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