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 "numeric.h"
  32 #include "inifcns.h"
  33 #include "relational.h"
  34 #include "symbol.h"
  35 #include "archive.h"
  36 #include "debugmsg.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  42
  43 typedef std::vector<int> intvector;
  44
  45 //////////
  46 // default ctor, dtor, copy ctor assignment operator and helpers
  47 //////////
  48
  49 // public
  50
  51 power::power() : basic(TINFO_power)
  52 {
  53         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
  54 }
  55
  56 // protected
  57
  58 void power::copy(const power & other)
  59 {
  60         inherited::copy(other);
  61         basis = other.basis;
  62         exponent = other.exponent;
  63 }
  64
  65 void power::destroy(bool call_parent)
  66 {
  67         if (call_parent) inherited::destroy(call_parent);
  68 }
  69
  70 //////////
  71 // other ctors
  72 //////////
  73
  74 // public
  75
  76 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  77 {
  78         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
  79         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  80 }
  81
  82 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  83 {
  84         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
  85         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  86 }
  87
  88 //////////
  89 // archiving
  90 //////////
  91
  92 /** Construct object from archive_node. */
  93 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  94 {
  95         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
  96         n.find_ex("basis", basis, sym_lst);
  97         n.find_ex("exponent", exponent, sym_lst);
  98 }
  99
 100 /** Unarchive the object. */
 101 ex power::unarchive(const archive_node &n, const lst &sym_lst)
 102 {
 103         return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
 104 }
 105
 106 /** Archive the object. */
 107 void power::archive(archive_node &n) const
 108 {
 109         inherited::archive(n);
 110         n.add_ex("basis", basis);
 111         n.add_ex("exponent", exponent);
 112 }
 113
 114 //////////
 115 // functions overriding virtual functions from bases classes
 116 //////////
 117
 118 // public
 119
 120 void power::print(std::ostream & os, unsigned upper_precedence) const
 121 {
 122         debugmsg("power print",LOGLEVEL_PRINT);
 123         if (exponent.is_equal(_ex1_2())) {
 124                 os << "sqrt(" << basis << ")";
 125         } else {
 126                 if (precedence<=upper_precedence) os << "(";
 127                 basis.print(os,precedence);
 128                 os << "^";
 129                 exponent.print(os,precedence);
 130                 if (precedence<=upper_precedence) os << ")";
 131         }
 132 }
 133
 134 void power::printraw(std::ostream & os) const
 135 {
 136         debugmsg("power printraw",LOGLEVEL_PRINT);
 137
 138         os << class_name() << "(";
 139         basis.printraw(os);
 140         os << ",";
 141         exponent.printraw(os);
 142         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
 143 }
 144
 145 void power::printtree(std::ostream & os, unsigned indent) const
 146 {
 147         debugmsg("power printtree",LOGLEVEL_PRINT);
 148
 149         os << std::string(indent,' ') << class_name()
 150            << ", hash=" << hashvalue
 151            << " (0x" << std::hex << hashvalue << std::dec << ")"
 152            << ", flags=" << flags << std::endl;
 153         basis.printtree(os, indent+delta_indent);
 154         exponent.printtree(os, indent+delta_indent);
 155 }
 156
 157 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
 158 {
 159         // Optimal output of integer powers of symbols to aid compiler CSE.
 160         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 161         // to learn why such a hack is really necessary.
 162         if (exp == 1) {
 163                 x.printcsrc(os, type, 0);
 164         } else if (exp == 2) {
 165                 x.printcsrc(os, type, 0);
 166                 os << "*";
 167                 x.printcsrc(os, type, 0);
 168         } else if (exp & 1) {
 169                 x.printcsrc(os, 0);
 170                 os << "*";
 171                 print_sym_pow(os, type, x, exp-1);
 172         } else {
 173                 os << "(";
 174                 print_sym_pow(os, type, x, exp >> 1);
 175                 os << ")*(";
 176                 print_sym_pow(os, type, x, exp >> 1);
 177                 os << ")";
 178         }
 179 }
 180
 181 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
 182 {
 183         debugmsg("power print csrc", LOGLEVEL_PRINT);
 184
 185         // Integer powers of symbols are printed in a special, optimized way
 186         if (exponent.info(info_flags::integer)
 187          && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
 188                 int exp = ex_to_numeric(exponent).to_int();
 189                 if (exp > 0)
 190                         os << "(";
 191                 else {
 192                         exp = -exp;
 193                         if (type == csrc_types::ctype_cl_N)
 194                                 os << "recip(";
 195                         else
 196                                 os << "1.0/(";
 197                 }
 198                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
 199                 os << ")";
 200
 201         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 202         } else if (exponent.compare(_num_1()) == 0) {
 203                 if (type == csrc_types::ctype_cl_N)
 204                         os << "recip(";
 205                 else
 206                         os << "1.0/(";
 207                 basis.bp->printcsrc(os, type, 0);
 208                 os << ")";
 209
 210         // Otherwise, use the pow() or expt() (CLN) functions
 211         } else {
 212                 if (type == csrc_types::ctype_cl_N)
 213                         os << "expt(";
 214                 else
 215                         os << "pow(";
 216                 basis.bp->printcsrc(os, type, 0);
 217                 os << ",";
 218                 exponent.bp->printcsrc(os, type, 0);
 219                 os << ")";
 220         }
 221 }
 222
 223 bool power::info(unsigned inf) const
 224 {
 225         switch (inf) {
 226                 case info_flags::polynomial:
 227                 case info_flags::integer_polynomial:
 228                 case info_flags::cinteger_polynomial:
 229                 case info_flags::rational_polynomial:
 230                 case info_flags::crational_polynomial:
 231                         return exponent.info(info_flags::nonnegint);
 232                 case info_flags::rational_function:
 233                         return exponent.info(info_flags::integer);
 234                 case info_flags::algebraic:
 235                         return (!exponent.info(info_flags::integer) ||
 236                                         basis.info(inf));
 237         }
 238         return inherited::info(inf);
 239 }
 240
 241 unsigned power::nops() const
 242 {
 243         return 2;
 244 }
 245
 246 ex & power::let_op(int i)
 247 {
 248         GINAC_ASSERT(i>=0);
 249         GINAC_ASSERT(i<2);
 250
 251         return i==0 ? basis : exponent;
 252 }
 253
 254 int power::degree(const symbol & s) const
 255 {
 256         if (is_exactly_of_type(*exponent.bp,numeric)) {
 257                 if ((*basis.bp).compare(s)==0) {
 258                         if (ex_to_numeric(exponent).is_integer())
 259                                 return ex_to_numeric(exponent).to_int();
 260                         else
 261                                 return 0;
 262                 } else
 263                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 264         }
 265         return 0;
 266 }
 267
 268 int power::ldegree(const symbol & s) const
 269 {
 270         if (is_exactly_of_type(*exponent.bp,numeric)) {
 271                 if ((*basis.bp).compare(s)==0) {
 272                         if (ex_to_numeric(exponent).is_integer())
 273                                 return ex_to_numeric(exponent).to_int();
 274                         else
 275                                 return 0;
 276                 } else
 277                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 278         }
 279         return 0;
 280 }
 281
 282 ex power::coeff(const symbol & s, int n) const
 283 {
 284         if ((*basis.bp).compare(s)!=0) {
 285                 // basis not equal to s
 286                 if (n == 0)
 287                         return *this;
 288                 else
 289                         return _ex0();
 290         } else {
 291                 // basis equal to s
 292                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 293                         // integer exponent
 294                         int int_exp = ex_to_numeric(exponent).to_int();
 295                         if (n == int_exp)
 296                                 return _ex1();
 297                         else
 298                                 return _ex0();
 299                 } else {
 300                         // non-integer exponents are treated as zero
 301                         if (n == 0)
 302                                 return *this;
 303                         else
 304                                 return _ex0();
 305                 }
 306         }
 307 }
 308
 309 ex power::eval(int level) const
 310 {
 311         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 312         //                  ^(x,1) -> x
 313         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 314         //                  ^(1,x) -> 1
 315         //                  ^(c1,c2) -> *(c1^n,c1^(c2-n)) (c1, c2 numeric(), 0<(c2-n)<1 except if c1,c2 are rational, but c1^c2 is not)
 316         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 317         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 318         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 319         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 320
 321         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 322
 323         if ((level==1) && (flags & status_flags::evaluated))
 324                 return *this;
 325         else if (level == -max_recursion_level)
 326                 throw(std::runtime_error("max recursion level reached"));
 327
 328         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 329         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 330
 331         bool basis_is_numerical = 0;
 332         bool exponent_is_numerical = 0;
 333         numeric * num_basis;
 334         numeric * num_exponent;
 335
 336         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 337                 basis_is_numerical = 1;
 338                 num_basis = static_cast<numeric *>(ebasis.bp);
 339         }
 340         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 341                 exponent_is_numerical = 1;
 342                 num_exponent = static_cast<numeric *>(eexponent.bp);
 343         }
 344
 345         // ^(x,0) -> 1 (0^0 also handled here)
 346         if (eexponent.is_zero()) {
 347                 if (ebasis.is_zero())
 348                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 349                 else
 350                         return _ex1();
 351         }
 352
 353         // ^(x,1) -> x
 354         if (eexponent.is_equal(_ex1()))
 355                 return ebasis;
 356
 357         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 358         if (ebasis.is_zero() && exponent_is_numerical) {
 359                 if ((num_exponent->real()).is_zero())
 360                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 361                 else if ((num_exponent->real()).is_negative())
 362                         throw (pole_error("power::eval(): division by zero",1));
 363                 else
 364                         return _ex0();
 365         }
 366
 367         // ^(1,x) -> 1
 368         if (ebasis.is_equal(_ex1()))
 369                 return _ex1();
 370
 371         if (basis_is_numerical && exponent_is_numerical) {
 372                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 373                 // except if c1,c2 are rational, but c1^c2 is not)
 374                 bool basis_is_crational = num_basis->is_crational();
 375                 bool exponent_is_crational = num_exponent->is_crational();
 376                 numeric res = num_basis->power(*num_exponent);
 377
 378                 if ((!basis_is_crational || !exponent_is_crational)
 379                         || res.is_crational()) {
 380                         return res;
 381                 }
 382                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 383                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 384                 if (basis_is_crational && exponent_is_crational
 385                         && num_exponent->is_real()
 386                         && !num_exponent->is_integer()) {
 387                         numeric n = num_exponent->numer();
 388                         numeric m = num_exponent->denom();
 389                         numeric r;
 390                         numeric q = iquo(n, m, r);
 391                         if (r.is_negative()) {
 392                                 r = r.add(m);
 393                                 q = q.sub(_num1());
 394                         }
 395                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 396                                 return this->hold();
 397                         else {
 398                                 epvector res;
 399                                 res.push_back(expair(ebasis,r.div(m)));
 400                                 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 401                         }
 402                 }
 403         }
 404
 405         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 406         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 407         // case c1==1 should not happen, see below!)
 408         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 409                 const power & sub_power = ex_to_power(ebasis);
 410                 const ex & sub_basis = sub_power.basis;
 411                 const ex & sub_exponent = sub_power.exponent;
 412                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 413                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 414                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 415                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
 416                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 417                 }
 418         }
 419
 420         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 421         if (exponent_is_numerical && num_exponent->is_integer() &&
 422                 is_ex_exactly_of_type(ebasis,mul)) {
 423                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 424         }
 425
 426         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 427         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 428         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 429                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 430                 const mul & mulref = ex_to_mul(ebasis);
 431                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 432                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
 433                         if (num_coeff.is_real()) {
 434                                 if (num_coeff.is_positive()) {
 435                                         mul * mulp = new mul(mulref);
 436                                         mulp->overall_coeff = _ex1();
 437                                         mulp->clearflag(status_flags::evaluated);
 438                                         mulp->clearflag(status_flags::hash_calculated);
 439                                         return (new mul(power(*mulp,exponent),
 440                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 441                                 } else {
 442                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 443                                         if (num_coeff.compare(_num_1())!=0) {
 444                                                 mul * mulp = new mul(mulref);
 445                                                 mulp->overall_coeff = _ex_1();
 446                                                 mulp->clearflag(status_flags::evaluated);
 447                                                 mulp->clearflag(status_flags::hash_calculated);
 448                                                 return (new mul(power(*mulp,exponent),
 449                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 450                                         }
 451                                 }
 452                         }
 453                 }
 454         }
 455
 456         if (are_ex_trivially_equal(ebasis,basis) &&
 457                 are_ex_trivially_equal(eexponent,exponent)) {
 458                 return this->hold();
 459         }
 460         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 461                                                                                                    status_flags::evaluated);
 462 }
 463
 464 ex power::evalf(int level) const
 465 {
 466         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 467
 468         ex ebasis;
 469         ex eexponent;
 470
 471         if (level==1) {
 472                 ebasis = basis;
 473                 eexponent = exponent;
 474         } else if (level == -max_recursion_level) {
 475                 throw(std::runtime_error("max recursion level reached"));
 476         } else {
 477                 ebasis = basis.evalf(level-1);
 478                 if (!is_ex_exactly_of_type(eexponent,numeric))
 479                         eexponent = exponent.evalf(level-1);
 480                 else
 481                         eexponent = exponent;
 482         }
 483
 484         return power(ebasis,eexponent);
 485 }
 486
 487 ex power::subs(const lst & ls, const lst & lr) const
 488 {
 489         const ex & subsed_basis=basis.subs(ls,lr);
 490         const ex & subsed_exponent=exponent.subs(ls,lr);
 491
 492         if (are_ex_trivially_equal(basis,subsed_basis)&&
 493                 are_ex_trivially_equal(exponent,subsed_exponent)) {
 494                 return *this;
 495         }
 496
 497         return power(subsed_basis, subsed_exponent);
 498 }
 499
 500 ex power::simplify_ncmul(const exvector & v) const
 501 {
 502         return inherited::simplify_ncmul(v);
 503 }
 504
 505 // protected
 506
 507 /** Implementation of ex::diff() for a power.
 508  *  @see ex::diff */
 509 ex power::derivative(const symbol & s) const
 510 {
 511         if (exponent.info(info_flags::real)) {
 512                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 513                 epvector newseq;
 514                 newseq.reserve(2);
 515                 newseq.push_back(expair(basis, exponent - _ex1()));
 516                 newseq.push_back(expair(basis.diff(s), _ex1()));
 517                 return mul(newseq, exponent);
 518         } else {
 519                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 520                 return mul(*this,
 521                            add(mul(exponent.diff(s), log(basis)),
 522                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 523         }
 524 }
 525
 526 int power::compare_same_type(const basic & other) const
 527 {
 528         GINAC_ASSERT(is_exactly_of_type(other, power));
 529         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 530
 531         int cmpval;
 532         cmpval=basis.compare(o.basis);
 533         if (cmpval==0) {
 534                 return exponent.compare(o.exponent);
 535         }
 536         return cmpval;
 537 }
 538
 539 unsigned power::return_type(void) const
 540 {
 541         return basis.return_type();
 542 }
 543
 544 unsigned power::return_type_tinfo(void) const
 545 {
 546         return basis.return_type_tinfo();
 547 }
 548
 549 ex power::expand(unsigned options) const
 550 {
 551         if (flags & status_flags::expanded)
 552                 return *this;
 553
 554         ex expanded_basis = basis.expand(options);
 555         ex expanded_exponent = exponent.expand(options);
 556
 557         // x^(a+b) -> x^a * x^b
 558         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 559                 const add &a = ex_to_add(expanded_exponent);
 560                 exvector distrseq;
 561                 distrseq.reserve(a.seq.size() + 1);
 562                 epvector::const_iterator last = a.seq.end();
 563                 epvector::const_iterator cit = a.seq.begin();
 564                 while (cit!=last) {
 565                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 566                         cit++;
 567                 }
 568
 569                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 570                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 571                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 572                         int int_exponent = num_exponent.to_int();
 573                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 574                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 575                         else
 576                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 577                 } else
 578                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 579
 580                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 581                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 582                 return r.expand();
 583         }
 584
 585         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 586                 !ex_to_numeric(expanded_exponent).is_integer()) {
 587                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 588                         return this->hold();
 589                 } else {
 590                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 591                 }
 592         }
 593
 594         // integer numeric exponent
 595         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 596         int int_exponent = num_exponent.to_int();
 597
 598         // (x+y)^n, n>0
 599         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 600                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 601
 602         // (x*y)^n -> x^n * y^n
 603         if (is_ex_exactly_of_type(expanded_basis,mul))
 604                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 605
 606         // cannot expand further
 607         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 608                 return this->hold();
 609         else
 610                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 611 }
 612
 613 //////////
 614 // new virtual functions which can be overridden by derived classes
 615 //////////
 616
 617 // none
 618
 619 //////////
 620 // non-virtual functions in this class
 621 //////////
 622
 623 /** expand a^n where a is an add and n is an integer.
 624  *  @see power::expand */
 625 ex power::expand_add(const add & a, int n) const
 626 {
 627         if (n==2)
 628                 return expand_add_2(a);
 629
 630         int m = a.nops();
 631         exvector sum;
 632         sum.reserve((n+1)*(m-1));
 633         intvector k(m-1);
 634         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 635         intvector upper_limit(m-1);
 636         int l;
 637
 638         for (int l=0; l<m-1; l++) {
 639                 k[l] = 0;
 640                 k_cum[l] = 0;
 641                 upper_limit[l] = n;
 642         }
 643
 644         while (1) {
 645                 exvector term;
 646                 term.reserve(m+1);
 647                 for (l=0; l<m-1; l++) {
 648                         const ex & b = a.op(l);
 649                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 650                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 651                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 652                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 653                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 654                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 655                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 656                         if (is_ex_exactly_of_type(b,mul))
 657                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 658                         else
 659                                 term.push_back(power(b,k[l]));
 660                 }
 661
 662                 const ex & b = a.op(l);
 663                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 664                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 665                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 666                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 667                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 668                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 669                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 670                 if (is_ex_exactly_of_type(b,mul))
 671                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 672                 else
 673                         term.push_back(power(b,n-k_cum[m-2]));
 674
 675                 numeric f = binomial(numeric(n),numeric(k[0]));
 676                 for (l=1; l<m-1; l++)
 677                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 678
 679                 term.push_back(f);
 680
 681                 /*
 682                 cout << "begin term" << endl;
 683                 for (int i=0; i<m-1; i++) {
 684                         cout << "k[" << i << "]=" << k[i] << endl;
 685                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 686                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 687                 }
 688                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 689                         cout << *cit << endl;
 690                 }
 691                 cout << "end term" << endl;
 692                 */
 693
 694                 // TODO: optimize this
 695                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 696
 697                 // increment k[]
 698                 l = m-2;
 699                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 700                         k[l] = 0;
 701                         l--;
 702                 }
 703                 if (l<0) break;
 704
 705                 // recalc k_cum[] and upper_limit[]
 706                 if (l==0)
 707                         k_cum[0] = k[0];
 708                 else
 709                         k_cum[l] = k_cum[l-1]+k[l];
 710
 711                 for (int i=l+1; i<m-1; i++)
 712                         k_cum[i] = k_cum[i-1]+k[i];
 713
 714                 for (int i=l+1; i<m-1; i++)
 715                         upper_limit[i] = n-k_cum[i-1];
 716         }
 717         return (new add(sum))->setflag(status_flags::dynallocated |
 718                                                                    status_flags::expanded );
 719 }
 720
 721
 722 /** Special case of power::expand_add. Expands a^2 where a is an add.
 723  *  @see power::expand_add */
 724 ex power::expand_add_2(const add & a) const
 725 {
 726         epvector sum;
 727         unsigned a_nops = a.nops();
 728         sum.reserve((a_nops*(a_nops+1))/2);
 729         epvector::const_iterator last = a.seq.end();
 730
 731         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 732         // first part: ignore overall_coeff and expand other terms
 733         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 734                 const ex & r = (*cit0).rest;
 735                 const ex & c = (*cit0).coeff;
 736
 737                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 738                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 739                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 740                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 741                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 742                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 743                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 744
 745                 if (are_ex_trivially_equal(c,_ex1())) {
 746                         if (is_ex_exactly_of_type(r,mul)) {
 747                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 748                                                      _ex1()));
 749                         } else {
 750                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 751                                                      _ex1()));
 752                         }
 753                 } else {
 754                         if (is_ex_exactly_of_type(r,mul)) {
 755                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 756                                                      ex_to_numeric(c).power_dyn(_num2())));
 757                         } else {
 758                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 759                                                      ex_to_numeric(c).power_dyn(_num2())));
 760                         }
 761                 }
 762
 763                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 764                         const ex & r1 = (*cit1).rest;
 765                         const ex & c1 = (*cit1).coeff;
 766                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 767                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 768                 }
 769         }
 770
 771         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 772
 773         // second part: add terms coming from overall_factor (if != 0)
 774         if (!a.overall_coeff.is_zero()) {
 775                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 776                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 777                 }
 778                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 779         }
 780
 781         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 782
 783         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 784 }
 785
 786 /** Expand factors of m in m^n where m is a mul and n is and integer
 787  *  @see power::expand */
 788 ex power::expand_mul(const mul & m, const numeric & n) const
 789 {
 790         if (n.is_zero())
 791                 return _ex1();
 792
 793         epvector distrseq;
 794         distrseq.reserve(m.seq.size());
 795         epvector::const_iterator last = m.seq.end();
 796         epvector::const_iterator cit = m.seq.begin();
 797         while (cit!=last) {
 798                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 799                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 800                 } else {
 801                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 802                         // since n is an integer
 803                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 804                 }
 805                 ++cit;
 806         }
 807         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 808 }
 809
 810 /*
 811 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 812                                unsigned options) const
 813 {
 814         // obsolete
 815
 816         exvector distrseq;
 817         epvector splitseq;
 818
 819         const add & addref=static_cast<const add &>(*basis.bp);
 820
 821         splitseq=addref.seq;
 822         splitseq.pop_back();
 823         ex first_operands=add(splitseq);
 824         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 825
 826         int n=exponent.to_int();
 827         for (int k=0; k<=n; k++) {
 828                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
 829                                                  * power(last_operand,numeric(n-k)));
 830         }
 831         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
 832 }
 833 */
 834
 835 /*
 836 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 837                                                                 unsigned options) const
 838 {
 839         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 840                         expand(options | expand_options::internal_do_not_expand_power_operands);
 841
 842         return ex(mul(rest_power,basis),0).
 843                expand(options | expand_options::internal_do_not_expand_mul_operands);
 844 }
 845 */
 846
 847 //////////
 848 // static member variables
 849 //////////
 850
 851 // protected
 852
 853 unsigned power::precedence = 60;
 854
 855 // helper function
 856
 857 ex sqrt(const ex & a)
 858 {
 859         return power(a,_ex1_2());
 860 }
 861
 862 } // namespace GiNaC