ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999 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 "relational.h"
  33 #include "symbol.h"
  34 #include "debugmsg.h"
  35
  36 namespace GiNaC {
  37
  38 typedef vector<int> intvector;
  39
  40 //////////
  41 // default constructor, destructor, copy constructor assignment operator and helpers
  42 //////////
  43
  44 // public
  45
  46 power::power() : basic(TINFO_power)
  47 {
  48     debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
  49 }
  50
  51 power::~power()
  52 {
  53     debugmsg("power destructor",LOGLEVEL_DESTRUCT);
  54     destroy(0);
  55 }
  56
  57 power::power(power const & other)
  58 {
  59     debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT);
  60     copy(other);
  61 }
  62
  63 power const & power::operator=(power const & other)
  64 {
  65     debugmsg("power operator=",LOGLEVEL_ASSIGNMENT);
  66     if (this != &other) {
  67         destroy(1);
  68         copy(other);
  69     }
  70     return *this;
  71 }
  72
  73 // protected
  74
  75 void power::copy(power const & other)
  76 {
  77     basic::copy(other);
  78     basis=other.basis;
  79     exponent=other.exponent;
  80 }
  81
  82 void power::destroy(bool call_parent)
  83 {
  84     if (call_parent) basic::destroy(call_parent);
  85 }
  86
  87 //////////
  88 // other constructors
  89 //////////
  90
  91 // public
  92
  93 power::power(ex const & lh, ex const & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  94 {
  95     debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
  96     ASSERT(basis.return_type()==return_types::commutative);
  97 }
  98
  99 power::power(ex const & lh, numeric const & rh) : basic(TINFO_power), basis(lh), exponent(rh)
 100 {
 101     debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
 102     ASSERT(basis.return_type()==return_types::commutative);
 103 }
 104
 105 //////////
 106 // functions overriding virtual functions from bases classes
 107 //////////
 108
 109 // public
 110
 111 basic * power::duplicate() const
 112 {
 113     debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
 114     return new power(*this);
 115 }
 116
 117 bool power::info(unsigned inf) const
 118 {
 119     if (inf==info_flags::polynomial || inf==info_flags::integer_polynomial || inf==info_flags::rational_polynomial) {
 120         return exponent.info(info_flags::nonnegint);
 121     } else if (inf==info_flags::rational_function) {
 122         return exponent.info(info_flags::integer);
 123     } else {
 124         return basic::info(inf);
 125     }
 126 }
 127
 128 int power::nops() const
 129 {
 130     return 2;
 131 }
 132
 133 ex & power::let_op(int const i)
 134 {
 135     ASSERT(i>=0);
 136     ASSERT(i<2);
 137
 138     return i==0 ? basis : exponent;
 139 }
 140
 141 int power::degree(symbol const & s) const
 142 {
 143     if (is_exactly_of_type(*exponent.bp,numeric)) {
 144         if ((*basis.bp).compare(s)==0)
 145             return ex_to_numeric(exponent).to_int();
 146         else
 147             return basis.degree(s) * ex_to_numeric(exponent).to_int();
 148     }
 149     return 0;
 150 }
 151
 152 int power::ldegree(symbol const & s) const
 153 {
 154     if (is_exactly_of_type(*exponent.bp,numeric)) {
 155         if ((*basis.bp).compare(s)==0)
 156             return ex_to_numeric(exponent).to_int();
 157         else
 158             return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 159     }
 160     return 0;
 161 }
 162
 163 ex power::coeff(symbol const & s, int const n) const
 164 {
 165     if ((*basis.bp).compare(s)!=0) {
 166         // basis not equal to s
 167         if (n==0) {
 168             return *this;
 169         } else {
 170             return exZERO();
 171         }
 172     } else if (is_exactly_of_type(*exponent.bp,numeric)&&
 173                (static_cast<numeric const &>(*exponent.bp).compare(numeric(n))==0)) {
 174         return exONE();
 175     }
 176
 177     return exZERO();
 178 }
 179
 180 ex power::eval(int level) const
 181 {
 182     // simplifications: ^(x,0) -> 1 (0^0 handled here)
 183     //                  ^(x,1) -> x
 184     //                  ^(0,x) -> 0 (except if x is real and negative, in which case an exception is thrown)
 185     //                  ^(1,x) -> 1
 186     //                  ^(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)
 187     //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 188     //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 189     //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 190     //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 191
 192     debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 193
 194     if ((level==1)&&(flags & status_flags::evaluated)) {
 195         return *this;
 196     } else if (level == -max_recursion_level) {
 197         throw(std::runtime_error("max recursion level reached"));
 198     }
 199
 200     ex const & ebasis    = level==1 ? basis    : basis.eval(level-1);
 201     ex const & eexponent = level==1 ? exponent : exponent.eval(level-1);
 202
 203     bool basis_is_numerical=0;
 204     bool exponent_is_numerical=0;
 205     numeric * num_basis;
 206     numeric * num_exponent;
 207
 208     if (is_exactly_of_type(*ebasis.bp,numeric)) {
 209         basis_is_numerical=1;
 210         num_basis=static_cast<numeric *>(ebasis.bp);
 211     }
 212     if (is_exactly_of_type(*eexponent.bp,numeric)) {
 213         exponent_is_numerical=1;
 214         num_exponent=static_cast<numeric *>(eexponent.bp);
 215     }
 216
 217     // ^(x,0) -> 1 (0^0 also handled here)
 218     if (eexponent.is_zero())
 219         return exONE();
 220
 221     // ^(x,1) -> x
 222     if (eexponent.is_equal(exONE()))
 223         return ebasis;
 224
 225     // ^(0,x) -> 0 (except if x is real and negative)
 226     if (ebasis.is_zero()) {
 227         if (exponent_is_numerical && num_exponent->is_negative()) {
 228             throw(std::overflow_error("power::eval(): division by zero"));
 229         } else
 230             return exZERO();
 231     }
 232
 233     // ^(1,x) -> 1
 234     if (ebasis.is_equal(exONE()))
 235         return exONE();
 236
 237     if (basis_is_numerical && exponent_is_numerical) {
 238         // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 239         // except if c1,c2 are rational, but c1^c2 is not)
 240         bool basis_is_rational = num_basis->is_rational();
 241         bool exponent_is_rational = num_exponent->is_rational();
 242         numeric res = (*num_basis).power(*num_exponent);
 243
 244         if ((!basis_is_rational || !exponent_is_rational)
 245             || res.is_rational()) {
 246             return res;
 247         }
 248         ASSERT(!num_exponent->is_integer());  // has been handled by now
 249         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 250         if (basis_is_rational && exponent_is_rational
 251             && num_exponent->is_real()
 252             && !num_exponent->is_integer()) {
 253             numeric r, q, n, m;
 254             n = num_exponent->numer();
 255             m = num_exponent->denom();
 256             q = iquo(n, m, r);
 257             if (r.is_negative()) {
 258                 r = r.add(m);
 259                 q = q.sub(numONE());
 260             }
 261             if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 262                 return this->hold();
 263             else {
 264                 epvector res(2);
 265                 res.push_back(expair(ebasis,r.div(m)));
 266                 res.push_back(expair(ex(num_basis->power(q)),exONE()));
 267                 return (new mul(res))->setflag(status_flags::dynallocated | status_flags::evaluated);
 268                 /*return mul(num_basis->power(q),
 269                            power(ex(*num_basis),ex(r.div(m)))).hold();
 270                 */
 271                 /* return (new mul(num_basis->power(q),
 272                    power(*num_basis,r.div(m)).hold()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 273                 */
 274             }
 275         }
 276     }
 277
 278     // ^(^(x,c1),c2) -> ^(x,c1*c2)
 279     // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 280     // case c1=1 should not happen, see below!)
 281     if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 282         power const & sub_power=ex_to_power(ebasis);
 283         ex const & sub_basis=sub_power.basis;
 284         ex const & sub_exponent=sub_power.exponent;
 285         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 286             numeric const & num_sub_exponent=ex_to_numeric(sub_exponent);
 287             ASSERT(num_sub_exponent!=numeric(1));
 288             if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
 289                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 290             }
 291         }
 292     }
 293
 294     // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 295     if (exponent_is_numerical && num_exponent->is_integer() &&
 296         is_ex_exactly_of_type(ebasis,mul)) {
 297         return expand_mul(ex_to_mul(ebasis), *num_exponent);
 298     }
 299
 300     // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 301     // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 302     if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 303         ASSERT(!num_exponent->is_integer()); // should have been handled above
 304         mul const & mulref=ex_to_mul(ebasis);
 305         if (!mulref.overall_coeff.is_equal(exONE())) {
 306             numeric const & num_coeff=ex_to_numeric(mulref.overall_coeff);
 307             if (num_coeff.is_real()) {
 308                 if (num_coeff.is_positive()>0) {
 309                     mul * mulp=new mul(mulref);
 310                     mulp->overall_coeff=exONE();
 311                     mulp->clearflag(status_flags::evaluated);
 312                     mulp->clearflag(status_flags::hash_calculated);
 313                     return (new mul(power(*mulp,exponent),
 314                                     power(num_coeff,*num_exponent)))->
 315                         setflag(status_flags::dynallocated);
 316                 } else {
 317                     ASSERT(num_coeff.compare(numZERO())<0);
 318                     if (num_coeff.compare(numMINUSONE())!=0) {
 319                         mul * mulp=new mul(mulref);
 320                         mulp->overall_coeff=exMINUSONE();
 321                         mulp->clearflag(status_flags::evaluated);
 322                         mulp->clearflag(status_flags::hash_calculated);
 323                         return (new mul(power(*mulp,exponent),
 324                                         power(abs(num_coeff),*num_exponent)))->
 325                             setflag(status_flags::dynallocated);
 326                     }
 327                 }
 328             }
 329         }
 330     }
 331
 332     if (are_ex_trivially_equal(ebasis,basis) &&
 333         are_ex_trivially_equal(eexponent,exponent)) {
 334         return this->hold();
 335     }
 336     return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 337                                                    status_flags::evaluated);
 338 }
 339
 340 ex power::evalf(int level) const
 341 {
 342     debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 343
 344     ex ebasis;
 345     ex eexponent;
 346
 347     if (level==1) {
 348         ebasis=basis;
 349         eexponent=exponent;
 350     } else if (level == -max_recursion_level) {
 351         throw(std::runtime_error("max recursion level reached"));
 352     } else {
 353         ebasis=basis.evalf(level-1);
 354         eexponent=exponent.evalf(level-1);
 355     }
 356
 357     return power(ebasis,eexponent);
 358 }
 359
 360 ex power::subs(lst const & ls, lst const & lr) const
 361 {
 362     ex const & subsed_basis=basis.subs(ls,lr);
 363     ex const & subsed_exponent=exponent.subs(ls,lr);
 364
 365     if (are_ex_trivially_equal(basis,subsed_basis)&&
 366         are_ex_trivially_equal(exponent,subsed_exponent)) {
 367         return *this;
 368     }
 369
 370     return power(subsed_basis, subsed_exponent);
 371 }
 372
 373 ex power::simplify_ncmul(exvector const & v) const
 374 {
 375     return basic::simplify_ncmul(v);
 376 }
 377
 378 // protected
 379
 380 int power::compare_same_type(basic const & other) const
 381 {
 382     ASSERT(is_exactly_of_type(other, power));
 383     power const & o=static_cast<power const &>(const_cast<basic &>(other));
 384
 385     int cmpval;
 386     cmpval=basis.compare(o.basis);
 387     if (cmpval==0) {
 388         return exponent.compare(o.exponent);
 389     }
 390     return cmpval;
 391 }
 392
 393 unsigned power::return_type(void) const
 394 {
 395     return basis.return_type();
 396 }
 397
 398 unsigned power::return_type_tinfo(void) const
 399 {
 400     return basis.return_type_tinfo();
 401 }
 402
 403 ex power::expand(unsigned options) const
 404 {
 405     ex expanded_basis=basis.expand(options);
 406
 407     if (!is_ex_exactly_of_type(exponent,numeric)||
 408         !ex_to_numeric(exponent).is_integer()) {
 409         if (are_ex_trivially_equal(basis,expanded_basis)) {
 410             return this->hold();
 411         } else {
 412             return (new power(expanded_basis,exponent))->
 413                     setflag(status_flags::dynallocated);
 414         }
 415     }
 416
 417     // integer numeric exponent
 418     numeric const & num_exponent=ex_to_numeric(exponent);
 419     int int_exponent = num_exponent.to_int();
 420
 421     if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
 422         return expand_add(ex_to_add(expanded_basis), int_exponent);
 423     }
 424
 425     if (is_ex_exactly_of_type(expanded_basis,mul)) {
 426         return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 427     }
 428
 429     // cannot expand further
 430     if (are_ex_trivially_equal(basis,expanded_basis)) {
 431         return this->hold();
 432     } else {
 433         return (new power(expanded_basis,exponent))->
 434                setflag(status_flags::dynallocated);
 435     }
 436 }
 437
 438 //////////
 439 // new virtual functions which can be overridden by derived classes
 440 //////////
 441
 442 // none
 443
 444 //////////
 445 // non-virtual functions in this class
 446 //////////
 447
 448 ex power::expand_add(add const & a, int const n) const
 449 {
 450     // expand a^n where a is an add and n is an integer
 451
 452     if (n==2) {
 453         return expand_add_2(a);
 454     }
 455
 456     int m=a.nops();
 457     exvector sum;
 458     sum.reserve((n+1)*(m-1));
 459     intvector k(m-1);
 460     intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 461     intvector upper_limit(m-1);
 462     int l;
 463
 464     for (int l=0; l<m-1; l++) {
 465         k[l]=0;
 466         k_cum[l]=0;
 467         upper_limit[l]=n;
 468     }
 469
 470     while (1) {
 471         exvector term;
 472         term.reserve(m+1);
 473         for (l=0; l<m-1; l++) {
 474             ex const & b=a.op(l);
 475             ASSERT(!is_ex_exactly_of_type(b,add));
 476             ASSERT(!is_ex_exactly_of_type(b,power)||
 477                    !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 478                    !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
 479             if (is_ex_exactly_of_type(b,mul)) {
 480                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 481             } else {
 482                 term.push_back(power(b,k[l]));
 483             }
 484         }
 485
 486         ex const & b=a.op(l);
 487         ASSERT(!is_ex_exactly_of_type(b,add));
 488         ASSERT(!is_ex_exactly_of_type(b,power)||
 489                !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 490                !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
 491         if (is_ex_exactly_of_type(b,mul)) {
 492             term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 493         } else {
 494             term.push_back(power(b,n-k_cum[m-2]));
 495         }
 496
 497         numeric f=binomial(numeric(n),numeric(k[0]));
 498         for (l=1; l<m-1; l++) {
 499             f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 500         }
 501         term.push_back(f);
 502
 503         /*
 504         cout << "begin term" << endl;
 505         for (int i=0; i<m-1; i++) {
 506             cout << "k[" << i << "]=" << k[i] << endl;
 507             cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 508             cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 509         }
 510         for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 511             cout << *cit << endl;
 512         }
 513         cout << "end term" << endl;
 514         */
 515
 516         // TODO: optimize this
 517         sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 518
 519         // increment k[]
 520         l=m-2;
 521         while ((l>=0)&&((++k[l])>upper_limit[l])) {
 522             k[l]=0;
 523             l--;
 524         }
 525         if (l<0) break;
 526
 527         // recalc k_cum[] and upper_limit[]
 528         if (l==0) {
 529             k_cum[0]=k[0];
 530         } else {
 531             k_cum[l]=k_cum[l-1]+k[l];
 532         }
 533         for (int i=l+1; i<m-1; i++) {
 534             k_cum[i]=k_cum[i-1]+k[i];
 535         }
 536
 537         for (int i=l+1; i<m-1; i++) {
 538             upper_limit[i]=n-k_cum[i-1];
 539         }
 540     }
 541     return (new add(sum))->setflag(status_flags::dynallocated);
 542 }
 543
 544 /*
 545 ex power::expand_add_2(add const & a) const
 546 {
 547     // special case: expand a^2 where a is an add
 548
 549     epvector sum;
 550     sum.reserve((a.seq.size()*(a.seq.size()+1))/2);
 551     epvector::const_iterator last=a.seq.end();
 552
 553     for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 554         ex const & b=a.recombine_pair_to_ex(*cit0);
 555         ASSERT(!is_ex_exactly_of_type(b,add));
 556         ASSERT(!is_ex_exactly_of_type(b,power)||
 557                !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 558                !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
 559         if (is_ex_exactly_of_type(b,mul)) {
 560             sum.push_back(a.split_ex_to_pair(expand_mul(ex_to_mul(b),numTWO())));
 561         } else {
 562             sum.push_back(a.split_ex_to_pair((new power(b,exTWO()))->
 563                                               setflag(status_flags::dynallocated)));
 564         }
 565         for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 566             sum.push_back(a.split_ex_to_pair((new mul(a.recombine_pair_to_ex(*cit0),
 567                                                       a.recombine_pair_to_ex(*cit1)))->
 568                                               setflag(status_flags::dynallocated),
 569                                              exTWO()));
 570         }
 571     }
 572
 573     ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 574
 575     return (new add(sum))->setflag(status_flags::dynallocated);
 576 }
 577 */
 578
 579 ex power::expand_add_2(add const & a) const
 580 {
 581     // special case: expand a^2 where a is an add
 582
 583     epvector sum;
 584     unsigned a_nops=a.nops();
 585     sum.reserve((a_nops*(a_nops+1))/2);
 586     epvector::const_iterator last=a.seq.end();
 587
 588     // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 589     // first part: ignore overall_coeff and expand other terms
 590     for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 591         ex const & r=(*cit0).rest;
 592         ex const & c=(*cit0).coeff;
 593
 594         ASSERT(!is_ex_exactly_of_type(r,add));
 595         ASSERT(!is_ex_exactly_of_type(r,power)||
 596                !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric)||
 597                !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer()||
 598                !is_ex_exactly_of_type(ex_to_power(r).basis,add)||
 599                !is_ex_exactly_of_type(ex_to_power(r).basis,mul)||
 600                !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 601
 602         if (are_ex_trivially_equal(c,exONE())) {
 603             if (is_ex_exactly_of_type(r,mul)) {
 604                 sum.push_back(expair(expand_mul(ex_to_mul(r),numTWO()),exONE()));
 605             } else {
 606                 sum.push_back(expair((new power(r,exTWO()))->setflag(status_flags::dynallocated),
 607                                      exONE()));
 608             }
 609         } else {
 610             if (is_ex_exactly_of_type(r,mul)) {
 611                 sum.push_back(expair(expand_mul(ex_to_mul(r),numTWO()),
 612                                      ex_to_numeric(c).power_dyn(numTWO())));
 613             } else {
 614                 sum.push_back(expair((new power(r,exTWO()))->setflag(status_flags::dynallocated),
 615                                      ex_to_numeric(c).power_dyn(numTWO())));
 616             }
 617         }
 618
 619         for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 620             ex const & r1=(*cit1).rest;
 621             ex const & c1=(*cit1).coeff;
 622             sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 623                                                           numTWO().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 624         }
 625     }
 626
 627     ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 628
 629     // second part: add terms coming from overall_factor (if != 0)
 630     if (!a.overall_coeff.is_equal(exZERO())) {
 631         for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 632             sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(numTWO())));
 633         }
 634         sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(numTWO()),exONE()));
 635     }
 636
 637     ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 638
 639     return (new add(sum))->setflag(status_flags::dynallocated);
 640 }
 641
 642 ex power::expand_mul(mul const & m, numeric const & n) const
 643 {
 644     // expand m^n where m is a mul and n is and integer
 645
 646     if (n.is_equal(numZERO())) {
 647         return exONE();
 648     }
 649
 650     epvector distrseq;
 651     distrseq.reserve(m.seq.size());
 652     epvector::const_iterator last=m.seq.end();
 653     epvector::const_iterator cit=m.seq.begin();
 654     while (cit!=last) {
 655         if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 656             distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 657         } else {
 658             // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 659             // since n is an integer
 660             distrseq.push_back(expair((*cit).rest,
 661                                       ex_to_numeric((*cit).coeff).mul(n)));
 662         }
 663         ++cit;
 664     }
 665     return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))
 666                  ->setflag(status_flags::dynallocated);
 667 }
 668
 669 /*
 670 ex power::expand_commutative_3(ex const & basis, numeric const & exponent,
 671                              unsigned options) const
 672 {
 673     // obsolete
 674
 675     exvector distrseq;
 676     epvector splitseq;
 677
 678     add const & addref=static_cast<add const &>(*basis.bp);
 679
 680     splitseq=addref.seq;
 681     splitseq.pop_back();
 682     ex first_operands=add(splitseq);
 683     ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 684
 685     int n=exponent.to_int();
 686     for (int k=0; k<=n; k++) {
 687         distrseq.push_back(binomial(n,k)*power(first_operands,numeric(k))*
 688                            power(last_operand,numeric(n-k)));
 689     }
 690     return ex((new add(distrseq))->setflag(status_flags::sub_expanded |
 691                                            status_flags::expanded |
 692                                            status_flags::dynallocated  )).
 693            expand(options);
 694 }
 695 */
 696
 697 /*
 698 ex power::expand_noncommutative(ex const & basis, numeric const & exponent,
 699                                 unsigned options) const
 700 {
 701     ex rest_power=ex(power(basis,exponent.add(numMINUSONE()))).
 702                   expand(options | expand_options::internal_do_not_expand_power_operands);
 703
 704     return ex(mul(rest_power,basis),0).
 705            expand(options | expand_options::internal_do_not_expand_mul_operands);
 706 }
 707 */
 708
 709 //////////
 710 // static member variables
 711 //////////
 712
 713 // protected
 714
 715 unsigned power::precedence=60;
 716
 717 //////////
 718 // global constants
 719 //////////
 720
 721 const power some_power;
 722 type_info const & typeid_power=typeid(some_power);
 723
 724 } // namespace GiNaC