ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include "power.h"
  24 #include "expairseq.h"
  25 #include "add.h"
  26 #include "mul.h"
  27 #include "ncmul.h"
  28 #include "numeric.h"
  29 #include "constant.h"
  30 #include "operators.h"
  31 #include "inifcns.h" // for log() in power::derivative()
  32 #include "matrix.h"
  33 #include "indexed.h"
  34 #include "symbol.h"
  35 #include "lst.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38 #include "relational.h"
  39 #include "compiler.h"
  40
  41 #include <iostream>
  42 #include <limits>
  43 #include <stdexcept>
  44 #include <vector>
  45 #include <algorithm>
  46
  47 namespace GiNaC {
  48
  49 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
  50   print_func<print_dflt>(&power::do_print_dflt).
  51   print_func<print_latex>(&power::do_print_latex).
  52   print_func<print_csrc>(&power::do_print_csrc).
  53   print_func<print_python>(&power::do_print_python).
  54   print_func<print_python_repr>(&power::do_print_python_repr).
  55   print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
  56
  57 //////////
  58 // default constructor
  59 //////////
  60
  61 power::power() { }
  62
  63 //////////
  64 // other constructors
  65 //////////
  66
  67 // all inlined
  68
  69 //////////
  70 // archiving
  71 //////////
  72
  73 void power::read_archive(const archive_node &n, lst &sym_lst)
  74 {
  75         inherited::read_archive(n, sym_lst);
  76         n.find_ex("basis", basis, sym_lst);
  77         n.find_ex("exponent", exponent, sym_lst);
  78 }
  79
  80 void power::archive(archive_node &n) const
  81 {
  82         inherited::archive(n);
  83         n.add_ex("basis", basis);
  84         n.add_ex("exponent", exponent);
  85 }
  86
  87 //////////
  88 // functions overriding virtual functions from base classes
  89 //////////
  90
  91 // public
  92
  93 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  94 {
  95         // Ordinary output of powers using '^' or '**'
  96         if (precedence() <= level)
  97                 c.s << openbrace << '(';
  98         basis.print(c, precedence());
  99         c.s << powersymbol;
 100         c.s << openbrace;
 101         exponent.print(c, precedence());
 102         c.s << closebrace;
 103         if (precedence() <= level)
 104                 c.s << ')' << closebrace;
 105 }
 106
 107 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 108 {
 109         if (exponent.is_equal(_ex1_2)) {
 110
 111                 // Square roots are printed in a special way
 112                 c.s << "sqrt(";
 113                 basis.print(c);
 114                 c.s << ')';
 115
 116         } else
 117                 print_power(c, "^", "", "", level);
 118 }
 119
 120 void power::do_print_latex(const print_latex & c, unsigned level) const
 121 {
 122         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 123
 124                 // Powers with negative numeric exponents are printed as fractions
 125                 c.s << "\\frac{1}{";
 126                 power(basis, -exponent).eval().print(c);
 127                 c.s << '}';
 128
 129         } else if (exponent.is_equal(_ex1_2)) {
 130
 131                 // Square roots are printed in a special way
 132                 c.s << "\\sqrt{";
 133                 basis.print(c);
 134                 c.s << '}';
 135
 136         } else
 137                 print_power(c, "^", "{", "}", level);
 138 }
 139
 140 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 141 {
 142         // Optimal output of integer powers of symbols to aid compiler CSE.
 143         // C.f. ISO/IEC 14882:2011, section 1.9 [intro execution], paragraph 15
 144         // to learn why such a parenthesation is really necessary.
 145         if (exp == 1) {
 146                 x.print(c);
 147         } else if (exp == 2) {
 148                 x.print(c);
 149                 c.s << "*";
 150                 x.print(c);
 151         } else if (exp & 1) {
 152                 x.print(c);
 153                 c.s << "*";
 154                 print_sym_pow(c, x, exp-1);
 155         } else {
 156                 c.s << "(";
 157                 print_sym_pow(c, x, exp >> 1);
 158                 c.s << ")*(";
 159                 print_sym_pow(c, x, exp >> 1);
 160                 c.s << ")";
 161         }
 162 }
 163
 164 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
 165 {
 166         if (exponent.is_equal(_ex_1)) {
 167                 c.s << "recip(";
 168                 basis.print(c);
 169                 c.s << ')';
 170                 return;
 171         }
 172         c.s << "expt(";
 173         basis.print(c);
 174         c.s << ", ";
 175         exponent.print(c);
 176         c.s << ')';
 177 }
 178
 179 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 180 {
 181         // Integer powers of symbols are printed in a special, optimized way
 182         if (exponent.info(info_flags::integer) &&
 183             (is_a<symbol>(basis) || is_a<constant>(basis))) {
 184                 int exp = ex_to<numeric>(exponent).to_int();
 185                 if (exp > 0)
 186                         c.s << '(';
 187                 else {
 188                         exp = -exp;
 189                         c.s << "1.0/(";
 190                 }
 191                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 192                 c.s << ')';
 193
 194         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 195         } else if (exponent.is_equal(_ex_1)) {
 196                 c.s << "1.0/(";
 197                 basis.print(c);
 198                 c.s << ')';
 199
 200         // Otherwise, use the pow() function
 201         } else {
 202                 c.s << "pow(";
 203                 basis.print(c);
 204                 c.s << ',';
 205                 exponent.print(c);
 206                 c.s << ')';
 207         }
 208 }
 209
 210 void power::do_print_python(const print_python & c, unsigned level) const
 211 {
 212         print_power(c, "**", "", "", level);
 213 }
 214
 215 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 216 {
 217         c.s << class_name() << '(';
 218         basis.print(c);
 219         c.s << ',';
 220         exponent.print(c);
 221         c.s << ')';
 222 }
 223
 224 bool power::info(unsigned inf) const
 225 {
 226         switch (inf) {
 227                 case info_flags::polynomial:
 228                 case info_flags::integer_polynomial:
 229                 case info_flags::cinteger_polynomial:
 230                 case info_flags::rational_polynomial:
 231                 case info_flags::crational_polynomial:
 232                         return exponent.info(info_flags::nonnegint) &&
 233                                basis.info(inf);
 234                 case info_flags::rational_function:
 235                         return exponent.info(info_flags::integer) &&
 236                                basis.info(inf);
 237                 case info_flags::algebraic:
 238                         return !exponent.info(info_flags::integer) ||
 239                                basis.info(inf);
 240                 case info_flags::expanded:
 241                         return (flags & status_flags::expanded);
 242                 case info_flags::positive:
 243                         return basis.info(info_flags::positive) && exponent.info(info_flags::real);
 244                 case info_flags::nonnegative:
 245                         return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
 246                                (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
 247                 case info_flags::has_indices: {
 248                         if (flags & status_flags::has_indices)
 249                                 return true;
 250                         else if (flags & status_flags::has_no_indices)
 251                                 return false;
 252                         else if (basis.info(info_flags::has_indices)) {
 253                                 setflag(status_flags::has_indices);
 254                                 clearflag(status_flags::has_no_indices);
 255                                 return true;
 256                         } else {
 257                                 clearflag(status_flags::has_indices);
 258                                 setflag(status_flags::has_no_indices);
 259                                 return false;
 260                         }
 261                 }
 262         }
 263         return inherited::info(inf);
 264 }
 265
 266 size_t power::nops() const
 267 {
 268         return 2;
 269 }
 270
 271 ex power::op(size_t i) const
 272 {
 273         GINAC_ASSERT(i<2);
 274
 275         return i==0 ? basis : exponent;
 276 }
 277
 278 ex power::map(map_function & f) const
 279 {
 280         const ex &mapped_basis = f(basis);
 281         const ex &mapped_exponent = f(exponent);
 282
 283         if (!are_ex_trivially_equal(basis, mapped_basis)
 284          || !are_ex_trivially_equal(exponent, mapped_exponent))
 285                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 286         else
 287                 return *this;
 288 }
 289
 290 bool power::is_polynomial(const ex & var) const
 291 {
 292         if (basis.is_polynomial(var)) {
 293                 if (basis.has(var))
 294                         // basis is non-constant polynomial in var
 295                         return exponent.info(info_flags::nonnegint);
 296                 else
 297                         // basis is constant in var
 298                         return !exponent.has(var);
 299         }
 300         // basis is a non-polynomial function of var
 301         return false;
 302 }
 303
 304 int power::degree(const ex & s) const
 305 {
 306         if (is_equal(ex_to<basic>(s)))
 307                 return 1;
 308         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 309                 if (basis.is_equal(s))
 310                         return ex_to<numeric>(exponent).to_int();
 311                 else
 312                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 313         } else if (basis.has(s))
 314                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 315         else
 316                 return 0;
 317 }
 318
 319 int power::ldegree(const ex & s) const
 320 {
 321         if (is_equal(ex_to<basic>(s)))
 322                 return 1;
 323         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 324                 if (basis.is_equal(s))
 325                         return ex_to<numeric>(exponent).to_int();
 326                 else
 327                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 328         } else if (basis.has(s))
 329                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 330         else
 331                 return 0;
 332 }
 333
 334 ex power::coeff(const ex & s, int n) const
 335 {
 336         if (is_equal(ex_to<basic>(s)))
 337                 return n==1 ? _ex1 : _ex0;
 338         else if (!basis.is_equal(s)) {
 339                 // basis not equal to s
 340                 if (n == 0)
 341                         return *this;
 342                 else
 343                         return _ex0;
 344         } else {
 345                 // basis equal to s
 346                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 347                         // integer exponent
 348                         int int_exp = ex_to<numeric>(exponent).to_int();
 349                         if (n == int_exp)
 350                                 return _ex1;
 351                         else
 352                                 return _ex0;
 353                 } else {
 354                         // non-integer exponents are treated as zero
 355                         if (n == 0)
 356                                 return *this;
 357                         else
 358                                 return _ex0;
 359                 }
 360         }
 361 }
 362
 363 /** Perform automatic term rewriting rules in this class.  In the following
 364  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 365  *  stand for such expressions that contain a plain number.
 366  *  - ^(x,0) -> 1  (also handles ^(0,0))
 367  *  - ^(x,1) -> x
 368  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 369  *  - ^(1,x) -> 1
 370  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 371  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 372  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0), case c1=1 should not happen, see below!)
 373  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 374  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 375  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 376  *
 377  *  @param level cut-off in recursive evaluation */
 378 ex power::eval(int level) const
 379 {
 380         if ((level==1) && (flags & status_flags::evaluated))
 381                 return *this;
 382         else if (level == -max_recursion_level)
 383                 throw(std::runtime_error("max recursion level reached"));
 384
 385         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 386         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 387
 388         const numeric *num_basis = nullptr;
 389         const numeric *num_exponent = nullptr;
 390
 391         if (is_exactly_a<numeric>(ebasis)) {
 392                 num_basis = &ex_to<numeric>(ebasis);
 393         }
 394         if (is_exactly_a<numeric>(eexponent)) {
 395                 num_exponent = &ex_to<numeric>(eexponent);
 396         }
 397
 398         // ^(x,0) -> 1  (0^0 also handled here)
 399         if (eexponent.is_zero()) {
 400                 if (ebasis.is_zero())
 401                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 402                 else
 403                         return _ex1;
 404         }
 405
 406         // ^(x,1) -> x
 407         if (eexponent.is_equal(_ex1))
 408                 return ebasis;
 409
 410         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 411         if ( ebasis.is_zero() && num_exponent ) {
 412                 if ((num_exponent->real()).is_zero())
 413                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 414                 else if ((num_exponent->real()).is_negative())
 415                         throw (pole_error("power::eval(): division by zero",1));
 416                 else
 417                         return _ex0;
 418         }
 419
 420         // ^(1,x) -> 1
 421         if (ebasis.is_equal(_ex1))
 422                 return _ex1;
 423
 424         // power of a function calculated by separate rules defined for this function
 425         if (is_exactly_a<function>(ebasis))
 426                 return ex_to<function>(ebasis).power(eexponent);
 427
 428         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 429         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 430                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 431
 432         if ( num_exponent ) {
 433
 434                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 435                 // except if c1,c2 are rational, but c1^c2 is not)
 436                 if ( num_basis ) {
 437                         const bool basis_is_crational = num_basis->is_crational();
 438                         const bool exponent_is_crational = num_exponent->is_crational();
 439                         if (!basis_is_crational || !exponent_is_crational) {
 440                                 // return a plain float
 441                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 442                                                                                                status_flags::evaluated |
 443                                                                                                status_flags::expanded);
 444                         }
 445
 446                         const numeric res = num_basis->power(*num_exponent);
 447                         if (res.is_crational()) {
 448                                 return res;
 449                         }
 450                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 451
 452                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 453                         if (basis_is_crational && exponent_is_crational
 454                             && num_exponent->is_real()
 455                             && !num_exponent->is_integer()) {
 456                                 const numeric n = num_exponent->numer();
 457                                 const numeric m = num_exponent->denom();
 458                                 numeric r;
 459                                 numeric q = iquo(n, m, r);
 460                                 if (r.is_negative()) {
 461                                         r += m;
 462                                         --q;
 463                                 }
 464                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 465                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 466                                                 // try it for numerator and denominator separately, in order to
 467                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 468                                                 const numeric bnum = num_basis->numer();
 469                                                 const numeric bden = num_basis->denom();
 470                                                 const numeric res_bnum = bnum.power(*num_exponent);
 471                                                 const numeric res_bden = bden.power(*num_exponent);
 472                                                 if (res_bnum.is_integer())
 473                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 474                                                 if (res_bden.is_integer())
 475                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 476                                         }
 477                                         return this->hold();
 478                                 } else {
 479                                         // assemble resulting product, but allowing for a re-evaluation,
 480                                         // because otherwise we'll end up with something like
 481                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 482                                         // instead of 7/16*7^(1/3).
 483                                         ex prod = power(*num_basis,r.div(m));
 484                                         return prod*power(*num_basis,q);
 485                                 }
 486                         }
 487                 }
 488
 489                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 490                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
 491                 // case c1==1 should not happen, see below!)
 492                 if (is_exactly_a<power>(ebasis)) {
 493                         const power & sub_power = ex_to<power>(ebasis);
 494                         const ex & sub_basis = sub_power.basis;
 495                         const ex & sub_exponent = sub_power.exponent;
 496                         if (is_exactly_a<numeric>(sub_exponent)) {
 497                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 498                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 499                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
 500                                     (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
 501                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 502                                 }
 503                         }
 504                 }
 505
 506                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 507                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 508                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 509                 }
 510
 511                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 512                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 513                         numeric icont = ebasis.integer_content();
 514                         const numeric lead_coeff =
 515                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 516
 517                         const bool canonicalizable = lead_coeff.is_integer();
 518                         const bool unit_normal = lead_coeff.is_pos_integer();
 519                         if (canonicalizable && (! unit_normal))
 520                                 icont = icont.mul(*_num_1_p);
 521
 522                         if (canonicalizable && (icont != *_num1_p)) {
 523                                 const add& addref = ex_to<add>(ebasis);
 524                                 add* addp = new add(addref);
 525                                 addp->setflag(status_flags::dynallocated);
 526                                 addp->clearflag(status_flags::hash_calculated);
 527                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 528                                 for (auto & i : addp->seq)
 529                                         i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
 530
 531                                 const numeric c = icont.power(*num_exponent);
 532                                 if (likely(c != *_num1_p))
 533                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 534                                 else
 535                                         return power(*addp, *num_exponent);
 536                         }
 537                 }
 538
 539                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 540                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 541                 if (is_exactly_a<mul>(ebasis)) {
 542                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 543                         const mul & mulref = ex_to<mul>(ebasis);
 544                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 545                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 546                                 if (num_coeff.is_real()) {
 547                                         if (num_coeff.is_positive()) {
 548                                                 mul *mulp = new mul(mulref);
 549                                                 mulp->overall_coeff = _ex1;
 550                                                 mulp->setflag(status_flags::dynallocated);
 551                                                 mulp->clearflag(status_flags::evaluated);
 552                                                 mulp->clearflag(status_flags::hash_calculated);
 553                                                 return (new mul(power(*mulp,exponent),
 554                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 555                                         } else {
 556                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 557                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 558                                                         mul *mulp = new mul(mulref);
 559                                                         mulp->overall_coeff = _ex_1;
 560                                                         mulp->setflag(status_flags::dynallocated);
 561                                                         mulp->clearflag(status_flags::evaluated);
 562                                                         mulp->clearflag(status_flags::hash_calculated);
 563                                                         return (new mul(power(*mulp,exponent),
 564                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 565                                                 }
 566                                         }
 567                                 }
 568                         }
 569                 }
 570
 571                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 572                 if (num_exponent->is_pos_integer() &&
 573                     ebasis.return_type() != return_types::commutative &&
 574                     !is_a<matrix>(ebasis)) {
 575                         return ncmul(exvector(num_exponent->to_int(), ebasis));
 576                 }
 577         }
 578
 579         if (are_ex_trivially_equal(ebasis,basis) &&
 580             are_ex_trivially_equal(eexponent,exponent)) {
 581                 return this->hold();
 582         }
 583         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 584                                                        status_flags::evaluated);
 585 }
 586
 587 ex power::evalf(int level) const
 588 {
 589         ex ebasis;
 590         ex eexponent;
 591
 592         if (level==1) {
 593                 ebasis = basis;
 594                 eexponent = exponent;
 595         } else if (level == -max_recursion_level) {
 596                 throw(std::runtime_error("max recursion level reached"));
 597         } else {
 598                 ebasis = basis.evalf(level-1);
 599                 if (!is_exactly_a<numeric>(exponent))
 600                         eexponent = exponent.evalf(level-1);
 601                 else
 602                         eexponent = exponent;
 603         }
 604
 605         return power(ebasis,eexponent);
 606 }
 607
 608 ex power::evalm() const
 609 {
 610         const ex ebasis = basis.evalm();
 611         const ex eexponent = exponent.evalm();
 612         if (is_a<matrix>(ebasis)) {
 613                 if (is_exactly_a<numeric>(eexponent)) {
 614                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 615                 }
 616         }
 617         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 618 }
 619
 620 bool power::has(const ex & other, unsigned options) const
 621 {
 622         if (!(options & has_options::algebraic))
 623                 return basic::has(other, options);
 624         if (!is_a<power>(other))
 625                 return basic::has(other, options);
 626         if (!exponent.info(info_flags::integer) ||
 627             !other.op(1).info(info_flags::integer))
 628                 return basic::has(other, options);
 629         if (exponent.info(info_flags::posint) &&
 630             other.op(1).info(info_flags::posint) &&
 631             ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
 632             basis.match(other.op(0)))
 633                 return true;
 634         if (exponent.info(info_flags::negint) &&
 635             other.op(1).info(info_flags::negint) &&
 636             ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
 637             basis.match(other.op(0)))
 638                 return true;
 639         return basic::has(other, options);
 640 }
 641
 642 // from mul.cpp
 643 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 644
 645 ex power::subs(const exmap & m, unsigned options) const
 646 {
 647         const ex &subsed_basis = basis.subs(m, options);
 648         const ex &subsed_exponent = exponent.subs(m, options);
 649
 650         if (!are_ex_trivially_equal(basis, subsed_basis)
 651          || !are_ex_trivially_equal(exponent, subsed_exponent))
 652                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 653
 654         if (!(options & subs_options::algebraic))
 655                 return subs_one_level(m, options);
 656
 657         for (auto & it : m) {
 658                 int nummatches = std::numeric_limits<int>::max();
 659                 exmap repls;
 660                 if (tryfactsubs(*this, it.first, nummatches, repls)) {
 661                         ex anum = it.second.subs(repls, subs_options::no_pattern);
 662                         ex aden = it.first.subs(repls, subs_options::no_pattern);
 663                         ex result = (*this)*power(anum/aden, nummatches);
 664                         return (ex_to<basic>(result)).subs_one_level(m, options);
 665                 }
 666         }
 667
 668         return subs_one_level(m, options);
 669 }
 670
 671 ex power::eval_ncmul(const exvector & v) const
 672 {
 673         return inherited::eval_ncmul(v);
 674 }
 675
 676 ex power::conjugate() const
 677 {
 678         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 679         // branch cut which runs along the negative real axis.
 680         if (basis.info(info_flags::positive)) {
 681                 ex newexponent = exponent.conjugate();
 682                 if (are_ex_trivially_equal(exponent, newexponent)) {
 683                         return *this;
 684                 }
 685                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
 686         }
 687         if (exponent.info(info_flags::integer)) {
 688                 ex newbasis = basis.conjugate();
 689                 if (are_ex_trivially_equal(basis, newbasis)) {
 690                         return *this;
 691                 }
 692                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
 693         }
 694         return conjugate_function(*this).hold();
 695 }
 696
 697 ex power::real_part() const
 698 {
 699         // basis == a+I*b, exponent == c+I*d
 700         const ex a = basis.real_part();
 701         const ex c = exponent.real_part();
 702         if (basis.is_equal(a) && exponent.is_equal(c)) {
 703                 // Re(a^c)
 704                 return *this;
 705         }
 706
 707         const ex b = basis.imag_part();
 708         if (exponent.info(info_flags::integer)) {
 709                 // Re((a+I*b)^c)  w/  c ∈ ℤ
 710                 long N = ex_to<numeric>(c).to_long();
 711                 // Use real terms in Binomial expansion to construct
 712                 // Re(expand(power(a+I*b, N))).
 713                 long NN = N > 0 ? N : -N;
 714                 ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
 715                 ex result = 0;
 716                 for (long n = 0; n <= NN; n += 2) {
 717                         ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
 718                         if (n % 4 == 0) {
 719                                 result += term;  // sign: I^n w/ n == 4*m
 720                         } else {
 721                                 result -= term;  // sign: I^n w/ n == 4*m+2
 722                         }
 723                 }
 724                 return result;
 725         }
 726
 727         // Re((a+I*b)^(c+I*d))
 728         const ex d = exponent.imag_part();
 729         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 730 }
 731
 732 ex power::imag_part() const
 733 {
 734         const ex a = basis.real_part();
 735         const ex c = exponent.real_part();
 736         if (basis.is_equal(a) && exponent.is_equal(c)) {
 737                 // Im(a^c)
 738                 return 0;
 739         }
 740
 741         const ex b = basis.imag_part();
 742         if (exponent.info(info_flags::integer)) {
 743                 // Im((a+I*b)^c)  w/  c ∈ ℤ
 744                 long N = ex_to<numeric>(c).to_long();
 745                 // Use imaginary terms in Binomial expansion to construct
 746                 // Im(expand(power(a+I*b, N))).
 747                 long p = N > 0 ? 1 : 3;  // modulus for positive sign
 748                 long NN = N > 0 ? N : -N;
 749                 ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
 750                 ex result = 0;
 751                 for (long n = 1; n <= NN; n += 2) {
 752                         ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
 753                         if (n % 4 == p) {
 754                                 result += term;  // sign: I^n w/ n == 4*m+p
 755                         } else {
 756                                 result -= term;  // sign: I^n w/ n == 4*m+2+p
 757                         }
 758                 }
 759                 return result;
 760         }
 761
 762         // Im((a+I*b)^(c+I*d))
 763         const ex d = exponent.imag_part();
 764         return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 765 }
 766
 767 // protected
 768
 769 /** Implementation of ex::diff() for a power.
 770  *  @see ex::diff */
 771 ex power::derivative(const symbol & s) const
 772 {
 773         if (is_a<numeric>(exponent)) {
 774                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 775                 epvector newseq;
 776                 newseq.reserve(2);
 777                 newseq.push_back(expair(basis, exponent - _ex1));
 778                 newseq.push_back(expair(basis.diff(s), _ex1));
 779                 return mul(std::move(newseq), exponent);
 780         } else {
 781                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 782                 return mul(*this,
 783                            add(mul(exponent.diff(s), log(basis)),
 784                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 785         }
 786 }
 787
 788 int power::compare_same_type(const basic & other) const
 789 {
 790         GINAC_ASSERT(is_exactly_a<power>(other));
 791         const power &o = static_cast<const power &>(other);
 792
 793         int cmpval = basis.compare(o.basis);
 794         if (cmpval)
 795                 return cmpval;
 796         else
 797                 return exponent.compare(o.exponent);
 798 }
 799
 800 unsigned power::return_type() const
 801 {
 802         return basis.return_type();
 803 }
 804
 805 return_type_t power::return_type_tinfo() const
 806 {
 807         return basis.return_type_tinfo();
 808 }
 809
 810 ex power::expand(unsigned options) const
 811 {
 812         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 813                 // A special case worth optimizing.
 814                 setflag(status_flags::expanded);
 815                 return *this;
 816         }
 817
 818         // (x*p)^c -> x^c * p^c, if p>0
 819         // makes sense before expanding the basis
 820         if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
 821                 const mul &m = ex_to<mul>(basis);
 822                 exvector prodseq;
 823                 epvector powseq;
 824                 prodseq.reserve(m.seq.size() + 1);
 825                 powseq.reserve(m.seq.size() + 1);
 826                 bool possign = true;
 827
 828                 // search for positive/negative factors
 829                 for (auto & cit : m.seq) {
 830                         ex e=m.recombine_pair_to_ex(cit);
 831                         if (e.info(info_flags::positive))
 832                                 prodseq.push_back(pow(e, exponent).expand(options));
 833                         else if (e.info(info_flags::negative)) {
 834                                 prodseq.push_back(pow(-e, exponent).expand(options));
 835                                 possign = !possign;
 836                         } else
 837                                 powseq.push_back(cit);
 838                 }
 839
 840                 // take care on the numeric coefficient
 841                 ex coeff=(possign? _ex1 : _ex_1);
 842                 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
 843                         prodseq.push_back(power(m.overall_coeff, exponent));
 844                 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
 845                         prodseq.push_back(power(-m.overall_coeff, exponent));
 846                 else
 847                         coeff *= m.overall_coeff;
 848
 849                 // If positive/negative factors are found, then extract them.
 850                 // In either case we set a flag to avoid the second run on a part
 851                 // which does not have positive/negative terms.
 852                 if (prodseq.size() > 0) {
 853                         ex newbasis = coeff*mul(std::move(powseq));
 854                         ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
 855                         return ((new mul(std::move(prodseq)))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
 856                 } else
 857                         ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
 858         }
 859
 860         const ex expanded_basis = basis.expand(options);
 861         const ex expanded_exponent = exponent.expand(options);
 862
 863         // x^(a+b) -> x^a * x^b
 864         if (is_exactly_a<add>(expanded_exponent)) {
 865                 const add &a = ex_to<add>(expanded_exponent);
 866                 exvector distrseq;
 867                 distrseq.reserve(a.seq.size() + 1);
 868                 for (auto & cit : a.seq) {
 869                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(cit)));
 870                 }
 871
 872                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 873                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 874                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 875                         long int_exponent = num_exponent.to_int();
 876                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 877                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 878                         else
 879                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 880                 } else
 881                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 882
 883                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 884                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 885                 return r.expand(options);
 886         }
 887
 888         if (!is_exactly_a<numeric>(expanded_exponent) ||
 889                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 890                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 891                         return this->hold();
 892                 } else {
 893                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 894                 }
 895         }
 896
 897         // integer numeric exponent
 898         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 899         long int_exponent = num_exponent.to_long();
 900
 901         // (x+y)^n, n>0
 902         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 903                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 904
 905         // (x*y)^n -> x^n * y^n
 906         if (is_exactly_a<mul>(expanded_basis))
 907                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 908
 909         // cannot expand further
 910         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 911                 return this->hold();
 912         else
 913                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 914 }
 915
 916 //////////
 917 // new virtual functions which can be overridden by derived classes
 918 //////////
 919
 920 // none
 921
 922 //////////
 923 // non-virtual functions in this class
 924 //////////
 925
 926 namespace {  // anonymous namespace for power::expand_add() helpers
 927
 928 /** Helper class to generate all bounded combinatorial partitions of an integer
 929  *  n with exactly m parts (including zero parts) in non-decreasing order.
 930  */
 931 class partition_generator {
 932 private:
 933         // Partitions n into m parts, not including zero parts.
 934         // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
 935         // FXT library)
 936         struct mpartition2
 937         {
 938                 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
 939                 std::vector<int> x;
 940                 int n;   // n>0
 941                 int m;   // 0<m<=n
 942                 mpartition2(unsigned n_, unsigned m_)
 943                   : x(m_+1), n(n_), m(m_)
 944                 {
 945                         for (int k=1; k<m; ++k)
 946                                 x[k] = 1;
 947                         x[m] = n - m + 1;
 948                 }
 949                 bool next_partition()
 950                 {
 951                         int u = x[m];  // last element
 952                         int k = m;
 953                         int s = u;
 954                         while (--k) {
 955                                 s += x[k];
 956                                 if (x[k] + 2 <= u)
 957                                         break;
 958                         }
 959                         if (k==0)
 960                                 return false;  // current is last
 961                         int f = x[k] + 1;
 962                         while (k < m) {
 963                                 x[k] = f;
 964                                 s -= f;
 965                                 ++k;
 966                         }
 967                         x[m] = s;
 968                         return true;
 969                 }
 970         } mpgen;
 971         int m;  // number of parts 0<m<=n
 972         mutable std::vector<int> partition;  // current partition
 973 public:
 974         partition_generator(unsigned n_, unsigned m_)
 975           : mpgen(n_, 1), m(m_), partition(m_)
 976         { }
 977         // returns current partition in non-decreasing order, padded with zeros
 978         const std::vector<int>& current() const
 979         {
 980                 for (int i = 0; i < m - mpgen.m; ++i)
 981                         partition[i] = 0;  // pad with zeros
 982
 983                 for (int i = m - mpgen.m; i < m; ++i)
 984                         partition[i] = mpgen.x[i - m + mpgen.m + 1];
 985
 986                 return partition;
 987         }
 988         bool next()
 989         {
 990                 if (!mpgen.next_partition()) {
 991                         if (mpgen.m == m || mpgen.m == mpgen.n)
 992                                 return false;  // current is last
 993                         // increment number of parts
 994                         mpgen = mpartition2(mpgen.n, mpgen.m + 1);
 995                 }
 996                 return true;
 997         }
 998 };
 999
1000 /** Helper class to generate all compositions of a partition of an integer n,
1001  *  starting with the compositions which has non-decreasing order.
1002  */
1003 class composition_generator {
1004 private:
1005         // Generates all distinct permutations of a multiset.
1006         // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
1007         // Multiset Permutations using a Constant Number of Variables by Prefix
1008         // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
1009         struct coolmulti {
1010                 // element of singly linked list
1011                 struct element {
1012                         int value;
1013                         element* next;
1014                         element(int val, element* n)
1015                           : value(val), next(n) {}
1016                         ~element()
1017                         {   // recurses down to the end of the singly linked list
1018                                 delete next;
1019                         }
1020                 };
1021                 element *head, *i, *after_i;
1022                 // NB: Partition must be sorted in non-decreasing order.
1023                 explicit coolmulti(const std::vector<int>& partition)
1024                   : head(nullptr), i(nullptr), after_i(nullptr)
1025                 {
1026                         for (unsigned n = 0; n < partition.size(); ++n) {
1027                                 head = new element(partition[n], head);
1028                                 if (n <= 1)
1029                                         i = head;
1030                         }
1031                         after_i = i->next;
1032                 }
1033                 ~coolmulti()
1034                 {   // deletes singly linked list
1035                         delete head;
1036                 }
1037                 void next_permutation()
1038                 {
1039                         element *before_k;
1040                         if (after_i->next != nullptr && i->value >= after_i->next->value)
1041                                 before_k = after_i;
1042                         else
1043                                 before_k = i;
1044                         element *k = before_k->next;
1045                         before_k->next = k->next;
1046                         k->next = head;
1047                         if (k->value < head->value)
1048                                 i = k;
1049                         after_i = i->next;
1050                         head = k;
1051                 }
1052                 bool finished() const
1053                 {
1054                         return after_i->next == nullptr && after_i->value >= head->value;
1055                 }
1056         } cmgen;
1057         bool atend;  // needed for simplifying iteration over permutations
1058         bool trivial;  // likewise, true if all elements are equal
1059         mutable std::vector<int> composition;  // current compositions
1060 public:
1061         explicit composition_generator(const std::vector<int>& partition)
1062           : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1063         {
1064                 for (unsigned i=1; i<partition.size(); ++i)
1065                         trivial = trivial && (partition[0] == partition[i]);
1066         }
1067         const std::vector<int>& current() const
1068         {
1069                 coolmulti::element* it = cmgen.head;
1070                 size_t i = 0;
1071                 while (it != nullptr) {
1072                         composition[i] = it->value;
1073                         it = it->next;
1074                         ++i;
1075                 }
1076                 return composition;
1077         }
1078         bool next()
1079         {
1080                 // This ugly contortion is needed because the original coolmulti
1081                 // algorithm requires code duplication of the payload procedure,
1082                 // one before the loop and one inside it.
1083                 if (trivial || atend)
1084                         return false;
1085                 cmgen.next_permutation();
1086                 atend = cmgen.finished();
1087                 return true;
1088         }
1089 };
1090
1091 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1092  *  where n = p1+p2+...+pk, i.e. p is a partition of n.
1093  */
1094 const numeric
1095 multinomial_coefficient(const std::vector<int> & p)
1096 {
1097         numeric n = 0, d = 1;
1098         for (auto & it : p) {
1099                 n += numeric(it);
1100                 d *= factorial(numeric(it));
1101         }
1102         return factorial(numeric(n)) / d;
1103 }
1104
1105 }  // anonymous namespace
1106
1107
1108 /** expand a^n where a is an add and n is a positive integer.
1109  *  @see power::expand */
1110 ex power::expand_add(const add & a, long n, unsigned options)
1111 {
1112         // The special case power(+(x,...y;x),2) can be optimized better.
1113         if (n==2)
1114                 return expand_add_2(a, options);
1115
1116         // method:
1117         //
1118         // Consider base as the sum of all symbolic terms and the overall numeric
1119         // coefficient and apply the binomial theorem:
1120         // S = power(+(x,...,z;c),n)
1121         //   = power(+(+(x,...,z;0);c),n)
1122         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1123         // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1124         // The multinomial theorem is computed by an outer loop over all
1125         // partitions of the exponent and an inner loop over all compositions of
1126         // that partition. This method makes the expansion a combinatorial
1127         // problem and allows us to directly construct the expanded sum and also
1128         // to re-use the multinomial coefficients (since they depend only on the
1129         // partition, not on the composition).
1130         //
1131         // multinomial power(+(x,y,z;0),3) example:
1132         // partition : compositions                : multinomial coefficient
1133         // [0,0,3]   : [3,0,0],[0,3,0],[0,0,3]     : 3!/(3!*0!*0!) = 1
1134         // [0,1,2]   : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1135         // [1,1,1]   : [1,1,1]                     : 3!/(1!*1!*1!) = 6
1136         //  =>  (x + y + z)^3 =
1137         //        x^3 + y^3 + z^3
1138         //      + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1139         //      + 6*x*y*z
1140         //
1141         // multinomial power(+(x,y,z;0),4) example:
1142         // partition : compositions                : multinomial coefficient
1143         // [0,0,4]   : [4,0,0],[0,4,0],[0,0,4]     : 4!/(4!*0!*0!) = 1
1144         // [0,1,3]   : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1145         // [0,2,2]   : [2,2,0],[2,0,2],[0,2,2]     : 4!/(2!*2!*0!) = 6
1146         // [1,1,2]   : [2,1,1],[1,2,1],[1,1,2]     : 4!/(2!*1!*1!) = 12
1147         // (no [1,1,1,1] partition since it has too many parts)
1148         //  =>  (x + y + z)^4 =
1149         //        x^4 + y^4 + z^4
1150         //      + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1151         //      + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1152         //      + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1153         //
1154         // Summary:
1155         // r = 0
1156         // for k from 0 to n:
1157         //     f = c^(n-k)*binomial(n,k)
1158         //     for p in all partitions of n with m parts (including zero parts):
1159         //         h = f * multinomial coefficient of p
1160         //         for c in all compositions of p:
1161         //             t = 1
1162         //             for e in all elements of c:
1163         //                 t = t * a[e]^e
1164         //             r = r + h*t
1165         // return r
1166
1167         epvector result;
1168         // The number of terms will be the number of combinatorial compositions,
1169         // i.e. the number of unordered arrangements of m nonnegative integers
1170         // which sum up to n.  It is frequently written as C_n(m) and directly
1171         // related with binomial coefficients: binomial(n+m-1,m-1).
1172         size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
1173         if (!a.overall_coeff.is_zero()) {
1174                 // the result's overall_coeff is one of the terms
1175                 --result_size;
1176         }
1177         result.reserve(result_size);
1178
1179         // Iterate over all terms in binomial expansion of
1180         // S = power(+(x,...,z;c),n)
1181         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1182         for (int k = 1; k <= n; ++k) {
1183                 numeric binomial_coefficient;  // binomial(n,k)*c^(n-k)
1184                 if (a.overall_coeff.is_zero()) {
1185                         // degenerate case with zero overall_coeff:
1186                         // apply multinomial theorem directly to power(+(x,...z;0),n)
1187                         binomial_coefficient = 1;
1188                         if (k < n) {
1189                                 continue;
1190                         }
1191                 } else {
1192                         binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1193                 }
1194
1195                 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1196                 // Iterate over all partitions of k with exactly as many parts as
1197                 // there are symbolic terms in the basis (including zero parts).
1198                 partition_generator partitions(k, a.seq.size());
1199                 do {
1200                         const std::vector<int>& partition = partitions.current();
1201                         // All monomials of this partition have the same number of terms and the same coefficient.
1202                         const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
1203                         const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1204
1205                         // Iterate over all compositions of the current partition.
1206                         composition_generator compositions(partition);
1207                         do {
1208                                 const std::vector<int>& exponent = compositions.current();
1209                                 exvector monomial;
1210                                 monomial.reserve(msize);
1211                                 numeric factor = coeff;
1212                                 for (unsigned i = 0; i < exponent.size(); ++i) {
1213                                         const ex & r = a.seq[i].rest;
1214                                         GINAC_ASSERT(!is_exactly_a<add>(r));
1215                                         GINAC_ASSERT(!is_exactly_a<power>(r) ||
1216                                                      !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1217                                                      !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1218                                                      !is_exactly_a<add>(ex_to<power>(r).basis) ||
1219                                                      !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1220                                                      !is_exactly_a<power>(ex_to<power>(r).basis));
1221                                         GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1222                                         const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1223                                         if (exponent[i] == 0) {
1224                                                 // optimize away
1225                                         } else if (exponent[i] == 1) {
1226                                                 // optimized
1227                                                 monomial.push_back(r);
1228                                                 if (c != *_num1_p)
1229                                                         factor = factor.mul(c);
1230                                         } else { // general case exponent[i] > 1
1231                                                 monomial.push_back((new power(r, exponent[i]))->setflag(status_flags::dynallocated));
1232                                                 if (c != *_num1_p)
1233                                                         factor = factor.mul(c.power(exponent[i]));
1234                                         }
1235                                 }
1236                                 result.push_back(a.combine_ex_with_coeff_to_pair(mul(monomial).expand(options), factor));
1237                         } while (compositions.next());
1238                 } while (partitions.next());
1239         }
1240
1241         GINAC_ASSERT(result.size() == result_size);
1242
1243         if (a.overall_coeff.is_zero()) {
1244                 return (new add(std::move(result)))->setflag(status_flags::dynallocated |
1245                                                              status_flags::expanded);
1246         } else {
1247                 return (new add(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated |
1248                                                                                                        status_flags::expanded);
1249         }
1250 }
1251
1252
1253 /** Special case of power::expand_add. Expands a^2 where a is an add.
1254  *  @see power::expand_add */
1255 ex power::expand_add_2(const add & a, unsigned options)
1256 {
1257         epvector result;
1258         size_t result_size = (a.nops() * (a.nops()+1)) / 2;
1259         if (!a.overall_coeff.is_zero()) {
1260                 // the result's overall_coeff is one of the terms
1261                 --result_size;
1262         }
1263         result.reserve(result_size);
1264
1265         epvector::const_iterator last = a.seq.end();
1266
1267         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1268         // first part: ignore overall_coeff and expand other terms
1269         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1270                 const ex & r = cit0->rest;
1271                 const ex & c = cit0->coeff;
1272
1273                 GINAC_ASSERT(!is_exactly_a<add>(r));
1274                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1275                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1276                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1277                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
1278                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1279                              !is_exactly_a<power>(ex_to<power>(r).basis));
1280
1281                 if (c.is_equal(_ex1)) {
1282                         if (is_exactly_a<mul>(r)) {
1283                                 result.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1284                                                                                  _ex1));
1285                         } else {
1286                                 result.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1287                                                                                  _ex1));
1288                         }
1289                 } else {
1290                         if (is_exactly_a<mul>(r)) {
1291                                 result.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1292                                                                                  ex_to<numeric>(c).power_dyn(*_num2_p)));
1293                         } else {
1294                                 result.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1295                                                                                  ex_to<numeric>(c).power_dyn(*_num2_p)));
1296                         }
1297                 }
1298
1299                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1300                         const ex & r1 = cit1->rest;
1301                         const ex & c1 = cit1->coeff;
1302                         result.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
1303                                                                          _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1304                 }
1305         }
1306
1307         // second part: add terms coming from overall_coeff (if != 0)
1308         if (!a.overall_coeff.is_zero()) {
1309                 for (auto & i : a.seq)
1310                         result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1311         }
1312
1313         GINAC_ASSERT(result.size() == result_size);
1314
1315         if (a.overall_coeff.is_zero()) {
1316                 return (new add(std::move(result)))->setflag(status_flags::dynallocated |
1317                                                              status_flags::expanded);
1318         } else {
1319                 return (new add(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)))->setflag(status_flags::dynallocated |
1320                                                                                                        status_flags::expanded);
1321         }
1322 }
1323
1324 /** Expand factors of m in m^n where m is a mul and n is an integer.
1325  *  @see power::expand */
1326 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
1327 {
1328         GINAC_ASSERT(n.is_integer());
1329
1330         if (n.is_zero()) {
1331                 return _ex1;
1332         }
1333
1334         // do not bother to rename indices if there are no any.
1335         if (!(options & expand_options::expand_rename_idx) &&
1336             m.info(info_flags::has_indices))
1337                 options |= expand_options::expand_rename_idx;
1338         // Leave it to multiplication since dummy indices have to be renamed
1339         if ((options & expand_options::expand_rename_idx) &&
1340             (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1341                 ex result = m;
1342                 exvector va = get_all_dummy_indices(m);
1343                 sort(va.begin(), va.end(), ex_is_less());
1344
1345                 for (int i=1; i < n.to_int(); i++)
1346                         result *= rename_dummy_indices_uniquely(va, m);
1347                 return result;
1348         }
1349
1350         epvector distrseq;
1351         distrseq.reserve(m.seq.size());
1352         bool need_reexpand = false;
1353
1354         for (auto & cit : m.seq) {
1355                 expair p = m.combine_pair_with_coeff_to_pair(cit, n);
1356                 if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1357                         // this happens when e.g. (a+b)^(1/2) gets squared and
1358                         // the resulting product needs to be reexpanded
1359                         need_reexpand = true;
1360                 }
1361                 distrseq.push_back(p);
1362         }
1363
1364         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1365         if (need_reexpand)
1366                 return ex(result).expand(options);
1367         if (from_expand)
1368                 return result.setflag(status_flags::expanded);
1369         return result;
1370 }
1371
1372 GINAC_BIND_UNARCHIVER(power);
1373
1374 } // namespace GiNaC