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