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