ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26
  27 #include "power.h"
  28 #include "expairseq.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "numeric.h"
  32 #include "inifcns.h"
  33 #include "relational.h"
  34 #include "symbol.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "debugmsg.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  43
  44 typedef std::vector<int> intvector;
  45
  46 //////////
  47 // default ctor, dtor, copy ctor assignment operator and helpers
  48 //////////
  49
  50 power::power() : basic(TINFO_power)
  51 {
  52         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
  53 }
  54
  55 void power::copy(const power & other)
  56 {
  57         inherited::copy(other);
  58         basis = other.basis;
  59         exponent = other.exponent;
  60 }
  61
  62 DEFAULT_DESTROY(power)
  63
  64 //////////
  65 // other ctors
  66 //////////
  67
  68 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  69 {
  70         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
  71         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  72 }
  73
  74 /** Ctor from an ex and a bare numeric.  This is somewhat more efficient than
  75  *  the normal ctor from two ex whenever it can be used. */
  76 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  77 {
  78         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
  79         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  80 }
  81
  82 //////////
  83 // archiving
  84 //////////
  85
  86 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  87 {
  88         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
  89         n.find_ex("basis", basis, sym_lst);
  90         n.find_ex("exponent", exponent, sym_lst);
  91 }
  92
  93 void power::archive(archive_node &n) const
  94 {
  95         inherited::archive(n);
  96         n.add_ex("basis", basis);
  97         n.add_ex("exponent", exponent);
  98 }
  99
 100 DEFAULT_UNARCHIVE(power)
 101
 102 //////////
 103 // functions overriding virtual functions from bases classes
 104 //////////
 105
 106 // public
 107
 108 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 109 {
 110         // Optimal output of integer powers of symbols to aid compiler CSE.
 111         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 112         // to learn why such a hack is really necessary.
 113         if (exp == 1) {
 114                 x.print(c);
 115         } else if (exp == 2) {
 116                 x.print(c);
 117                 c.s << "*";
 118                 x.print(c);
 119         } else if (exp & 1) {
 120                 x.print(c);
 121                 c.s << "*";
 122                 print_sym_pow(c, x, exp-1);
 123         } else {
 124                 c.s << "(";
 125                 print_sym_pow(c, x, exp >> 1);
 126                 c.s << ")*(";
 127                 print_sym_pow(c, x, exp >> 1);
 128                 c.s << ")";
 129         }
 130 }
 131
 132 void power::print(const print_context & c, unsigned level) const
 133 {
 134         debugmsg("power print", LOGLEVEL_PRINT);
 135
 136         if (is_of_type(c, print_tree)) {
 137
 138                 inherited::print(c, level);
 139
 140         } else if (is_of_type(c, print_csrc)) {
 141
 142                 // Integer powers of symbols are printed in a special, optimized way
 143                 if (exponent.info(info_flags::integer)
 144                  && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
 145                         int exp = ex_to_numeric(exponent).to_int();
 146                         if (exp > 0)
 147                                 c.s << '(';
 148                         else {
 149                                 exp = -exp;
 150                                 if (is_of_type(c, print_csrc_cl_N))
 151                                         c.s << "recip(";
 152                                 else
 153                                         c.s << "1.0/(";
 154                         }
 155                         print_sym_pow(c, ex_to_symbol(basis), exp);
 156                         c.s << ')';
 157
 158                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 159                 } else if (exponent.compare(_num_1()) == 0) {
 160                         if (is_of_type(c, print_csrc_cl_N))
 161                                 c.s << "recip(";
 162                         else
 163                                 c.s << "1.0/(";
 164                         basis.print(c);
 165                         c.s << ')';
 166
 167                 // Otherwise, use the pow() or expt() (CLN) functions
 168                 } else {
 169                         if (is_of_type(c, print_csrc_cl_N))
 170                                 c.s << "expt(";
 171                         else
 172                                 c.s << "pow(";
 173                         basis.print(c);
 174                         c.s << ',';
 175                         exponent.print(c);
 176                         c.s << ')';
 177                 }
 178
 179         } else {
 180
 181                 if (exponent.is_equal(_ex1_2())) {
 182                         if (is_of_type(c, print_latex))
 183                                 c.s << "\\sqrt{";
 184                         else
 185                                 c.s << "sqrt(";
 186                         basis.print(c);
 187                         if (is_of_type(c, print_latex))
 188                                 c.s << '}';
 189                         else
 190                                 c.s << ')';
 191                 } else {
 192                         if (precedence() <= level) {
 193                                 if (is_of_type(c, print_latex))
 194                                         c.s << "{(";
 195                                 else
 196                                         c.s << "(";
 197                         }
 198                         basis.print(c, precedence());
 199                         c.s << '^';
 200                         if (is_of_type(c, print_latex))
 201                                 c.s << '{';
 202                         exponent.print(c, precedence());
 203                         if (is_of_type(c, print_latex))
 204                                 c.s << '}';
 205                         if (precedence() <= level) {
 206                                 if (is_of_type(c, print_latex))
 207                                         c.s << ")}";
 208                                 else
 209                                         c.s << ')';
 210                         }
 211                 }
 212         }
 213 }
 214
 215 bool power::info(unsigned inf) const
 216 {
 217         switch (inf) {
 218                 case info_flags::polynomial:
 219                 case info_flags::integer_polynomial:
 220                 case info_flags::cinteger_polynomial:
 221                 case info_flags::rational_polynomial:
 222                 case info_flags::crational_polynomial:
 223                         return exponent.info(info_flags::nonnegint);
 224                 case info_flags::rational_function:
 225                         return exponent.info(info_flags::integer);
 226                 case info_flags::algebraic:
 227                         return (!exponent.info(info_flags::integer) ||
 228                                         basis.info(inf));
 229         }
 230         return inherited::info(inf);
 231 }
 232
 233 unsigned power::nops() const
 234 {
 235         return 2;
 236 }
 237
 238 ex & power::let_op(int i)
 239 {
 240         GINAC_ASSERT(i>=0);
 241         GINAC_ASSERT(i<2);
 242
 243         return i==0 ? basis : exponent;
 244 }
 245
 246 int power::degree(const ex & s) const
 247 {
 248         if (is_exactly_of_type(*exponent.bp,numeric)) {
 249                 if (basis.is_equal(s)) {
 250                         if (ex_to_numeric(exponent).is_integer())
 251                                 return ex_to_numeric(exponent).to_int();
 252                         else
 253                                 return 0;
 254                 } else
 255                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 256         }
 257         return 0;
 258 }
 259
 260 int power::ldegree(const ex & s) const
 261 {
 262         if (is_exactly_of_type(*exponent.bp,numeric)) {
 263                 if (basis.is_equal(s)) {
 264                         if (ex_to_numeric(exponent).is_integer())
 265                                 return ex_to_numeric(exponent).to_int();
 266                         else
 267                                 return 0;
 268                 } else
 269                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 270         }
 271         return 0;
 272 }
 273
 274 ex power::coeff(const ex & s, int n) const
 275 {
 276         if (!basis.is_equal(s)) {
 277                 // basis not equal to s
 278                 if (n == 0)
 279                         return *this;
 280                 else
 281                         return _ex0();
 282         } else {
 283                 // basis equal to s
 284                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 285                         // integer exponent
 286                         int int_exp = ex_to_numeric(exponent).to_int();
 287                         if (n == int_exp)
 288                                 return _ex1();
 289                         else
 290                                 return _ex0();
 291                 } else {
 292                         // non-integer exponents are treated as zero
 293                         if (n == 0)
 294                                 return *this;
 295                         else
 296                                 return _ex0();
 297                 }
 298         }
 299 }
 300
 301 ex power::eval(int level) const
 302 {
 303         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 304         //                  ^(x,1) -> x
 305         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 306         //                  ^(1,x) -> 1
 307         //                  ^(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)
 308         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 309         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 310         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 311         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 312
 313         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 314
 315         if ((level==1) && (flags & status_flags::evaluated))
 316                 return *this;
 317         else if (level == -max_recursion_level)
 318                 throw(std::runtime_error("max recursion level reached"));
 319
 320         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 321         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 322
 323         bool basis_is_numerical = 0;
 324         bool exponent_is_numerical = 0;
 325         numeric * num_basis;
 326         numeric * num_exponent;
 327
 328         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 329                 basis_is_numerical = 1;
 330                 num_basis = static_cast<numeric *>(ebasis.bp);
 331         }
 332         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 333                 exponent_is_numerical = 1;
 334                 num_exponent = static_cast<numeric *>(eexponent.bp);
 335         }
 336
 337         // ^(x,0) -> 1 (0^0 also handled here)
 338         if (eexponent.is_zero()) {
 339                 if (ebasis.is_zero())
 340                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 341                 else
 342                         return _ex1();
 343         }
 344
 345         // ^(x,1) -> x
 346         if (eexponent.is_equal(_ex1()))
 347                 return ebasis;
 348
 349         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 350         if (ebasis.is_zero() && exponent_is_numerical) {
 351                 if ((num_exponent->real()).is_zero())
 352                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 353                 else if ((num_exponent->real()).is_negative())
 354                         throw (pole_error("power::eval(): division by zero",1));
 355                 else
 356                         return _ex0();
 357         }
 358
 359         // ^(1,x) -> 1
 360         if (ebasis.is_equal(_ex1()))
 361                 return _ex1();
 362
 363         if (basis_is_numerical && exponent_is_numerical) {
 364                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 365                 // except if c1,c2 are rational, but c1^c2 is not)
 366                 bool basis_is_crational = num_basis->is_crational();
 367                 bool exponent_is_crational = num_exponent->is_crational();
 368                 numeric res = num_basis->power(*num_exponent);
 369
 370                 if ((!basis_is_crational || !exponent_is_crational)
 371                         || res.is_crational()) {
 372                         return res;
 373                 }
 374                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 375                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 376                 if (basis_is_crational && exponent_is_crational
 377                         && num_exponent->is_real()
 378                         && !num_exponent->is_integer()) {
 379                         numeric n = num_exponent->numer();
 380                         numeric m = num_exponent->denom();
 381                         numeric r;
 382                         numeric q = iquo(n, m, r);
 383                         if (r.is_negative()) {
 384                                 r = r.add(m);
 385                                 q = q.sub(_num1());
 386                         }
 387                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 388                                 return this->hold();
 389                         else {
 390                                 epvector res;
 391                                 res.push_back(expair(ebasis,r.div(m)));
 392                                 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 393                         }
 394                 }
 395         }
 396
 397         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 398         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 399         // case c1==1 should not happen, see below!)
 400         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 401                 const power & sub_power = ex_to_power(ebasis);
 402                 const ex & sub_basis = sub_power.basis;
 403                 const ex & sub_exponent = sub_power.exponent;
 404                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 405                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 406                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 407                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
 408                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 409                 }
 410         }
 411
 412         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 413         if (exponent_is_numerical && num_exponent->is_integer() &&
 414                 is_ex_exactly_of_type(ebasis,mul)) {
 415                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 416         }
 417
 418         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 419         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 420         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 421                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 422                 const mul & mulref = ex_to_mul(ebasis);
 423                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 424                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
 425                         if (num_coeff.is_real()) {
 426                                 if (num_coeff.is_positive()) {
 427                                         mul * mulp = new mul(mulref);
 428                                         mulp->overall_coeff = _ex1();
 429                                         mulp->clearflag(status_flags::evaluated);
 430                                         mulp->clearflag(status_flags::hash_calculated);
 431                                         return (new mul(power(*mulp,exponent),
 432                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 433                                 } else {
 434                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 435                                         if (num_coeff.compare(_num_1())!=0) {
 436                                                 mul * mulp = new mul(mulref);
 437                                                 mulp->overall_coeff = _ex_1();
 438                                                 mulp->clearflag(status_flags::evaluated);
 439                                                 mulp->clearflag(status_flags::hash_calculated);
 440                                                 return (new mul(power(*mulp,exponent),
 441                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 442                                         }
 443                                 }
 444                         }
 445                 }
 446         }
 447
 448         if (are_ex_trivially_equal(ebasis,basis) &&
 449                 are_ex_trivially_equal(eexponent,exponent)) {
 450                 return this->hold();
 451         }
 452         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 453                                                                                                    status_flags::evaluated);
 454 }
 455
 456 ex power::evalf(int level) const
 457 {
 458         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 459
 460         ex ebasis;
 461         ex eexponent;
 462
 463         if (level==1) {
 464                 ebasis = basis;
 465                 eexponent = exponent;
 466         } else if (level == -max_recursion_level) {
 467                 throw(std::runtime_error("max recursion level reached"));
 468         } else {
 469                 ebasis = basis.evalf(level-1);
 470                 if (!is_ex_exactly_of_type(eexponent,numeric))
 471                         eexponent = exponent.evalf(level-1);
 472                 else
 473                         eexponent = exponent;
 474         }
 475
 476         return power(ebasis,eexponent);
 477 }
 478
 479 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 480 {
 481         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 482         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 483
 484         if (are_ex_trivially_equal(basis, subsed_basis)
 485          && are_ex_trivially_equal(exponent, subsed_exponent))
 486                 return basic::subs(ls, lr, no_pattern);
 487         else
 488                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
 489 }
 490
 491 ex power::simplify_ncmul(const exvector & v) const
 492 {
 493         return inherited::simplify_ncmul(v);
 494 }
 495
 496 // protected
 497
 498 /** Implementation of ex::diff() for a power.
 499  *  @see ex::diff */
 500 ex power::derivative(const symbol & s) const
 501 {
 502         if (exponent.info(info_flags::real)) {
 503                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 504                 epvector newseq;
 505                 newseq.reserve(2);
 506                 newseq.push_back(expair(basis, exponent - _ex1()));
 507                 newseq.push_back(expair(basis.diff(s), _ex1()));
 508                 return mul(newseq, exponent);
 509         } else {
 510                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 511                 return mul(*this,
 512                            add(mul(exponent.diff(s), log(basis)),
 513                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 514         }
 515 }
 516
 517 int power::compare_same_type(const basic & other) const
 518 {
 519         GINAC_ASSERT(is_exactly_of_type(other, power));
 520         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 521
 522         int cmpval;
 523         cmpval=basis.compare(o.basis);
 524         if (cmpval==0) {
 525                 return exponent.compare(o.exponent);
 526         }
 527         return cmpval;
 528 }
 529
 530 unsigned power::return_type(void) const
 531 {
 532         return basis.return_type();
 533 }
 534
 535 unsigned power::return_type_tinfo(void) const
 536 {
 537         return basis.return_type_tinfo();
 538 }
 539
 540 ex power::expand(unsigned options) const
 541 {
 542         if (flags & status_flags::expanded)
 543                 return *this;
 544
 545         ex expanded_basis = basis.expand(options);
 546         ex expanded_exponent = exponent.expand(options);
 547
 548         // x^(a+b) -> x^a * x^b
 549         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 550                 const add &a = ex_to_add(expanded_exponent);
 551                 exvector distrseq;
 552                 distrseq.reserve(a.seq.size() + 1);
 553                 epvector::const_iterator last = a.seq.end();
 554                 epvector::const_iterator cit = a.seq.begin();
 555                 while (cit!=last) {
 556                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 557                         cit++;
 558                 }
 559
 560                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 561                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 562                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 563                         int int_exponent = num_exponent.to_int();
 564                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 565                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 566                         else
 567                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 568                 } else
 569                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 570
 571                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 572                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 573                 return r.expand();
 574         }
 575
 576         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 577                 !ex_to_numeric(expanded_exponent).is_integer()) {
 578                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 579                         return this->hold();
 580                 } else {
 581                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 582                 }
 583         }
 584
 585         // integer numeric exponent
 586         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 587         int int_exponent = num_exponent.to_int();
 588
 589         // (x+y)^n, n>0
 590         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 591                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 592
 593         // (x*y)^n -> x^n * y^n
 594         if (is_ex_exactly_of_type(expanded_basis,mul))
 595                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 596
 597         // cannot expand further
 598         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 599                 return this->hold();
 600         else
 601                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 602 }
 603
 604 //////////
 605 // new virtual functions which can be overridden by derived classes
 606 //////////
 607
 608 // none
 609
 610 //////////
 611 // non-virtual functions in this class
 612 //////////
 613
 614 /** expand a^n where a is an add and n is an integer.
 615  *  @see power::expand */
 616 ex power::expand_add(const add & a, int n) const
 617 {
 618         if (n==2)
 619                 return expand_add_2(a);
 620
 621         int m = a.nops();
 622         exvector sum;
 623         sum.reserve((n+1)*(m-1));
 624         intvector k(m-1);
 625         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 626         intvector upper_limit(m-1);
 627         int l;
 628
 629         for (int l=0; l<m-1; l++) {
 630                 k[l] = 0;
 631                 k_cum[l] = 0;
 632                 upper_limit[l] = n;
 633         }
 634
 635         while (1) {
 636                 exvector term;
 637                 term.reserve(m+1);
 638                 for (l=0; l<m-1; l++) {
 639                         const ex & b = a.op(l);
 640                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 641                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 642                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 643                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 644                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 645                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 646                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 647                         if (is_ex_exactly_of_type(b,mul))
 648                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 649                         else
 650                                 term.push_back(power(b,k[l]));
 651                 }
 652
 653                 const ex & b = a.op(l);
 654                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 655                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 656                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 657                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 658                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 659                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 660                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 661                 if (is_ex_exactly_of_type(b,mul))
 662                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 663                 else
 664                         term.push_back(power(b,n-k_cum[m-2]));
 665
 666                 numeric f = binomial(numeric(n),numeric(k[0]));
 667                 for (l=1; l<m-1; l++)
 668                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 669
 670                 term.push_back(f);
 671
 672                 /*
 673                 cout << "begin term" << endl;
 674                 for (int i=0; i<m-1; i++) {
 675                         cout << "k[" << i << "]=" << k[i] << endl;
 676                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 677                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 678                 }
 679                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
 680                 cout << "end term" << endl;
 681                 */
 682
 683                 // TODO: optimize this
 684                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 685
 686                 // increment k[]
 687                 l = m-2;
 688                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 689                         k[l] = 0;
 690                         l--;
 691                 }
 692                 if (l<0) break;
 693
 694                 // recalc k_cum[] and upper_limit[]
 695                 if (l==0)
 696                         k_cum[0] = k[0];
 697                 else
 698                         k_cum[l] = k_cum[l-1]+k[l];
 699
 700                 for (int i=l+1; i<m-1; i++)
 701                         k_cum[i] = k_cum[i-1]+k[i];
 702
 703                 for (int i=l+1; i<m-1; i++)
 704                         upper_limit[i] = n-k_cum[i-1];
 705         }
 706         return (new add(sum))->setflag(status_flags::dynallocated |
 707                                                                    status_flags::expanded );
 708 }
 709
 710
 711 /** Special case of power::expand_add. Expands a^2 where a is an add.
 712  *  @see power::expand_add */
 713 ex power::expand_add_2(const add & a) const
 714 {
 715         epvector sum;
 716         unsigned a_nops = a.nops();
 717         sum.reserve((a_nops*(a_nops+1))/2);
 718         epvector::const_iterator last = a.seq.end();
 719
 720         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 721         // first part: ignore overall_coeff and expand other terms
 722         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 723                 const ex & r = (*cit0).rest;
 724                 const ex & c = (*cit0).coeff;
 725
 726                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 727                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 728                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 729                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 730                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 731                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 732                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 733
 734                 if (are_ex_trivially_equal(c,_ex1())) {
 735                         if (is_ex_exactly_of_type(r,mul)) {
 736                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 737                                                      _ex1()));
 738                         } else {
 739                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 740                                                      _ex1()));
 741                         }
 742                 } else {
 743                         if (is_ex_exactly_of_type(r,mul)) {
 744                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 745                                                      ex_to_numeric(c).power_dyn(_num2())));
 746                         } else {
 747                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 748                                                      ex_to_numeric(c).power_dyn(_num2())));
 749                         }
 750                 }
 751
 752                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 753                         const ex & r1 = (*cit1).rest;
 754                         const ex & c1 = (*cit1).coeff;
 755                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 756                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 757                 }
 758         }
 759
 760         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 761
 762         // second part: add terms coming from overall_factor (if != 0)
 763         if (!a.overall_coeff.is_zero()) {
 764                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 765                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 766                 }
 767                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 768         }
 769
 770         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 771
 772         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 773 }
 774
 775 /** Expand factors of m in m^n where m is a mul and n is and integer
 776  *  @see power::expand */
 777 ex power::expand_mul(const mul & m, const numeric & n) const
 778 {
 779         if (n.is_zero())
 780                 return _ex1();
 781
 782         epvector distrseq;
 783         distrseq.reserve(m.seq.size());
 784         epvector::const_iterator last = m.seq.end();
 785         epvector::const_iterator cit = m.seq.begin();
 786         while (cit!=last) {
 787                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 788                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 789                 } else {
 790                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 791                         // since n is an integer
 792                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 793                 }
 794                 ++cit;
 795         }
 796         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 797 }
 798
 799 /*
 800 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 801                                                                 unsigned options) const
 802 {
 803         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 804                         expand(options | expand_options::internal_do_not_expand_power_operands);
 805
 806         return ex(mul(rest_power,basis),0).
 807                expand(options | expand_options::internal_do_not_expand_mul_operands);
 808 }
 809 */
 810
 811 // helper function
 812
 813 ex sqrt(const ex & a)
 814 {
 815         return power(a,_ex1_2());
 816 }
 817
 818 } // namespace GiNaC