ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26
  27 #include "power.h"
  28 #include "expairseq.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "numeric.h"
  33 #include "constant.h"
  34 #include "inifcns.h" // for log() in power::derivative()
  35 #include "matrix.h"
  36 #include "symbol.h"
  37 #include "print.h"
  38 #include "archive.h"
  39 #include "debugmsg.h"
  40 #include "utils.h"
  41
  42 namespace GiNaC {
  43
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  45
  46 typedef std::vector<int> intvector;
  47
  48 //////////
  49 // default ctor, dtor, copy ctor assignment operator and helpers
  50 //////////
  51
  52 power::power() : inherited(TINFO_power)
  53 {
  54         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
  55 }
  56
  57 void power::copy(const power & other)
  58 {
  59         inherited::copy(other);
  60         basis = other.basis;
  61         exponent = other.exponent;
  62 }
  63
  64 DEFAULT_DESTROY(power)
  65
  66 //////////
  67 // other ctors
  68 //////////
  69
  70 // all inlined
  71
  72 //////////
  73 // archiving
  74 //////////
  75
  76 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  77 {
  78         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
  79         n.find_ex("basis", basis, sym_lst);
  80         n.find_ex("exponent", exponent, sym_lst);
  81 }
  82
  83 void power::archive(archive_node &n) const
  84 {
  85         inherited::archive(n);
  86         n.add_ex("basis", basis);
  87         n.add_ex("exponent", exponent);
  88 }
  89
  90 DEFAULT_UNARCHIVE(power)
  91
  92 //////////
  93 // functions overriding virtual functions from base classes
  94 //////////
  95
  96 // public
  97
  98 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
  99 {
 100         // Optimal output of integer powers of symbols to aid compiler CSE.
 101         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 102         // to learn why such a parenthisation is really necessary.
 103         if (exp == 1) {
 104                 x.print(c);
 105         } else if (exp == 2) {
 106                 x.print(c);
 107                 c.s << "*";
 108                 x.print(c);
 109         } else if (exp & 1) {
 110                 x.print(c);
 111                 c.s << "*";
 112                 print_sym_pow(c, x, exp-1);
 113         } else {
 114                 c.s << "(";
 115                 print_sym_pow(c, x, exp >> 1);
 116                 c.s << ")*(";
 117                 print_sym_pow(c, x, exp >> 1);
 118                 c.s << ")";
 119         }
 120 }
 121
 122 void power::print(const print_context & c, unsigned level) const
 123 {
 124         debugmsg("power print", LOGLEVEL_PRINT);
 125
 126         if (is_a<print_tree>(c)) {
 127
 128                 inherited::print(c, level);
 129
 130         } else if (is_a<print_csrc>(c)) {
 131
 132                 // Integer powers of symbols are printed in a special, optimized way
 133                 if (exponent.info(info_flags::integer)
 134                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
 135                         int exp = ex_to<numeric>(exponent).to_int();
 136                         if (exp > 0)
 137                                 c.s << '(';
 138                         else {
 139                                 exp = -exp;
 140                                 if (is_a<print_csrc_cl_N>(c))
 141                                         c.s << "recip(";
 142                                 else
 143                                         c.s << "1.0/(";
 144                         }
 145                         print_sym_pow(c, ex_to<symbol>(basis), exp);
 146                         c.s << ')';
 147
 148                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 149                 } else if (exponent.compare(_num_1()) == 0) {
 150                         if (is_a<print_csrc_cl_N>(c))
 151                                 c.s << "recip(";
 152                         else
 153                                 c.s << "1.0/(";
 154                         basis.print(c);
 155                         c.s << ')';
 156
 157                 // Otherwise, use the pow() or expt() (CLN) functions
 158                 } else {
 159                         if (is_a<print_csrc_cl_N>(c))
 160                                 c.s << "expt(";
 161                         else
 162                                 c.s << "pow(";
 163                         basis.print(c);
 164                         c.s << ',';
 165                         exponent.print(c);
 166                         c.s << ')';
 167                 }
 168
 169         } else {
 170
 171                 if (exponent.is_equal(_ex1_2())) {
 172                         if (is_a<print_latex>(c))
 173                                 c.s << "\\sqrt{";
 174                         else
 175                                 c.s << "sqrt(";
 176                         basis.print(c);
 177                         if (is_a<print_latex>(c))
 178                                 c.s << '}';
 179                         else
 180                                 c.s << ')';
 181                 } else {
 182                         if (precedence() <= level) {
 183                                 if (is_a<print_latex>(c))
 184                                         c.s << "{(";
 185                                 else
 186                                         c.s << "(";
 187                         }
 188                         basis.print(c, precedence());
 189                         c.s << '^';
 190                         if (is_a<print_latex>(c))
 191                                 c.s << '{';
 192                         exponent.print(c, precedence());
 193                         if (is_a<print_latex>(c))
 194                                 c.s << '}';
 195                         if (precedence() <= level) {
 196                                 if (is_a<print_latex>(c))
 197                                         c.s << ")}";
 198                                 else
 199                                         c.s << ')';
 200                         }
 201                 }
 202         }
 203 }
 204
 205 bool power::info(unsigned inf) const
 206 {
 207         switch (inf) {
 208                 case info_flags::polynomial:
 209                 case info_flags::integer_polynomial:
 210                 case info_flags::cinteger_polynomial:
 211                 case info_flags::rational_polynomial:
 212                 case info_flags::crational_polynomial:
 213                         return exponent.info(info_flags::nonnegint);
 214                 case info_flags::rational_function:
 215                         return exponent.info(info_flags::integer);
 216                 case info_flags::algebraic:
 217                         return (!exponent.info(info_flags::integer) ||
 218                                         basis.info(inf));
 219         }
 220         return inherited::info(inf);
 221 }
 222
 223 unsigned power::nops() const
 224 {
 225         return 2;
 226 }
 227
 228 ex & power::let_op(int i)
 229 {
 230         GINAC_ASSERT(i>=0);
 231         GINAC_ASSERT(i<2);
 232
 233         return i==0 ? basis : exponent;
 234 }
 235
 236 ex power::map(map_function & f) const
 237 {
 238         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 239 }
 240
 241 int power::degree(const ex & s) const
 242 {
 243         if (is_exactly_of_type(*exponent.bp,numeric)) {
 244                 if (basis.is_equal(s)) {
 245                         if (ex_to<numeric>(exponent).is_integer())
 246                                 return ex_to<numeric>(exponent).to_int();
 247                         else
 248                                 return 0;
 249                 } else
 250                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 251         }
 252         return 0;
 253 }
 254
 255 int power::ldegree(const ex & s) const
 256 {
 257         if (is_exactly_of_type(*exponent.bp,numeric)) {
 258                 if (basis.is_equal(s)) {
 259                         if (ex_to<numeric>(exponent).is_integer())
 260                                 return ex_to<numeric>(exponent).to_int();
 261                         else
 262                                 return 0;
 263                 } else
 264                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 265         }
 266         return 0;
 267 }
 268
 269 ex power::coeff(const ex & s, int n) const
 270 {
 271         if (!basis.is_equal(s)) {
 272                 // basis not equal to s
 273                 if (n == 0)
 274                         return *this;
 275                 else
 276                         return _ex0();
 277         } else {
 278                 // basis equal to s
 279                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
 280                         // integer exponent
 281                         int int_exp = ex_to<numeric>(exponent).to_int();
 282                         if (n == int_exp)
 283                                 return _ex1();
 284                         else
 285                                 return _ex0();
 286                 } else {
 287                         // non-integer exponents are treated as zero
 288                         if (n == 0)
 289                                 return *this;
 290                         else
 291                                 return _ex0();
 292                 }
 293         }
 294 }
 295
 296 /** Perform automatic term rewriting rules in this class.  In the following
 297  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 298  *  stand for such expressions that contain a plain number.
 299  *  - ^(x,0) -> 1  (also handles ^(0,0))
 300  *  - ^(x,1) -> x
 301  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 302  *  - ^(1,x) -> 1
 303  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 304  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 305  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 306  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 307  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 308  *
 309  *  @param level cut-off in recursive evaluation */
 310 ex power::eval(int level) const
 311 {
 312         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 313
 314         if ((level==1) && (flags & status_flags::evaluated))
 315                 return *this;
 316         else if (level == -max_recursion_level)
 317                 throw(std::runtime_error("max recursion level reached"));
 318
 319         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 320         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 321
 322         bool basis_is_numerical = false;
 323         bool exponent_is_numerical = false;
 324         const numeric *num_basis;
 325         const numeric *num_exponent;
 326
 327         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 328                 basis_is_numerical = true;
 329                 num_basis = static_cast<const numeric *>(ebasis.bp);
 330         }
 331         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 332                 exponent_is_numerical = true;
 333                 num_exponent = static_cast<const numeric *>(eexponent.bp);
 334         }
 335
 336         // ^(x,0) -> 1  (0^0 also handled here)
 337         if (eexponent.is_zero()) {
 338                 if (ebasis.is_zero())
 339                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 340                 else
 341                         return _ex1();
 342         }
 343
 344         // ^(x,1) -> x
 345         if (eexponent.is_equal(_ex1()))
 346                 return ebasis;
 347
 348         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 349         if (ebasis.is_zero() && exponent_is_numerical) {
 350                 if ((num_exponent->real()).is_zero())
 351                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 352                 else if ((num_exponent->real()).is_negative())
 353                         throw (pole_error("power::eval(): division by zero",1));
 354                 else
 355                         return _ex0();
 356         }
 357
 358         // ^(1,x) -> 1
 359         if (ebasis.is_equal(_ex1()))
 360                 return _ex1();
 361
 362         if (exponent_is_numerical) {
 363
 364                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 365                 // except if c1,c2 are rational, but c1^c2 is not)
 366                 if (basis_is_numerical) {
 367                         const bool basis_is_crational = num_basis->is_crational();
 368                         const bool exponent_is_crational = num_exponent->is_crational();
 369                         if (!basis_is_crational || !exponent_is_crational) {
 370                                 // return a plain float
 371                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 372                                                                                                status_flags::evaluated |
 373                                                                                                status_flags::expanded);
 374                         }
 375
 376                         const numeric res = num_basis->power(*num_exponent);
 377                         if (res.is_crational()) {
 378                                 return res;
 379                         }
 380                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 381
 382                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 383                         if (basis_is_crational && exponent_is_crational
 384                             && num_exponent->is_real()
 385                             && !num_exponent->is_integer()) {
 386                                 const numeric n = num_exponent->numer();
 387                                 const numeric m = num_exponent->denom();
 388                                 numeric r;
 389                                 numeric q = iquo(n, m, r);
 390                                 if (r.is_negative()) {
 391                                         r += m;
 392                                         --q;
 393                                 }
 394                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 395                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 396                                                 // try it for numerator and denominator separately, in order to
 397                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 398                                                 const numeric bnum = num_basis->numer();
 399                                                 const numeric bden = num_basis->denom();
 400                                                 const numeric res_bnum = bnum.power(*num_exponent);
 401                                                 const numeric res_bden = bden.power(*num_exponent);
 402                                                 if (res_bnum.is_integer())
 403                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 404                                                 if (res_bden.is_integer())
 405                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 406                                         }
 407                                         return this->hold();
 408                                 } else {
 409                                         // assemble resulting product, but allowing for a re-evaluation,
 410                                         // because otherwise we'll end up with something like
 411                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 412                                         // instead of 7/16*7^(1/3).
 413                                         ex prod = power(*num_basis,r.div(m));
 414                                         return prod*power(*num_basis,q);
 415                                 }
 416                         }
 417                 }
 418
 419                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 420                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 421                 // case c1==1 should not happen, see below!)
 422                 if (is_ex_exactly_of_type(ebasis,power)) {
 423                         const power & sub_power = ex_to<power>(ebasis);
 424                         const ex & sub_basis = sub_power.basis;
 425                         const ex & sub_exponent = sub_power.exponent;
 426                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 427                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 428                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 429                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
 430                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 431                         }
 432                 }
 433
 434                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 435                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
 436                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 437                 }
 438
 439                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 440                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 441                 if (is_ex_exactly_of_type(ebasis,mul)) {
 442                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 443                         const mul & mulref = ex_to<mul>(ebasis);
 444                         if (!mulref.overall_coeff.is_equal(_ex1())) {
 445                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 446                                 if (num_coeff.is_real()) {
 447                                         if (num_coeff.is_positive()) {
 448                                                 mul *mulp = new mul(mulref);
 449                                                 mulp->overall_coeff = _ex1();
 450                                                 mulp->clearflag(status_flags::evaluated);
 451                                                 mulp->clearflag(status_flags::hash_calculated);
 452                                                 return (new mul(power(*mulp,exponent),
 453                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 454                                         } else {
 455                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
 456                                                 if (num_coeff.compare(_num_1())!=0) {
 457                                                         mul *mulp = new mul(mulref);
 458                                                         mulp->overall_coeff = _ex_1();
 459                                                         mulp->clearflag(status_flags::evaluated);
 460                                                         mulp->clearflag(status_flags::hash_calculated);
 461                                                         return (new mul(power(*mulp,exponent),
 462                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 463                                                 }
 464                                         }
 465                                 }
 466                         }
 467                 }
 468
 469                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 470                 if (num_exponent->is_pos_integer() &&
 471                     ebasis.return_type() != return_types::commutative &&
 472                     !is_ex_of_type(ebasis,matrix)) {
 473                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 474                 }
 475         }
 476
 477         if (are_ex_trivially_equal(ebasis,basis) &&
 478             are_ex_trivially_equal(eexponent,exponent)) {
 479                 return this->hold();
 480         }
 481         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 482                                                        status_flags::evaluated);
 483 }
 484
 485 ex power::evalf(int level) const
 486 {
 487         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 488
 489         ex ebasis;
 490         ex eexponent;
 491
 492         if (level==1) {
 493                 ebasis = basis;
 494                 eexponent = exponent;
 495         } else if (level == -max_recursion_level) {
 496                 throw(std::runtime_error("max recursion level reached"));
 497         } else {
 498                 ebasis = basis.evalf(level-1);
 499                 if (!is_ex_exactly_of_type(eexponent,numeric))
 500                         eexponent = exponent.evalf(level-1);
 501                 else
 502                         eexponent = exponent;
 503         }
 504
 505         return power(ebasis,eexponent);
 506 }
 507
 508 ex power::evalm(void) const
 509 {
 510         const ex ebasis = basis.evalm();
 511         const ex eexponent = exponent.evalm();
 512         if (is_ex_of_type(ebasis,matrix)) {
 513                 if (is_ex_of_type(eexponent,numeric)) {
 514                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 515                 }
 516         }
 517         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 518 }
 519
 520 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 521 {
 522         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 523         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 524
 525         if (are_ex_trivially_equal(basis, subsed_basis)
 526          && are_ex_trivially_equal(exponent, subsed_exponent))
 527                 return basic::subs(ls, lr, no_pattern);
 528         else
 529                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
 530 }
 531
 532 ex power::simplify_ncmul(const exvector & v) const
 533 {
 534         return inherited::simplify_ncmul(v);
 535 }
 536
 537 // protected
 538
 539 /** Implementation of ex::diff() for a power.
 540  *  @see ex::diff */
 541 ex power::derivative(const symbol & s) const
 542 {
 543         if (exponent.info(info_flags::real)) {
 544                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 545                 epvector newseq;
 546                 newseq.reserve(2);
 547                 newseq.push_back(expair(basis, exponent - _ex1()));
 548                 newseq.push_back(expair(basis.diff(s), _ex1()));
 549                 return mul(newseq, exponent);
 550         } else {
 551                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 552                 return mul(*this,
 553                            add(mul(exponent.diff(s), log(basis)),
 554                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 555         }
 556 }
 557
 558 int power::compare_same_type(const basic & other) const
 559 {
 560         GINAC_ASSERT(is_exactly_of_type(other, power));
 561         const power &o = static_cast<const power &>(other);
 562
 563         int cmpval = basis.compare(o.basis);
 564         if (cmpval)
 565                 return cmpval;
 566         else
 567                 return exponent.compare(o.exponent);
 568 }
 569
 570 unsigned power::return_type(void) const
 571 {
 572         return basis.return_type();
 573 }
 574
 575 unsigned power::return_type_tinfo(void) const
 576 {
 577         return basis.return_type_tinfo();
 578 }
 579
 580 ex power::expand(unsigned options) const
 581 {
 582         if (options == 0 && (flags & status_flags::expanded))
 583                 return *this;
 584
 585         const ex expanded_basis = basis.expand(options);
 586         const ex expanded_exponent = exponent.expand(options);
 587
 588         // x^(a+b) -> x^a * x^b
 589         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 590                 const add &a = ex_to<add>(expanded_exponent);
 591                 exvector distrseq;
 592                 distrseq.reserve(a.seq.size() + 1);
 593                 epvector::const_iterator last = a.seq.end();
 594                 epvector::const_iterator cit = a.seq.begin();
 595                 while (cit!=last) {
 596                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 597                         ++cit;
 598                 }
 599
 600                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 601                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 602                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 603                         int int_exponent = num_exponent.to_int();
 604                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 605                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 606                         else
 607                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 608                 } else
 609                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 610
 611                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 612                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 613                 return r.expand();
 614         }
 615
 616         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 617                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 618                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 619                         return this->hold();
 620                 } else {
 621                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 622                 }
 623         }
 624
 625         // integer numeric exponent
 626         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 627         int int_exponent = num_exponent.to_int();
 628
 629         // (x+y)^n, n>0
 630         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 631                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 632
 633         // (x*y)^n -> x^n * y^n
 634         if (is_ex_exactly_of_type(expanded_basis,mul))
 635                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 636
 637         // cannot expand further
 638         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 639                 return this->hold();
 640         else
 641                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 642 }
 643
 644 //////////
 645 // new virtual functions which can be overridden by derived classes
 646 //////////
 647
 648 // none
 649
 650 //////////
 651 // non-virtual functions in this class
 652 //////////
 653
 654 /** expand a^n where a is an add and n is an integer.
 655  *  @see power::expand */
 656 ex power::expand_add(const add & a, int n) const
 657 {
 658         if (n==2)
 659                 return expand_add_2(a);
 660
 661         int m = a.nops();
 662         exvector sum;
 663         sum.reserve((n+1)*(m-1));
 664         intvector k(m-1);
 665         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 666         intvector upper_limit(m-1);
 667         int l;
 668
 669         for (int l=0; l<m-1; l++) {
 670                 k[l] = 0;
 671                 k_cum[l] = 0;
 672                 upper_limit[l] = n;
 673         }
 674
 675         while (true) {
 676                 exvector term;
 677                 term.reserve(m+1);
 678                 for (l=0; l<m-1; l++) {
 679                         const ex & b = a.op(l);
 680                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 681                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 682                                      !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
 683                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 684                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
 685                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
 686                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
 687                         if (is_ex_exactly_of_type(b,mul))
 688                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 689                         else
 690                                 term.push_back(power(b,k[l]));
 691                 }
 692
 693                 const ex & b = a.op(l);
 694                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 695                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 696                              !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
 697                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 698                              !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
 699                              !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
 700                              !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
 701                 if (is_ex_exactly_of_type(b,mul))
 702                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 703                 else
 704                         term.push_back(power(b,n-k_cum[m-2]));
 705
 706                 numeric f = binomial(numeric(n),numeric(k[0]));
 707                 for (l=1; l<m-1; l++)
 708                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 709
 710                 term.push_back(f);
 711
 712                 // TODO: Can we optimize this?  Alex seemed to think so...
 713                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 714
 715                 // increment k[]
 716                 l = m-2;
 717                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 718                         k[l] = 0;
 719                         --l;
 720                 }
 721                 if (l<0) break;
 722
 723                 // recalc k_cum[] and upper_limit[]
 724                 if (l==0)
 725                         k_cum[0] = k[0];
 726                 else
 727                         k_cum[l] = k_cum[l-1]+k[l];
 728
 729                 for (int i=l+1; i<m-1; i++)
 730                         k_cum[i] = k_cum[i-1]+k[i];
 731
 732                 for (int i=l+1; i<m-1; i++)
 733                         upper_limit[i] = n-k_cum[i-1];
 734         }
 735         return (new add(sum))->setflag(status_flags::dynallocated |
 736                                        status_flags::expanded );
 737 }
 738
 739
 740 /** Special case of power::expand_add. Expands a^2 where a is an add.
 741  *  @see power::expand_add */
 742 ex power::expand_add_2(const add & a) const
 743 {
 744         epvector sum;
 745         unsigned a_nops = a.nops();
 746         sum.reserve((a_nops*(a_nops+1))/2);
 747         epvector::const_iterator last = a.seq.end();
 748
 749         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 750         // first part: ignore overall_coeff and expand other terms
 751         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 752                 const ex & r = cit0->rest;
 753                 const ex & c = cit0->coeff;
 754
 755                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 756                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 757                              !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
 758                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 759                              !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
 760                              !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
 761                              !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
 762
 763                 if (are_ex_trivially_equal(c,_ex1())) {
 764                         if (is_ex_exactly_of_type(r,mul)) {
 765                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
 766                                                      _ex1()));
 767                         } else {
 768                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 769                                                      _ex1()));
 770                         }
 771                 } else {
 772                         if (is_ex_exactly_of_type(r,mul)) {
 773                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
 774                                                      ex_to<numeric>(c).power_dyn(_num2())));
 775                         } else {
 776                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 777                                                      ex_to<numeric>(c).power_dyn(_num2())));
 778                         }
 779                 }
 780
 781                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 782                         const ex & r1 = cit1->rest;
 783                         const ex & c1 = cit1->coeff;
 784                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 785                                                                       _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 786                 }
 787         }
 788
 789         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 790
 791         // second part: add terms coming from overall_factor (if != 0)
 792         if (!a.overall_coeff.is_zero()) {
 793                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 794                 while (i != end) {
 795                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
 796                         ++i;
 797                 }
 798                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
 799         }
 800
 801         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 802
 803         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 804 }
 805
 806 /** Expand factors of m in m^n where m is a mul and n is and integer
 807  *  @see power::expand */
 808 ex power::expand_mul(const mul & m, const numeric & n) const
 809 {
 810         if (n.is_zero())
 811                 return _ex1();
 812
 813         epvector distrseq;
 814         distrseq.reserve(m.seq.size());
 815         epvector::const_iterator last = m.seq.end();
 816         epvector::const_iterator cit = m.seq.begin();
 817         while (cit!=last) {
 818                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 819                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 820                 } else {
 821                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 822                         // since n is an integer
 823                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
 824                 }
 825                 ++cit;
 826         }
 827         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 828 }
 829
 830 // helper function
 831
 832 ex sqrt(const ex & a)
 833 {
 834         return power(a,_ex1_2());
 835 }
 836
 837 } // namespace GiNaC