain number.
 358  *  - ^(x,0) -> 1  (also handles ^(0,0))
 359  *  - ^(x,1) -> x
 360  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 361  *  - ^(1,x) -> 1
 362  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 363  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 364  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 365  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 366  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 367  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 368  *
 369  *  @param level cut-off in recursive evaluation */
 370 ex power::eval(int level) const
 371 {
 372         if ((level==1) && (flags & status_flags::evaluated))
 373                 return *this;
 374         else if (level == -max_recursion_level)
 375                 throw(std::runtime_error("max recursion level reached"));
 376
 377         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 378         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 379
 380         bool basis_is_numerical = false;
 381         bool exponent_is_numerical = false;
 382         const numeric *num_basis;
 383         const numeric *num_exponent;
 384
 385         if (is_exactly_a<numeric>(ebasis)) {
 386                 basis_is_numerical = true;
 387                 num_basis = &ex_to<numeric>(ebasis);
 388         }
 389         if (is_exactly_a<numeric>(eexponent)) {
 390                 exponent_is_numerical = true;
 391                 num_exponent = &ex_to<numeric>(eexponent);
 392         }
 393
 394         // ^(x,0) -> 1  (0^0 also handled here)
 395         if (eexponent.is_zero()) {
 396                 if (ebasis.is_zero())
 397                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 398                 else
 399                         return _ex1;
 400         }
 401
 402         // ^(x,1) -> x
 403         if (eexponent.is_equal(_ex1))
 404                 return ebasis;
 405
 406         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 407         if (ebasis.is_zero() && exponent_is_numerical) {
 408                 if ((num_exponent->real()).is_zero())
 409                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 410                 else if ((num_exponent->real()).is_negative())
 411                         throw (pole_error("power::eval(): division by zero",1));
 412                 else
 413                         return _ex0;
 414         }
 415
 416         // ^(1,x) -> 1
 417         if (ebasis.is_equal(_ex1))
 418                 return _ex1;
 419
 420         // power of a function calculated by separate rules defined for this function
 421         if (is_exactly_a<function>(ebasis))
 422                 return ex_to<function>(ebasis).power(eexponent);
 423
 424         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 425         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 426                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 427
 428         if (exponent_is_numerical) {
 429
 430                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 431                 // except if c1,c2 are rational, but c1^c2 is not)
 432                 if (basis_is_numerical) {
 433                         const bool basis_is_crational = num_basis->is_crational();
 434                         const bool exponent_is_crational = num_exponent->is_crational();
 435                         if (!basis_is_crational || !exponent_is_crational) {
 436                                 // return a plain float
 437                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 438                                                                                                status_flags::evaluated |
 439                                                                                                status_flags::expanded);
 440                         }
 441
 442                         const numeric res = num_basis->power(*num_exponent);
 443                         if (res.is_crational()) {
 444                                 return res;
 445                         }
 446                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 447
 448                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 449                         if (basis_is_crational && exponent_is_crational
 450                             && num_exponent->is_real()
 451                             && !num_exponent->is_integer()) {
 452                                 const numeric n = num_exponent->numer();
 453                                 const numeric m = num_exponent->denom();
 454                                 numeric r;
 455                                 numeric q = iquo(n, m, r);
 456                                 if (r.is_negative()) {
 457                                         r += m;
 458                                         --q;
 459                                 }
 460                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 461                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 462                                                 // try it for numerator and denominator separately, in order to
 463                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 464                                                 const numeric bnum = num_basis->numer();
 465                                                 const numeric bden = num_basis->denom();
 466                                                 const numeric res_bnum = bnum.power(*num_exponent);
 467                                                 const numeric res_bden = bden.power(*num_exponent);
 468                                                 if (res_bnum.is_integer())
 469                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 470                                                 if (res_bden.is_integer())
 471                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 472                                         }
 473                                         return this->hold();
 474                                 } else {
 475                                         // assemble resulting product, but allowing for a re-evaluation,
 476                                         // because otherwise we'll end up with something like
 477                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 478                                         // instead of 7/16*7^(1/3).
 479                                         ex prod = power(*num_basis,r.div(m));
 480                                         return prod*power(*num_basis,q);
 481                                 }
 482                         }
 483                 }
 484
 485                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 486                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 487                 // case c1==1 should not happen, see below!)
 488                 if (is_exactly_a<power>(ebasis)) {
 489                         const power & sub_power = ex_to<power>(ebasis);
 490                         const ex & sub_basis = sub_power.basis;
 491                         const ex & sub_exponent = sub_power.exponent;
 492                         if (is_exactly_a<numeric>(sub_exponent)) {
 493                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 494                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 495                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
 496                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 497                                 }
 498                         }
 499                 }
 500
 501                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 502                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 503                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 504                 }
 505
 506                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 507                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 508                         numeric icont = ebasis.integer_content();
 509                         const numeric lead_coeff =
 510                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 511
 512                         const bool canonicalizable = lead_coeff.is_integer();
 513                         const bool unit_normal = lead_coeff.is_pos_integer();
 514                         if (canonicalizable && (! unit_normal))
 515                                 icont = icont.mul(*_num_1_p);
 516
 517                         if (canonicalizable && (icont != *_num1_p)) {
 518                                 const add& addref = ex_to<add>(ebasis);
 519                                 add* addp = new add(addref);
 520                                 addp->setflag(status_flags::dynallocated);
 521                                 addp->clearflag(status_flags::hash_calculated);
 522                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 523                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 524                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 525
 526                                 const numeric c = icont.power(*num_exponent);
 527                                 if (likely(c != *_num1_p))
 528                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 529                                 else
 530                                         return power(*addp, *num_exponent);
 531                         }
 532                 }
 533
 534                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 535                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 536                 if (is_exactly_a<mul>(ebasis)) {
 537                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 538                         const mul & mulref = ex_to<mul>(ebasis);
 539                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 540                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 541                                 if (num_coeff.is_real()) {
 542                                         if (num_coeff.is_positive()) {
 543                                                 mul *mulp = new mul(mulref);
 544                                                 mulp->overall_coeff = _ex1;
 545                                                 mulp->clearflag(status_flags::evaluated);
 546                                                 mulp->clearflag(status_flags::hash_calculated);
 547                                                 return (new mul(power(*mulp,exponent),
 548                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 549                                         } else {
 550                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 551                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 552                                                         mul *mulp = new mul(mulref);
 553                                                         mulp->overall_coeff = _ex_1;
 554                                                         mulp->clearflag(status_flags::evaluated);
 555                                                         mulp->clearflag(status_flags::hash_calculated);
 556                                                         return (new mul(power(*mulp,exponent),
 557                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 558                                                 }
 559                                         }
 560                                 }
 561                         }
 562                 }
 563
 564                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 565                 if (num_exponent->is_pos_integer() &&
 566                     ebasis.return_type() != return_types::commutative &&
 567                     !is_a<matrix>(ebasis)) {
 568                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 569                 }
 570         }
 571
 572         if (are_ex_trivially_equal(ebasis,basis) &&
 573             are_ex_trivially_equal(eexponent,exponent)) {
 574                 return this->hold();
 575         }
 576         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 577                                                        status_flags::evaluated);
 578 }
 579
 580 ex power::evalf(int level) const
 581 {
 582         ex ebasis;
 583         ex eexponent;
 584
 585         if (level==1) {
 586                 ebasis = basis;
 587                 eexponent = exponent;
 588         } else if (level == -max_recursion_level) {
 589                 throw(std::runtime_error("max recursion level reached"));
 590         } else {
 591                 ebasis = basis.evalf(level-1);
 592                 if (!is_exactly_a<numeric>(exponent))
 593                         eexponent = exponent.evalf(level-1);
 594                 else
 595                         eexponent = exponent;
 596         }
 597
 598         return power(ebasis,eexponent);
 599 }
 600
 601 ex power::evalm() const
 602 {
 603         const ex ebasis = basis.evalm();
 604         const ex eexponent = exponent.evalm();
 605         if (is_a<matrix>(ebasis)) {
 606                 if (is_exactly_a<numeric>(eexponent)) {
 607                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 608                 }
 609         }
 610         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 611 }
 612
 613 bool power::has(const ex & other, unsigned options) const
 614 {
 615         if (!(options & has_options::algebraic))
 616                 return basic::has(other, options);
 617         if (!is_a<power>(other))
 618                 return basic::has(other, options);
 619         if (!exponent.info(info_flags::integer)
 620                         || !other.op(1).info(info_flags::integer))
 621                 return basic::has(other, options);
 622         if (exponent.info(info_flags::posint)
 623                         && other.op(1).info(info_flags::posint)
 624                         && ex_to<numeric>(exponent).to_int()
 625                                         > ex_to<numeric>(other.op(1)).to_int()
 626                         && basis.match(other.op(0)))
 627                 return true;
 628         if (exponent.info(info_flags::negint)
 629                         && other.op(1).info(info_flags::negint)
 630                         && ex_to<numeric>(exponent).to_int()
 631                                         < ex_to<numeric>(other.op(1)).to_int()
 632                         && basis.match(other.op(0)))
 633                 return true;
 634         return basic::has(other, options);
 635 }
 636
 637 // from mul.cpp
 638 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 639
 640 ex power::subs(const exmap & m, unsigned options) const
 641 {
 642         const ex &subsed_basis = basis.subs(m, options);
 643         const ex &subsed_exponent = exponent.subs(m, options);
 644
 645         if (!are_ex_trivially_equal(basis, subsed_basis)
 646          || !are_ex_trivially_equal(exponent, subsed_exponent))
 647                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 648
 649         if (!(options & subs_options::algebraic))
 650                 return subs_one_level(m, options);
 651
 652         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 653                 int nummatches = std::numeric_limits<int>::max();
 654                 exmap repls;
 655                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 656                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 657                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 658                         ex result = (*this)*power(anum/aden, nummatches);
 659                         return (ex_to<basic>(result)).subs_one_level(m, options);
 660                 }
 661         }
 662
 663         return subs_one_level(m, options);
 664 }
 665
 666 ex power::eval_ncmul(const exvector & v) const
 667 {
 668         return inherited::eval_ncmul(v);
 669 }
 670
 671 ex power::conjugate() const
 672 {
 673         ex newbasis = basis.conjugate();
 674         ex newexponent = exponent.conjugate();
 675         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
 676                 return *this;
 677         }
 678         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
 679 }
 680
 681 ex power::real_part() const
 682 {
 683         if (exponent.info(info_flags::integer)) {
 684                 ex basis_real = basis.real_part();
 685                 if (basis_real == basis)
 686                         return *this;
 687                 realsymbol a("a"),b("b");
 688                 ex result;
 689                 if (exponent.info(info_flags::posint))
 690                         result = power(a+I*b,exponent);
 691                 else
 692                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 693                 result = result.expand();
 694                 result = result.real_part();
 695                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 696                 return result;
 697         }
 698
 699         ex a = basis.real_part();
 700         ex b = basis.imag_part();
 701         ex c = exponent.real_part();
 702         ex d = exponent.imag_part();
 703         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 704 }
 705
 706 ex power::imag_part() const
 707 {
 708         if (exponent.info(info_flags::integer)) {
 709                 ex basis_real = basis.real_part();
 710                 if (basis_real == basis)
 711                         return 0;
 712                 realsymbol a("a"),b("b");
 713                 ex result;
 714                 if (exponent.info(info_flags::posint))
 715                         result = power(a+I*b,exponent);
 716                 else
 717                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 718                 result = result.expand();
 719                 result = result.imag_part();
 720                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 721                 return result;
 722         }
 723
 724         ex a=basis.real_part();
 725         ex b=basis.imag_part();
 726         ex c=exponent.real_part();
 727         ex d=exponent.imag_part();
 728         return
 729                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 730 }
 731
 732 // protected
 733
 734 // protected
 735
 736 /** Implementation of ex::diff() for a power.
 737  *  @see ex::diff */
 738 ex power::derivative(const symbol & s) const
 739 {
 740         if (is_a<numeric>(exponent)) {
 741                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 742                 epvector newseq;
 743                 newseq.reserve(2);
 744                 newseq.push_back(expair(basis, exponent - _ex1));
 745                 newseq.push_back(expair(basis.diff(s), _ex1));
 746                 return mul(newseq, exponent);
 747         } else {
 748                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 749                 return mul(*this,
 750                            add(mul(exponent.diff(s), log(basis)),
 751                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 752         }
 753 }
 754
 755 int power::compare_same_type(const basic & other) const
 756 {
 757         GINAC_ASSERT(is_exactly_a<power>(other));
 758         const power &o = static_cast<const power &>(other);
 759
 760         int cmpval = basis.compare(o.basis);
 761         if (cmpval)
 762                 return cmpval;
 763         else
 764                 return exponent.compare(o.exponent);
 765 }
 766
 767 unsigned power::return_type() const
 768 {
 769         return basis.return_type();
 770 }
 771
 772 tinfo_t power::return_type_tinfo() const
 773 {
 774         return basis.return_type_tinfo();
 775 }
 776
 777 ex power::expand(unsigned options) const
 778 {
 779         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 780                 // A special case worth optimizing.
 781                 setflag(status_flags::expanded);
 782                 return *this;
 783         }
 784
 785         const ex expanded_basis = basis.expand(options);
 786         const ex expanded_exponent = exponent.expand(options);
 787
 788         // x^(a+b) -> x^a * x^b
 789         if (is_exactly_a<add>(expanded_exponent)) {
 790                 const add &a = ex_to<add>(expanded_exponent);
 791                 exvector distrseq;
 792                 distrseq.reserve(a.seq.size() + 1);
 793                 epvector::const_iterator last = a.seq.end();
 794                 epvector::const_iterator cit = a.seq.begin();
 795                 while (cit!=last) {
 796                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 797                         ++cit;
 798                 }
 799
 800                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 801                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 802                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 803                         int int_exponent = num_exponent.to_int();
 804                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 805                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 806                         else
 807                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 808                 } else
 809                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 810
 811                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 812                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 813                 return r.expand(options);
 814         }
 815
 816         if (!is_exactly_a<numeric>(expanded_exponent) ||
 817                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 818                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 819                         return this->hold();
 820                 } else {
 821                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 822                 }
 823         }
 824
 825         // integer numeric exponent
 826         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 827         int int_exponent = num_exponent.to_int();
 828
 829         // (x+y)^n, n>0
 830         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 831                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 832
 833         // (x*y)^n -> x^n * y^n
 834         if (is_exactly_a<mul>(expanded_basis))
 835                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 836
 837         // cannot expand further
 838         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 839                 return this->hold();
 840         else
 841                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 842 }
 843
 844 //////////
 845 // new virtual functions which can be overridden by derived classes
 846 //////////
 847
 848 // none
 849
 850 //////////
 851 // non-virtual functions in this class
 852 //////////
 853
 854 /** expand a^n where a is an add and n is a positive integer.
 855  *  @see power::expand */
 856 ex power::expand_add(const add & a, int n, unsigned options) const
 857 {
 858         if (n==2)
 859                 return expand_add_2(a, options);
 860
 861         const size_t m = a.nops();
 862         exvector result;
 863         // The number of terms will be the number of combinatorial compositions,
 864         // i.e. the number of unordered arrangements of m nonnegative integers
 865         // which sum up to n.  It is frequently written as C_n(m) and directly
 866         // related with binomial coefficients:
 867         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 868         intvector k(m-1);
 869         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 870         intvector upper_limit(m-1);
 871
 872         for (size_t l=0; l<m-1; ++l) {
 873                 k[l] = 0;
 874                 k_cum[l] = 0;
 875                 upper_limit[l] = n;
 876         }
 877
 878         while (true) {
 879                 exvector term;
 880                 term.reserve(m+1);
 881                 for (std::size_t l = 0; l < m - 1; ++l) {
 882                         const ex & b = a.op(l);
 883                         GINAC_ASSERT(!is_exactly_a<add>(b));
 884                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 885                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 886                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 887                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 888                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 889                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 890                         if (is_exactly_a<mul>(b))
 891                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 892                         else
 893                                 term.push_back(power(b,k[l]));
 894                 }
 895
 896                 const ex & b = a.op(m - 1);
 897                 GINAC_ASSERT(!is_exactly_a<add>(b));
 898                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 899                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 900                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 901                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 902                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 903                              !is_exactly_a<power>(ex_to<power>(b).basis));
 904                 if (is_exactly_a<mul>(b))
 905                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 906                 else
 907                         term.push_back(power(b,n-k_cum[m-2]));
 908
 909                 numeric f = binomial(numeric(n),numeric(k[0]));
 910                 for (std::size_t l = 1; l < m - 1; ++l)
 911                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 912
 913                 term.push_back(f);
 914
 915                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 916
 917                 // increment k[]
 918                 bool done = false;
 919                 std::size_t l = m - 2;
 920                 while ((++k[l]) > upper_limit[l]) {
 921                         k[l] = 0;
 922                         if (l != 0)
 923                                 --l;
 924                         else {
 925                                 done = true;
 926                                 break;
 927                         }
 928                 }
 929                 if (done)
 930                         break;
 931
 932                 // recalc k_cum[] and upper_limit[]
 933                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 934
 935                 for (size_t i=l+1; i<m-1; ++i)
 936                         k_cum[i] = k_cum[i-1]+k[i];
 937
 938                 for (size_t i=l+1; i<m-1; ++i)
 939                         upper_limit[i] = n-k_cum[i-1];
 940         }
 941
 942         return (new add(result))->setflag(status_flags::dynallocated |
 943                                           status_flags::expanded);
 944 }
 945
 946
 947 /** Special case of power::expand_add. Expands a^2 where a is an add.
 948  *  @see power::expand_add */
 949 ex power::expand_add_2(const add & a, unsigned options) const
 950 {
 951         epvector sum;
 952         size_t a_nops = a.nops();
 953         sum.reserve((a_nops*(a_nops+1))/2);
 954         epvector::const_iterator last = a.seq.end();
 955
 956         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 957         // first part: ignore overall_coeff and expand other terms
 958         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 959                 const ex & r = cit0->rest;
 960                 const ex & c = cit0->coeff;
 961
 962                 GINAC_ASSERT(!is_exactly_a<add>(r));
 963                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 964                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 965                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 966                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 967                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 968                              !is_exactly_a<power>(ex_to<power>(r).basis));
 969
 970                 if (c.is_equal(_ex1)) {
 971                         if (is_exactly_a<mul>(r)) {
 972                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 973                                                      _ex1));
 974                         } else {
 975                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 976                                                      _ex1));
 977                         }
 978                 } else {
 979                         if (is_exactly_a<mul>(r)) {
 980                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 981                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 982                         } else {
 983                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 984                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 985                         }
 986                 }
 987
 988                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 989                         const ex & r1 = cit1->rest;
 990                         const ex & c1 = cit1->coeff;
 991                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 992                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 993                 }
 994         }
 995
 996         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 997
 998         // second part: add terms coming from overall_factor (if != 0)
 999         if (!a.overall_coeff.is_zero()) {
1000                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1001                 while (i != end) {
1002                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1003                         ++i;
1004                 }
1005                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1006         }
1007
1008         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1009
1010         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1011 }
1012
1013 /** Expand factors of m in m^n where m is a mul and n is an integer.
1014  *  @see power::expand */
1015 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1016 {
1017         GINAC_ASSERT(n.is_integer());
1018
1019         if (n.is_zero()) {
1020                 return _ex1;
1021         }
1022
1023         // do not bother to rename indices if there are no any.
1024         if ((!(options & expand_options::expand_rename_idx))
1025                         && m.info(info_flags::has_indices))
1026                 options |= expand_options::expand_rename_idx;
1027         // Leave it to multiplication since dummy indices have to be renamed
1028         if ((options & expand_options::expand_rename_idx) &&
1029                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1030                 ex result = m;
1031                 exvector va = get_all_dummy_indices(m);
1032                 sort(va.begin(), va.end(), ex_is_less());
1033
1034                 for (int i=1; i < n.to_int(); i++)
1035                         result *= rename_dummy_indices_uniquely(va, m);
1036                 return result;
1037         }
1038
1039         epvector distrseq;
1040         distrseq.reserve(m.seq.size());
1041         bool need_reexpand = false;
1042
1043         epvector::const_iterator last = m.seq.end();
1044         epvector::const_iterator cit = m.seq.begin();
1045         while (cit!=last) {
1046                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1047                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1048                         // this happens when e.g. (a+b)^(1/2) gets squared and
1049                         // the resulting product needs to be reexpanded
1050                         need_reexpand = true;
1051                 }
1052                 distrseq.push_back(p);
1053                 ++cit;
1054         }
1055
1056         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1057         if (need_reexpand)
1058                 return ex(result).expand(options);
1059         if (from_expand)
1060                 return result.setflag(status_flags::expanded);
1061         return result;
1062 }
1063
1064 } // namespace GiNaC