ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent).
   4  *
   5  *  GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
   6  *
   7  *  This program is free software; you can redistribute it and/or modify
   8  *  it under the terms of the GNU General Public License as published by
   9  *  the Free Software Foundation; either version 2 of the License, or
  10  *  (at your option) any later version.
  11  *
  12  *  This program is distributed in the hope that it will be useful,
  13  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  14  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  15  *  GNU General Public License for more details.
  16  *
  17  *  You should have received a copy of the GNU General Public License
  18  *  along with this program; if not, write to the Free Software
  19  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  20  */
  21
  22 #include <vector>
  23 #include <iostream>
  24 #include <stdexcept>
  25
  26 #include "power.h"
  27 #include "expairseq.h"
  28 #include "add.h"
  29 #include "mul.h"
  30 #include "numeric.h"
  31 #include "relational.h"
  32 #include "symbol.h"
  33
  34 typedef vector<int> intvector;
  35
  36 //////////
  37 // default constructor, destructor, copy constructor assignment operator and helpers
  38 //////////
  39
  40 // public
  41
  42 power::power() : basic(TINFO_power)
  43 {
  44     debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
  45 }
  46
  47 power::~power()
  48 {
  49     debugmsg("power destructor",LOGLEVEL_DESTRUCT);
  50     destroy(0);
  51 }
  52
  53 power::power(power const & other)
  54 {
  55     debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT);
  56     copy(other);
  57 }
  58
  59 power const & power::operator=(power const & other)
  60 {
  61     debugmsg("power operator=",LOGLEVEL_ASSIGNMENT);
  62     if (this != &other) {
  63         destroy(1);
  64         copy(other);
  65     }
  66     return *this;
  67 }
  68
  69 // protected
  70
  71 void power::copy(power const & other)
  72 {
  73     basic::copy(other);
  74     basis=other.basis;
  75     exponent=other.exponent;
  76 }
  77
  78 void power::destroy(bool call_parent)
  79 {
  80     if (call_parent) basic::destroy(call_parent);
  81 }
  82
  83 //////////
  84 // other constructors
  85 //////////
  86
  87 // public
  88
  89 power::power(ex const & lh, ex const & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  90 {
  91     debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
  92     ASSERT(basis.return_type()==return_types::commutative);
  93 }
  94
  95 power::power(ex const & lh, numeric const & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  96 {
  97     debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
  98     ASSERT(basis.return_type()==return_types::commutative);
  99 }
 100
 101 //////////
 102 // functions overriding virtual functions from bases classes
 103 //////////
 104
 105 // public
 106
 107 basic * power::duplicate() const
 108 {
 109     debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
 110     return new power(*this);
 111 }
 112
 113 bool power::info(unsigned inf) const
 114 {
 115     if (inf==info_flags::polynomial || inf==info_flags::integer_polynomial || inf==info_flags::rational_polynomial) {
 116         return exponent.info(info_flags::nonnegint);
 117     } else if (inf==info_flags::rational_function) {
 118         return exponent.info(info_flags::integer);
 119     } else {
 120         return basic::info(inf);
 121     }
 122 }
 123
 124 int power::nops() const
 125 {
 126     return 2;
 127 }
 128
 129 ex & power::let_op(int const i)
 130 {
 131     ASSERT(i>=0);
 132     ASSERT(i<2);
 133
 134     return i==0 ? basis : exponent;
 135 }
 136
 137 int power::degree(symbol const & s) const
 138 {
 139     if (is_exactly_of_type(*exponent.bp,numeric)) {
 140         if ((*basis.bp).compare(s)==0)
 141             return ex_to_numeric(exponent).to_int();
 142         else
 143             return basis.degree(s) * ex_to_numeric(exponent).to_int();
 144     }
 145     return 0;
 146 }
 147
 148 int power::ldegree(symbol const & s) const
 149 {
 150     if (is_exactly_of_type(*exponent.bp,numeric)) {
 151         if ((*basis.bp).compare(s)==0)
 152             return ex_to_numeric(exponent).to_int();
 153         else
 154             return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 155     }
 156     return 0;
 157 }
 158
 159 ex power::coeff(symbol const & s, int const n) const
 160 {
 161     if ((*basis.bp).compare(s)!=0) {
 162         // basis not equal to s
 163         if (n==0) {
 164             return *this;
 165         } else {
 166             return exZERO();
 167         }
 168     } else if (is_exactly_of_type(*exponent.bp,numeric)&&
 169                (static_cast<numeric const &>(*exponent.bp).compare(numeric(n))==0)) {
 170         return exONE();
 171     }
 172
 173     return exZERO();
 174 }
 175
 176 ex power::eval(int level) const
 177 {
 178     // simplifications: ^(x,0) -> 1 (0^0 handled here)
 179     //                  ^(x,1) -> x
 180     //                  ^(0,x) -> 0 (except if x is real and negative, in which case an exception is thrown)
 181     //                  ^(1,x) -> 1
 182     //                  ^(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)
 183     //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 184     //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 185     //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 186     //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 187
 188     debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 189
 190     if ((level==1)&&(flags & status_flags::evaluated)) {
 191         return *this;
 192     } else if (level == -max_recursion_level) {
 193         throw(std::runtime_error("max recursion level reached"));
 194     }
 195
 196     ex const & ebasis    = level==1 ? basis    : basis.eval(level-1);
 197     ex const & eexponent = level==1 ? exponent : exponent.eval(level-1);
 198
 199     bool basis_is_numerical=0;
 200     bool exponent_is_numerical=0;
 201     numeric * num_basis;
 202     numeric * num_exponent;
 203
 204     if (is_exactly_of_type(*ebasis.bp,numeric)) {
 205         basis_is_numerical=1;
 206         num_basis=static_cast<numeric *>(ebasis.bp);
 207     }
 208     if (is_exactly_of_type(*eexponent.bp,numeric)) {
 209         exponent_is_numerical=1;
 210         num_exponent=static_cast<numeric *>(eexponent.bp);
 211     }
 212
 213     // ^(x,0) -> 1 (0^0 also handled here)
 214     if (eexponent.is_zero())
 215         return exONE();
 216
 217     // ^(x,1) -> x
 218     if (eexponent.is_equal(exONE()))
 219         return ebasis;
 220
 221     // ^(0,x) -> 0 (except if x is real and negative)
 222     if (ebasis.is_zero()) {
 223         if (exponent_is_numerical && num_exponent->is_negative()) {
 224             throw(std::overflow_error("power::eval(): division by zero"));
 225         } else
 226             return exZERO();
 227     }
 228
 229     // ^(1,x) -> 1
 230     if (ebasis.is_equal(exONE()))
 231         return exONE();
 232
 233     if (basis_is_numerical && exponent_is_numerical) {
 234         // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 235         // except if c1,c2 are rational, but c1^c2 is not)
 236         bool basis_is_rational = num_basis->is_rational();
 237         bool exponent_is_rational = num_exponent->is_rational();
 238         numeric res = (*num_basis).power(*num_exponent);
 239
 240         if ((!basis_is_rational || !exponent_is_rational)
 241             || res.is_rational()) {
 242             return res;
 243         }
 244         ASSERT(!num_exponent->is_integer());  // has been handled by now
 245         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 246         if (basis_is_rational && exponent_is_rational
 247             && num_exponent->is_real()
 248             && !num_exponent->is_integer()) {
 249             numeric r, q, n, m;
 250             n = num_exponent->numer();
 251             m = num_exponent->denom();
 252             q = iquo(n, m, r);
 253             if (r.is_negative()) {
 254                 r = r.add(m);
 255                 q = q.sub(numONE());
 256             }
 257             if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 258                 return this->hold();
 259             else {
 260                 epvector res(2);
 261                 res.push_back(expair(ebasis,r.div(m)));
 262                 res.push_back(expair(ex(num_basis->power(q)),exONE()));
 263                 return (new mul(res))->setflag(status_flags::dynallocated | status_flags::evaluated);
 264                 /*return mul(num_basis->power(q),
 265                            power(ex(*num_basis),ex(r.div(m)))).hold();
 266                 */
 267                 /* return (new mul(num_basis->power(q),
 268                    power(*num_basis,r.div(m)).hold()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 269                 */
 270             }
 271         }
 272     }
 273
 274     // ^(^(x,c1),c2) -> ^(x,c1*c2)
 275     // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 276     // case c1=1 should not happen, see below!)
 277     if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 278         power const & sub_power=ex_to_power(ebasis);
 279         ex const & sub_basis=sub_power.basis;
 280         ex const & sub_exponent=sub_power.exponent;
 281         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 282             numeric const & num_sub_exponent=ex_to_numeric(sub_exponent);
 283             ASSERT(num_sub_exponent!=numeric(1));
 284             if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
 285                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 286             }
 287         }
 288     }
 289
 290     // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 291     if (exponent_is_numerical && num_exponent->is_integer() &&
 292         is_ex_exactly_of_type(ebasis,mul)) {
 293         return expand_mul(ex_to_mul(ebasis), *num_exponent);
 294     }
 295
 296     // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 297     // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 298     if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 299         ASSERT(!num_exponent->is_integer()); // should have been handled above
 300         mul const & mulref=ex_to_mul(ebasis);
 301         if (!mulref.overall_coeff.is_equal(exONE())) {
 302             numeric const & num_coeff=ex_to_numeric(mulref.overall_coeff);
 303             if (num_coeff.is_real()) {
 304                 if (num_coeff.is_positive()>0) {
 305                     mul * mulp=new mul(mulref);
 306                     mulp->overall_coeff=exONE();
 307                     mulp->clearflag(status_flags::evaluated);
 308                     mulp->clearflag(status_flags::hash_calculated);
 309                     return (new mul(power(*mulp,exponent),
 310                                     power(num_coeff,*num_exponent)))->
 311                         setflag(status_flags::dynallocated);
 312                 } else {
 313                     ASSERT(num_coeff.compare(numZERO())<0);
 314                     if (num_coeff.compare(numMINUSONE())!=0) {
 315                         mul * mulp=new mul(mulref);
 316                         mulp->overall_coeff=exMINUSONE();
 317                         mulp->clearflag(status_flags::evaluated);
 318                         mulp->clearflag(status_flags::hash_calculated);
 319                         return (new mul(power(*mulp,exponent),
 320                                         power(abs(num_coeff),*num_exponent)))->
 321                             setflag(status_flags::dynallocated);
 322                     }
 323                 }
 324             }
 325         }
 326     }
 327
 328     if (are_ex_trivially_equal(ebasis,basis) &&
 329         are_ex_trivially_equal(eexponent,exponent)) {
 330         return this->hold();
 331     }
 332     return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 333                                                    status_flags::evaluated);
 334 }
 335
 336 ex power::evalf(int level) const
 337 {
 338     debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 339
 340     ex ebasis;
 341     ex eexponent;
 342
 343     if (level==1) {
 344         ebasis=basis;
 345         eexponent=exponent;
 346     } else if (level == -max_recursion_level) {
 347         throw(std::runtime_error("max recursion level reached"));
 348     } else {
 349         ebasis=basis.evalf(level-1);
 350         eexponent=exponent.evalf(level-1);
 351     }
 352
 353     return power(ebasis,eexponent);
 354 }
 355
 356 ex power::subs(lst const & ls, lst const & lr) const
 357 {
 358     ex const & subsed_basis=basis.subs(ls,lr);
 359     ex const & subsed_exponent=exponent.subs(ls,lr);
 360
 361     if (are_ex_trivially_equal(basis,subsed_basis)&&
 362         are_ex_trivially_equal(exponent,subsed_exponent)) {
 363         return *this;
 364     }
 365
 366     return power(subsed_basis, subsed_exponent);
 367 }
 368
 369 ex power::simplify_ncmul(exvector const & v) const
 370 {
 371     return basic::simplify_ncmul(v);
 372 }
 373
 374 // protected
 375
 376 int power::compare_same_type(basic const & other) const
 377 {
 378     ASSERT(is_exactly_of_type(other, power));
 379     power const & o=static_cast<power const &>(const_cast<basic &>(other));
 380
 381     int cmpval;
 382     cmpval=basis.compare(o.basis);
 383     if (cmpval==0) {
 384         return exponent.compare(o.exponent);
 385     }
 386     return cmpval;
 387 }
 388
 389 unsigned power::return_type(void) const
 390 {
 391     return basis.return_type();
 392 }
 393
 394 unsigned power::return_type_tinfo(void) const
 395 {
 396     return basis.return_type_tinfo();
 397 }
 398
 399 ex power::expand(unsigned options) const
 400 {
 401     ex expanded_basis=basis.expand(options);
 402
 403     if (!is_ex_exactly_of_type(exponent,numeric)||
 404         !ex_to_numeric(exponent).is_integer()) {
 405         if (are_ex_trivially_equal(basis,expanded_basis)) {
 406             return this->hold();
 407         } else {
 408             return (new power(expanded_basis,exponent))->
 409                     setflag(status_flags::dynallocated);
 410         }
 411     }
 412
 413     // integer numeric exponent
 414     numeric const & num_exponent=ex_to_numeric(exponent);
 415     int int_exponent = num_exponent.to_int();
 416
 417     if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
 418         return expand_add(ex_to_add(expanded_basis), int_exponent);
 419     }
 420
 421     if (is_ex_exactly_of_type(expanded_basis,mul)) {
 422         return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 423     }
 424
 425     // cannot expand further
 426     if (are_ex_trivially_equal(basis,expanded_basis)) {
 427         return this->hold();
 428     } else {
 429         return (new power(expanded_basis,exponent))->
 430                setflag(status_flags::dynallocated);
 431     }
 432 }
 433
 434 //////////
 435 // new virtual functions which can be overridden by derived classes
 436 //////////
 437
 438 // none
 439
 440 //////////
 441 // non-virtual functions in this class
 442 //////////
 443
 444 ex power::expand_add(add const & a, int const n) const
 445 {
 446     // expand a^n where a is an add and n is an integer
 447
 448     if (n==2) {
 449         return expand_add_2(a);
 450     }
 451
 452     int m=a.nops();
 453     exvector sum;
 454     sum.reserve((n+1)*(m-1));
 455     intvector k(m-1);
 456     intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 457     intvector upper_limit(m-1);
 458     int l;
 459
 460     for (int l=0; l<m-1; l++) {
 461         k[l]=0;
 462         k_cum[l]=0;
 463         upper_limit[l]=n;
 464     }
 465
 466     while (1) {
 467         exvector term;
 468         term.reserve(m+1);
 469         for (l=0; l<m-1; l++) {
 470             ex const & b=a.op(l);
 471             ASSERT(!is_ex_exactly_of_type(b,add));
 472             ASSERT(!is_ex_exactly_of_type(b,power)||
 473                    !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 474                    !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
 475             if (is_ex_exactly_of_type(b,mul)) {
 476                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 477             } else {
 478                 term.push_back(power(b,k[l]));
 479             }
 480         }
 481
 482         ex const & b=a.op(l);
 483         ASSERT(!is_ex_exactly_of_type(b,add));
 484         ASSERT(!is_ex_exactly_of_type(b,power)||
 485                !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 486                !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
 487         if (is_ex_exactly_of_type(b,mul)) {
 488             term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 489         } else {
 490             term.push_back(power(b,n-k_cum[m-2]));
 491         }
 492
 493         numeric f=binomial(numeric(n),numeric(k[0]));
 494         for (l=1; l<m-1; l++) {
 495             f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 496         }
 497         term.push_back(f);
 498
 499         /*
 500         cout << "begin term" << endl;
 501         for (int i=0; i<m-1; i++) {
 502             cout << "k[" << i << "]=" << k[i] << endl;
 503             cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 504             cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 505         }
 506         for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 507             cout << *cit << endl;
 508         }
 509         cout << "end term" << endl;
 510         */
 511
 512         // TODO: optimize!!!!!!!!
 513         sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 514
 515         // increment k[]
 516         l=m-2;
 517         while ((l>=0)&&((++k[l])>upper_limit[l])) {
 518             k[l]=0;
 519             l--;
 520         }
 521         if (l<0) break;
 522
 523         // recalc k_cum[] and upper_limit[]
 524         if (l==0) {
 525             k_cum[0]=k[0];
 526         } else {
 527             k_cum[l]=k_cum[l-1]+k[l];
 528         }
 529         for (int i=l+1; i<m-1; i++) {
 530             k_cum[i]=k_cum[i-1]+k[i];
 531         }
 532
 533         for (int i=l+1; i<m-1; i++) {
 534             upper_limit[i]=n-k_cum[i-1];
 535         }
 536     }
 537     return (new add(sum))->setflag(status_flags::dynallocated);
 538 }
 539
 540 /*
 541 ex power::expand_add_2(add const & a) const
 542 {
 543     // special case: expand a^2 where a is an add
 544
 545     epvector sum;
 546     sum.reserve((a.seq.size()*(a.seq.size()+1))/2);
 547     epvector::const_iterator last=a.seq.end();
 548
 549     for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 550         ex const & b=a.recombine_pair_to_ex(*cit0);
 551         ASSERT(!is_ex_exactly_of_type(b,add));
 552         ASSERT(!is_ex_exactly_of_type(b,power)||
 553                !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 554                !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
 555         if (is_ex_exactly_of_type(b,mul)) {
 556             sum.push_back(a.split_ex_to_pair(expand_mul(ex_to_mul(b),numTWO())));
 557         } else {
 558             sum.push_back(a.split_ex_to_pair((new power(b,exTWO()))->
 559                                               setflag(status_flags::dynallocated)));
 560         }
 561         for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 562             sum.push_back(a.split_ex_to_pair((new mul(a.recombine_pair_to_ex(*cit0),
 563                                                       a.recombine_pair_to_ex(*cit1)))->
 564                                               setflag(status_flags::dynallocated),
 565                                              exTWO()));
 566         }
 567     }
 568
 569     ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 570
 571     return (new add(sum))->setflag(status_flags::dynallocated);
 572 }
 573 */
 574
 575 ex power::expand_add_2(add const & a) const
 576 {
 577     // special case: expand a^2 where a is an add
 578
 579     epvector sum;
 580     unsigned a_nops=a.nops();
 581     sum.reserve((a_nops*(a_nops+1))/2);
 582     epvector::const_iterator last=a.seq.end();
 583
 584     // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 585     // first part: ignore overall_coeff and expand other terms
 586     for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 587         ex const & r=(*cit0).rest;
 588         ex const & c=(*cit0).coeff;
 589
 590         ASSERT(!is_ex_exactly_of_type(r,add));
 591         ASSERT(!is_ex_exactly_of_type(r,power)||
 592                !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric)||
 593                !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer()||
 594                !is_ex_exactly_of_type(ex_to_power(r).basis,add)||
 595                !is_ex_exactly_of_type(ex_to_power(r).basis,mul)||
 596                !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 597
 598         if (are_ex_trivially_equal(c,exONE())) {
 599             if (is_ex_exactly_of_type(r,mul)) {
 600                 sum.push_back(expair(expand_mul(ex_to_mul(r),numTWO()),exONE()));
 601             } else {
 602                 sum.push_back(expair((new power(r,exTWO()))->setflag(status_flags::dynallocated),
 603                                      exONE()));
 604             }
 605         } else {
 606             if (is_ex_exactly_of_type(r,mul)) {
 607                 sum.push_back(expair(expand_mul(ex_to_mul(r),numTWO()),
 608                                      ex_to_numeric(c).power_dyn(numTWO())));
 609             } else {
 610                 sum.push_back(expair((new power(r,exTWO()))->setflag(status_flags::dynallocated),
 611                                      ex_to_numeric(c).power_dyn(numTWO())));
 612             }
 613         }
 614
 615         for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 616             ex const & r1=(*cit1).rest;
 617             ex const & c1=(*cit1).coeff;
 618             sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 619                                                           numTWO().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 620         }
 621     }
 622
 623     ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 624
 625     // second part: add terms coming from overall_factor (if != 0)
 626     if (!a.overall_coeff.is_equal(exZERO())) {
 627         for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 628             sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(numTWO())));
 629         }
 630         sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(numTWO()),exONE()));
 631     }
 632
 633     ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 634
 635     return (new add(sum))->setflag(status_flags::dynallocated);
 636 }
 637
 638 ex power::expand_mul(mul const & m, numeric const & n) const
 639 {
 640     // expand m^n where m is a mul and n is and integer
 641
 642     if (n.is_equal(numZERO())) {
 643         return exONE();
 644     }
 645
 646     epvector distrseq;
 647     distrseq.reserve(m.seq.size());
 648     epvector::const_iterator last=m.seq.end();
 649     epvector::const_iterator cit=m.seq.begin();
 650     while (cit!=last) {
 651         if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 652             distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 653         } else {
 654             // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 655             // since n is an integer
 656             distrseq.push_back(expair((*cit).rest,
 657                                       ex_to_numeric((*cit).coeff).mul(n)));
 658         }
 659         ++cit;
 660     }
 661     return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))
 662                  ->setflag(status_flags::dynallocated);
 663 }
 664
 665 /*
 666 ex power::expand_commutative_3(ex const & basis, numeric const & exponent,
 667                              unsigned options) const
 668 {
 669     // obsolete
 670
 671     exvector distrseq;
 672     epvector splitseq;
 673
 674     add const & addref=static_cast<add const &>(*basis.bp);
 675
 676     splitseq=addref.seq;
 677     splitseq.pop_back();
 678     ex first_operands=add(splitseq);
 679     ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 680
 681     int n=exponent.to_int();
 682     for (int k=0; k<=n; k++) {
 683         distrseq.push_back(binomial(n,k)*power(first_operands,numeric(k))*
 684                            power(last_operand,numeric(n-k)));
 685     }
 686     return ex((new add(distrseq))->setflag(status_flags::sub_expanded |
 687                                            status_flags::expanded |
 688                                            status_flags::dynallocated  )).
 689            expand(options);
 690 }
 691 */
 692
 693 /*
 694 ex power::expand_noncommutative(ex const & basis, numeric const & exponent,
 695                                 unsigned options) const
 696 {
 697     ex rest_power=ex(power(basis,exponent.add(numMINUSONE()))).
 698                   expand(options | expand_options::internal_do_not_expand_power_operands);
 699
 700     return ex(mul(rest_power,basis),0).
 701            expand(options | expand_options::internal_do_not_expand_mul_operands);
 702 }
 703 */
 704
 705 //////////
 706 // static member variables
 707 //////////
 708
 709 // protected
 710
 711 unsigned power::precedence=60;
 712
 713 //////////
 714 // global constants
 715 //////////
 716
 717 const power some_power;
 718 type_info const & typeid_power=typeid(some_power);