* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
- * GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2017 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
-#include <vector>
-#include <iostream>
-#include <stdexcept>
-#include <limits>
-
#include "power.h"
#include "expairseq.h"
#include "add.h"
#include "relational.h"
#include "compiler.h"
+#include <iostream>
+#include <limits>
+#include <stdexcept>
+#include <vector>
+#include <algorithm>
+
namespace GiNaC {
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
print_func<print_python_repr>(&power::do_print_python_repr).
print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
-typedef std::vector<int> intvector;
-
//////////
// default constructor
//////////
static void print_sym_pow(const print_context & c, const symbol &x, int exp)
{
// Optimal output of integer powers of symbols to aid compiler CSE.
- // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
+ // C.f. ISO/IEC 14882:2011, section 1.9 [intro execution], paragraph 15
// to learn why such a parenthesation is really necessary.
if (exp == 1) {
x.print(c);
void power::do_print_csrc(const print_csrc & c, unsigned level) const
{
// Integer powers of symbols are printed in a special, optimized way
- if (exponent.info(info_flags::integer)
- && (is_a<symbol>(basis) || is_a<constant>(basis))) {
+ if (exponent.info(info_flags::integer) &&
+ (is_a<symbol>(basis) || is_a<constant>(basis))) {
int exp = ex_to<numeric>(exponent).to_int();
if (exp > 0)
c.s << '(';
case info_flags::cinteger_polynomial:
case info_flags::rational_polynomial:
case info_flags::crational_polynomial:
- return exponent.info(info_flags::nonnegint) &&
- basis.info(inf);
+ return basis.info(inf) && exponent.info(info_flags::nonnegint);
case info_flags::rational_function:
- return exponent.info(info_flags::integer) &&
- basis.info(inf);
- case info_flags::algebraic:
- return !exponent.info(info_flags::integer) ||
- basis.info(inf);
+ return basis.info(inf) && exponent.info(info_flags::integer);
+ case info_flags::real:
+ return basis.info(inf) && exponent.info(info_flags::integer);
case info_flags::expanded:
return (flags & status_flags::expanded);
+ case info_flags::positive:
+ return basis.info(info_flags::positive) && exponent.info(info_flags::real);
+ case info_flags::nonnegative:
+ return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
+ (basis.info(info_flags::real) && exponent.info(info_flags::even));
case info_flags::has_indices: {
if (flags & status_flags::has_indices)
return true;
if (!are_ex_trivially_equal(basis, mapped_basis)
|| !are_ex_trivially_equal(exponent, mapped_exponent))
- return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
+ return dynallocate<power>(mapped_basis, mapped_exponent);
else
return *this;
}
bool power::is_polynomial(const ex & var) const
{
- if (exponent.has(var))
- return false;
- if (!exponent.info(info_flags::nonnegint))
- return false;
- return basis.is_polynomial(var);
+ if (basis.is_polynomial(var)) {
+ if (basis.has(var))
+ // basis is non-constant polynomial in var
+ return exponent.info(info_flags::nonnegint);
+ else
+ // basis is constant in var
+ return !exponent.has(var);
+ }
+ // basis is a non-polynomial function of var
+ return false;
}
int power::degree(const ex & s) const
* - ^(1,x) -> 1
* - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
* - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
- * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
+ * - ^(^(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!)
* - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
* - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
* - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
- *
- * @param level cut-off in recursive evaluation */
-ex power::eval(int level) const
+ */
+ex power::eval() const
{
- if ((level==1) && (flags & status_flags::evaluated))
+ if (flags & status_flags::evaluated)
return *this;
- else if (level == -max_recursion_level)
- throw(std::runtime_error("max recursion level reached"));
-
- const ex & ebasis = level==1 ? basis : basis.eval(level-1);
- const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
-
- bool basis_is_numerical = false;
- bool exponent_is_numerical = false;
- const numeric *num_basis;
- const numeric *num_exponent;
-
- if (is_exactly_a<numeric>(ebasis)) {
- basis_is_numerical = true;
- num_basis = &ex_to<numeric>(ebasis);
+
+ const numeric *num_basis = nullptr;
+ const numeric *num_exponent = nullptr;
+
+ if (is_exactly_a<numeric>(basis)) {
+ num_basis = &ex_to<numeric>(basis);
}
- if (is_exactly_a<numeric>(eexponent)) {
- exponent_is_numerical = true;
- num_exponent = &ex_to<numeric>(eexponent);
+ if (is_exactly_a<numeric>(exponent)) {
+ num_exponent = &ex_to<numeric>(exponent);
}
// ^(x,0) -> 1 (0^0 also handled here)
- if (eexponent.is_zero()) {
- if (ebasis.is_zero())
+ if (exponent.is_zero()) {
+ if (basis.is_zero())
throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
else
return _ex1;
}
// ^(x,1) -> x
- if (eexponent.is_equal(_ex1))
- return ebasis;
+ if (exponent.is_equal(_ex1))
+ return basis;
// ^(0,c1) -> 0 or exception (depending on real value of c1)
- if (ebasis.is_zero() && exponent_is_numerical) {
+ if (basis.is_zero() && num_exponent) {
if ((num_exponent->real()).is_zero())
throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
else if ((num_exponent->real()).is_negative())
}
// ^(1,x) -> 1
- if (ebasis.is_equal(_ex1))
+ if (basis.is_equal(_ex1))
return _ex1;
// power of a function calculated by separate rules defined for this function
- if (is_exactly_a<function>(ebasis))
- return ex_to<function>(ebasis).power(eexponent);
+ if (is_exactly_a<function>(basis))
+ return ex_to<function>(basis).power(exponent);
// Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
- if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
- return power(ebasis.op(0), ebasis.op(1) * eexponent);
+ if (is_exactly_a<power>(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real))
+ return dynallocate<power>(basis.op(0), basis.op(1) * exponent);
- if (exponent_is_numerical) {
+ if ( num_exponent ) {
// ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
// except if c1,c2 are rational, but c1^c2 is not)
- if (basis_is_numerical) {
+ if ( num_basis ) {
const bool basis_is_crational = num_basis->is_crational();
const bool exponent_is_crational = num_exponent->is_crational();
if (!basis_is_crational || !exponent_is_crational) {
// return a plain float
- return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
- status_flags::evaluated |
- status_flags::expanded);
+ return dynallocate<numeric>(num_basis->power(*num_exponent));
}
const numeric res = num_basis->power(*num_exponent);
const numeric res_bnum = bnum.power(*num_exponent);
const numeric res_bden = bden.power(*num_exponent);
if (res_bnum.is_integer())
- return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ return dynallocate<mul>(dynallocate<power>(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated);
if (res_bden.is_integer())
- return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
+ return dynallocate<mul>(dynallocate<power>(bnum,*num_exponent),res_bden.inverse()).setflag(status_flags::evaluated);
}
return this->hold();
} else {
// because otherwise we'll end up with something like
// (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
// instead of 7/16*7^(1/3).
- ex prod = power(*num_basis,r.div(m));
- return prod*power(*num_basis,q);
+ return pow(basis, r.div(m)) * pow(basis, q);
}
}
}
// ^(^(x,c1),c2) -> ^(x,c1*c2)
- // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
+ // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
// case c1==1 should not happen, see below!)
- if (is_exactly_a<power>(ebasis)) {
- const power & sub_power = ex_to<power>(ebasis);
+ if (is_exactly_a<power>(basis)) {
+ const power & sub_power = ex_to<power>(basis);
const ex & sub_basis = sub_power.basis;
const ex & sub_exponent = sub_power.exponent;
if (is_exactly_a<numeric>(sub_exponent)) {
const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
GINAC_ASSERT(num_sub_exponent!=numeric(1));
- if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
- return power(sub_basis,num_sub_exponent.mul(*num_exponent));
+ if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
+ (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
+ return dynallocate<power>(sub_basis, num_sub_exponent.mul(*num_exponent));
}
}
}
// ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
- if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
- return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
+ if (num_exponent->is_integer() && is_exactly_a<mul>(basis)) {
+ return expand_mul(ex_to<mul>(basis), *num_exponent, false);
}
// (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
- if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
- numeric icont = ebasis.integer_content();
+ if (num_exponent->is_integer() && is_exactly_a<add>(basis)) {
+ numeric icont = basis.integer_content();
const numeric lead_coeff =
- ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
+ ex_to<numeric>(ex_to<add>(basis).seq.begin()->coeff).div(icont);
const bool canonicalizable = lead_coeff.is_integer();
const bool unit_normal = lead_coeff.is_pos_integer();
icont = icont.mul(*_num_1_p);
if (canonicalizable && (icont != *_num1_p)) {
- const add& addref = ex_to<add>(ebasis);
- add* addp = new add(addref);
- addp->setflag(status_flags::dynallocated);
- addp->clearflag(status_flags::hash_calculated);
- addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
- for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
- i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
+ const add& addref = ex_to<add>(basis);
+ add & addp = dynallocate<add>(addref);
+ addp.clearflag(status_flags::hash_calculated);
+ addp.overall_coeff = ex_to<numeric>(addp.overall_coeff).div_dyn(icont);
+ for (auto & i : addp.seq)
+ i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
const numeric c = icont.power(*num_exponent);
if (likely(c != *_num1_p))
- return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
+ return dynallocate<mul>(dynallocate<power>(addp, *num_exponent), c);
else
- return power(*addp, *num_exponent);
+ return dynallocate<power>(addp, *num_exponent);
}
}
// ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
// ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
- if (is_exactly_a<mul>(ebasis)) {
+ if (is_exactly_a<mul>(basis)) {
GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
- const mul & mulref = ex_to<mul>(ebasis);
+ const mul & mulref = ex_to<mul>(basis);
if (!mulref.overall_coeff.is_equal(_ex1)) {
const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
if (num_coeff.is_real()) {
if (num_coeff.is_positive()) {
- mul *mulp = new mul(mulref);
- mulp->overall_coeff = _ex1;
- mulp->clearflag(status_flags::evaluated);
- mulp->clearflag(status_flags::hash_calculated);
- return (new mul(power(*mulp,exponent),
- power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
+ mul & mulp = dynallocate<mul>(mulref);
+ mulp.overall_coeff = _ex1;
+ mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
+ return dynallocate<mul>(dynallocate<power>(mulp, exponent),
+ dynallocate<power>(num_coeff, *num_exponent));
} else {
GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
if (!num_coeff.is_equal(*_num_1_p)) {
- mul *mulp = new mul(mulref);
- mulp->overall_coeff = _ex_1;
- mulp->clearflag(status_flags::evaluated);
- mulp->clearflag(status_flags::hash_calculated);
- return (new mul(power(*mulp,exponent),
- power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
+ mul & mulp = dynallocate<mul>(mulref);
+ mulp.overall_coeff = _ex_1;
+ mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
+ return dynallocate<mul>(dynallocate<power>(mulp, exponent),
+ dynallocate<power>(abs(num_coeff), *num_exponent));
}
}
}
// ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
if (num_exponent->is_pos_integer() &&
- ebasis.return_type() != return_types::commutative &&
- !is_a<matrix>(ebasis)) {
- return ncmul(exvector(num_exponent->to_int(), ebasis), true);
+ basis.return_type() != return_types::commutative &&
+ !is_a<matrix>(basis)) {
+ return ncmul(exvector(num_exponent->to_int(), basis));
}
}
-
- if (are_ex_trivially_equal(ebasis,basis) &&
- are_ex_trivially_equal(eexponent,exponent)) {
- return this->hold();
- }
- return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
- status_flags::evaluated);
+
+ return this->hold();
}
-ex power::evalf(int level) const
+ex power::evalf() const
{
- ex ebasis;
+ ex ebasis = basis.evalf();
ex eexponent;
- if (level==1) {
- ebasis = basis;
+ if (!is_exactly_a<numeric>(exponent))
+ eexponent = exponent.evalf();
+ else
eexponent = exponent;
- } else if (level == -max_recursion_level) {
- throw(std::runtime_error("max recursion level reached"));
- } else {
- ebasis = basis.evalf(level-1);
- if (!is_exactly_a<numeric>(exponent))
- eexponent = exponent.evalf(level-1);
- else
- eexponent = exponent;
- }
- return power(ebasis,eexponent);
+ return dynallocate<power>(ebasis, eexponent);
}
ex power::evalm() const
const ex eexponent = exponent.evalm();
if (is_a<matrix>(ebasis)) {
if (is_exactly_a<numeric>(eexponent)) {
- return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
+ return dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
}
}
- return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
+ return dynallocate<power>(ebasis, eexponent);
}
bool power::has(const ex & other, unsigned options) const
return basic::has(other, options);
if (!is_a<power>(other))
return basic::has(other, options);
- if (!exponent.info(info_flags::integer)
- || !other.op(1).info(info_flags::integer))
+ if (!exponent.info(info_flags::integer) ||
+ !other.op(1).info(info_flags::integer))
return basic::has(other, options);
- if (exponent.info(info_flags::posint)
- && other.op(1).info(info_flags::posint)
- && ex_to<numeric>(exponent).to_int()
- > ex_to<numeric>(other.op(1)).to_int()
- && basis.match(other.op(0)))
+ if (exponent.info(info_flags::posint) &&
+ other.op(1).info(info_flags::posint) &&
+ ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
+ basis.match(other.op(0)))
return true;
- if (exponent.info(info_flags::negint)
- && other.op(1).info(info_flags::negint)
- && ex_to<numeric>(exponent).to_int()
- < ex_to<numeric>(other.op(1)).to_int()
- && basis.match(other.op(0)))
+ if (exponent.info(info_flags::negint) &&
+ other.op(1).info(info_flags::negint) &&
+ ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
+ basis.match(other.op(0)))
return true;
return basic::has(other, options);
}
if (!(options & subs_options::algebraic))
return subs_one_level(m, options);
- for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
+ for (auto & it : m) {
int nummatches = std::numeric_limits<int>::max();
exmap repls;
- if (tryfactsubs(*this, it->first, nummatches, repls)) {
- ex anum = it->second.subs(repls, subs_options::no_pattern);
- ex aden = it->first.subs(repls, subs_options::no_pattern);
- ex result = (*this)*power(anum/aden, nummatches);
+ if (tryfactsubs(*this, it.first, nummatches, repls)) {
+ ex anum = it.second.subs(repls, subs_options::no_pattern);
+ ex aden = it.first.subs(repls, subs_options::no_pattern);
+ ex result = (*this) * pow(anum/aden, nummatches);
return (ex_to<basic>(result)).subs_one_level(m, options);
}
}
ex power::conjugate() const
{
- ex newbasis = basis.conjugate();
- ex newexponent = exponent.conjugate();
- if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
- return *this;
+ // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
+ // branch cut which runs along the negative real axis.
+ if (basis.info(info_flags::positive)) {
+ ex newexponent = exponent.conjugate();
+ if (are_ex_trivially_equal(exponent, newexponent)) {
+ return *this;
+ }
+ return dynallocate<power>(basis, newexponent);
}
- return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
+ if (exponent.info(info_flags::integer)) {
+ ex newbasis = basis.conjugate();
+ if (are_ex_trivially_equal(basis, newbasis)) {
+ return *this;
+ }
+ return dynallocate<power>(newbasis, exponent);
+ }
+ return conjugate_function(*this).hold();
}
ex power::real_part() const
{
+ // basis == a+I*b, exponent == c+I*d
+ const ex a = basis.real_part();
+ const ex c = exponent.real_part();
+ if (basis.is_equal(a) && exponent.is_equal(c)) {
+ // Re(a^c)
+ return *this;
+ }
+
+ const ex b = basis.imag_part();
if (exponent.info(info_flags::integer)) {
- ex basis_real = basis.real_part();
- if (basis_real == basis)
- return *this;
- realsymbol a("a"),b("b");
- ex result;
- if (exponent.info(info_flags::posint))
- result = power(a+I*b,exponent);
- else
- result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
- result = result.expand();
- result = result.real_part();
- result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
+ // Re((a+I*b)^c) w/ c ∈ ℤ
+ long N = ex_to<numeric>(c).to_long();
+ // Use real terms in Binomial expansion to construct
+ // Re(expand(pow(a+I*b, N))).
+ long NN = N > 0 ? N : -N;
+ ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
+ ex result = 0;
+ for (long n = 0; n <= NN; n += 2) {
+ ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
+ if (n % 4 == 0) {
+ result += term; // sign: I^n w/ n == 4*m
+ } else {
+ result -= term; // sign: I^n w/ n == 4*m+2
+ }
+ }
return result;
}
-
- ex a = basis.real_part();
- ex b = basis.imag_part();
- ex c = exponent.real_part();
- ex d = exponent.imag_part();
- return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
+
+ // Re((a+I*b)^(c+I*d))
+ const ex d = exponent.imag_part();
+ return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
}
ex power::imag_part() const
{
+ // basis == a+I*b, exponent == c+I*d
+ const ex a = basis.real_part();
+ const ex c = exponent.real_part();
+ if (basis.is_equal(a) && exponent.is_equal(c)) {
+ // Im(a^c)
+ return 0;
+ }
+
+ const ex b = basis.imag_part();
if (exponent.info(info_flags::integer)) {
- ex basis_real = basis.real_part();
- if (basis_real == basis)
- return 0;
- realsymbol a("a"),b("b");
- ex result;
- if (exponent.info(info_flags::posint))
- result = power(a+I*b,exponent);
- else
- result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
- result = result.expand();
- result = result.imag_part();
- result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
+ // Im((a+I*b)^c) w/ c ∈ ℤ
+ long N = ex_to<numeric>(c).to_long();
+ // Use imaginary terms in Binomial expansion to construct
+ // Im(expand(pow(a+I*b, N))).
+ long p = N > 0 ? 1 : 3; // modulus for positive sign
+ long NN = N > 0 ? N : -N;
+ ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
+ ex result = 0;
+ for (long n = 1; n <= NN; n += 2) {
+ ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
+ if (n % 4 == p) {
+ result += term; // sign: I^n w/ n == 4*m+p
+ } else {
+ result -= term; // sign: I^n w/ n == 4*m+2+p
+ }
+ }
return result;
}
-
- ex a=basis.real_part();
- ex b=basis.imag_part();
- ex c=exponent.real_part();
- ex d=exponent.imag_part();
- return
- power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
-}
-// protected
+ // Im((a+I*b)^(c+I*d))
+ const ex d = exponent.imag_part();
+ return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
+}
// protected
{
if (is_a<numeric>(exponent)) {
// D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
- epvector newseq;
- newseq.reserve(2);
- newseq.push_back(expair(basis, exponent - _ex1));
- newseq.push_back(expair(basis.diff(s), _ex1));
- return mul(newseq, exponent);
+ const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
+ return dynallocate<mul>(std::move(newseq), exponent);
} else {
// D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
- return mul(*this,
- add(mul(exponent.diff(s), log(basis)),
- mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
+ return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1));
}
}
return *this;
}
+ // (x*p)^c -> x^c * p^c, if p>0
+ // makes sense before expanding the basis
+ if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
+ const mul &m = ex_to<mul>(basis);
+ exvector prodseq;
+ epvector powseq;
+ prodseq.reserve(m.seq.size() + 1);
+ powseq.reserve(m.seq.size() + 1);
+ bool possign = true;
+
+ // search for positive/negative factors
+ for (auto & cit : m.seq) {
+ ex e=m.recombine_pair_to_ex(cit);
+ if (e.info(info_flags::positive))
+ prodseq.push_back(pow(e, exponent).expand(options));
+ else if (e.info(info_flags::negative)) {
+ prodseq.push_back(pow(-e, exponent).expand(options));
+ possign = !possign;
+ } else
+ powseq.push_back(cit);
+ }
+
+ // take care on the numeric coefficient
+ ex coeff=(possign? _ex1 : _ex_1);
+ if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
+ prodseq.push_back(pow(m.overall_coeff, exponent));
+ else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
+ prodseq.push_back(pow(-m.overall_coeff, exponent));
+ else
+ coeff *= m.overall_coeff;
+
+ // If positive/negative factors are found, then extract them.
+ // In either case we set a flag to avoid the second run on a part
+ // which does not have positive/negative terms.
+ if (prodseq.size() > 0) {
+ ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
+ ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
+ return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
+ } else
+ ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
+ }
+
const ex expanded_basis = basis.expand(options);
const ex expanded_exponent = exponent.expand(options);
const add &a = ex_to<add>(expanded_exponent);
exvector distrseq;
distrseq.reserve(a.seq.size() + 1);
- epvector::const_iterator last = a.seq.end();
- epvector::const_iterator cit = a.seq.begin();
- while (cit!=last) {
- distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
- ++cit;
+ for (auto & cit : a.seq) {
+ distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(cit)));
}
// Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
if (ex_to<numeric>(a.overall_coeff).is_integer()) {
const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
- int int_exponent = num_exponent.to_int();
+ long int_exponent = num_exponent.to_int();
if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
else
- distrseq.push_back(power(expanded_basis, a.overall_coeff));
+ distrseq.push_back(pow(expanded_basis, a.overall_coeff));
} else
- distrseq.push_back(power(expanded_basis, a.overall_coeff));
+ distrseq.push_back(pow(expanded_basis, a.overall_coeff));
// Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
- ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
+ ex r = dynallocate<mul>(distrseq);
return r.expand(options);
}
if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
return this->hold();
} else {
- return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
+ return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
}
}
// integer numeric exponent
const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
- int int_exponent = num_exponent.to_int();
+ long int_exponent = num_exponent.to_long();
// (x+y)^n, n>0
if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
return this->hold();
else
- return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
+ return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
}
//////////
/** expand a^n where a is an add and n is a positive integer.
* @see power::expand */
-ex power::expand_add(const add & a, int n, unsigned options) const
+ex power::expand_add(const add & a, long n, unsigned options)
{
+ // The special case power(+(x,...y;x),2) can be optimized better.
if (n==2)
return expand_add_2(a, options);
- const size_t m = a.nops();
- exvector result;
+ // method:
+ //
+ // Consider base as the sum of all symbolic terms and the overall numeric
+ // coefficient and apply the binomial theorem:
+ // S = power(+(x,...,z;c),n)
+ // = power(+(+(x,...,z;0);c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
+ // The multinomial theorem is computed by an outer loop over all
+ // partitions of the exponent and an inner loop over all compositions of
+ // that partition. This method makes the expansion a combinatorial
+ // problem and allows us to directly construct the expanded sum and also
+ // to re-use the multinomial coefficients (since they depend only on the
+ // partition, not on the composition).
+ //
+ // multinomial power(+(x,y,z;0),3) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
+ // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
+ // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
+ // => (x + y + z)^3 =
+ // x^3 + y^3 + z^3
+ // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
+ // + 6*x*y*z
+ //
+ // multinomial power(+(x,y,z;0),4) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
+ // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
+ // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
+ // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
+ // (no [1,1,1,1] partition since it has too many parts)
+ // => (x + y + z)^4 =
+ // x^4 + y^4 + z^4
+ // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
+ // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
+ // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
+ //
+ // Summary:
+ // r = 0
+ // for k from 0 to n:
+ // f = c^(n-k)*binomial(n,k)
+ // for p in all partitions of n with m parts (including zero parts):
+ // h = f * multinomial coefficient of p
+ // for c in all compositions of p:
+ // t = 1
+ // for e in all elements of c:
+ // t = t * a[e]^e
+ // r = r + h*t
+ // return r
+
+ epvector result;
// The number of terms will be the number of combinatorial compositions,
// i.e. the number of unordered arrangements of m nonnegative integers
// which sum up to n. It is frequently written as C_n(m) and directly
- // related with binomial coefficients:
- result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
- intvector k(m-1);
- intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
- intvector upper_limit(m-1);
-
- for (size_t l=0; l<m-1; ++l) {
- k[l] = 0;
- k_cum[l] = 0;
- upper_limit[l] = n;
+ // related with binomial coefficients: binomial(n+m-1,m-1).
+ size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
+ if (!a.overall_coeff.is_zero()) {
+ // the result's overall_coeff is one of the terms
+ --result_size;
}
-
- while (true) {
- exvector term;
- term.reserve(m+1);
- for (std::size_t l = 0; l < m - 1; ++l) {
- const ex & b = a.op(l);
- GINAC_ASSERT(!is_exactly_a<add>(b));
- GINAC_ASSERT(!is_exactly_a<power>(b) ||
- !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
- !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_exactly_a<add>(ex_to<power>(b).basis) ||
- !is_exactly_a<mul>(ex_to<power>(b).basis) ||
- !is_exactly_a<power>(ex_to<power>(b).basis));
- if (is_exactly_a<mul>(b))
- term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
- else
- term.push_back(power(b,k[l]));
- }
-
- const ex & b = a.op(m - 1);
- GINAC_ASSERT(!is_exactly_a<add>(b));
- GINAC_ASSERT(!is_exactly_a<power>(b) ||
- !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
- !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_exactly_a<add>(ex_to<power>(b).basis) ||
- !is_exactly_a<mul>(ex_to<power>(b).basis) ||
- !is_exactly_a<power>(ex_to<power>(b).basis));
- if (is_exactly_a<mul>(b))
- term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
- else
- term.push_back(power(b,n-k_cum[m-2]));
-
- numeric f = binomial(numeric(n),numeric(k[0]));
- for (std::size_t l = 1; l < m - 1; ++l)
- f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
-
- term.push_back(f);
-
- result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
-
- // increment k[]
- bool done = false;
- std::size_t l = m - 2;
- while ((++k[l]) > upper_limit[l]) {
- k[l] = 0;
- if (l != 0)
- --l;
- else {
- done = true;
- break;
+ result.reserve(result_size);
+
+ // Iterate over all terms in binomial expansion of
+ // S = power(+(x,...,z;c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ for (int k = 1; k <= n; ++k) {
+ numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
+ if (a.overall_coeff.is_zero()) {
+ // degenerate case with zero overall_coeff:
+ // apply multinomial theorem directly to power(+(x,...z;0),n)
+ binomial_coefficient = 1;
+ if (k < n) {
+ continue;
}
+ } else {
+ binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
}
- if (done)
- break;
-
- // recalc k_cum[] and upper_limit[]
- k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
-
- for (size_t i=l+1; i<m-1; ++i)
- k_cum[i] = k_cum[i-1]+k[i];
- for (size_t i=l+1; i<m-1; ++i)
- upper_limit[i] = n-k_cum[i-1];
+ // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
+ // Iterate over all partitions of k with exactly as many parts as
+ // there are symbolic terms in the basis (including zero parts).
+ partition_generator partitions(k, a.seq.size());
+ do {
+ const std::vector<int>& partition = partitions.current();
+ // All monomials of this partition have the same number of terms and the same coefficient.
+ const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
+ const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
+
+ // Iterate over all compositions of the current partition.
+ composition_generator compositions(partition);
+ do {
+ const std::vector<int>& exponent = compositions.current();
+ epvector monomial;
+ monomial.reserve(msize);
+ numeric factor = coeff;
+ for (unsigned i = 0; i < exponent.size(); ++i) {
+ const ex & r = a.seq[i].rest;
+ GINAC_ASSERT(!is_exactly_a<add>(r));
+ GINAC_ASSERT(!is_exactly_a<power>(r) ||
+ !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
+ !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
+ !is_exactly_a<add>(ex_to<power>(r).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(r).basis) ||
+ !is_exactly_a<power>(ex_to<power>(r).basis));
+ GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
+ const numeric & c = ex_to<numeric>(a.seq[i].coeff);
+ if (exponent[i] == 0) {
+ // optimize away
+ } else if (exponent[i] == 1) {
+ // optimized
+ monomial.push_back(expair(r, _ex1));
+ if (c != *_num1_p)
+ factor = factor.mul(c);
+ } else { // general case exponent[i] > 1
+ monomial.push_back(expair(r, exponent[i]));
+ if (c != *_num1_p)
+ factor = factor.mul(c.power(exponent[i]));
+ }
+ }
+ result.push_back(expair(mul(std::move(monomial)).expand(options), factor));
+ } while (compositions.next());
+ } while (partitions.next());
}
- return (new add(result))->setflag(status_flags::dynallocated |
- status_flags::expanded);
+ GINAC_ASSERT(result.size() == result_size);
+ if (a.overall_coeff.is_zero()) {
+ return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
+ } else {
+ return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
+ }
}
/** Special case of power::expand_add. Expands a^2 where a is an add.
* @see power::expand_add */
-ex power::expand_add_2(const add & a, unsigned options) const
+ex power::expand_add_2(const add & a, unsigned options)
{
- epvector sum;
- size_t a_nops = a.nops();
- sum.reserve((a_nops*(a_nops+1))/2);
- epvector::const_iterator last = a.seq.end();
+ epvector result;
+ size_t result_size = (a.nops() * (a.nops()+1)) / 2;
+ if (!a.overall_coeff.is_zero()) {
+ // the result's overall_coeff is one of the terms
+ --result_size;
+ }
+ result.reserve(result_size);
+
+ auto last = a.seq.end();
// power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
// first part: ignore overall_coeff and expand other terms
- for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
+ for (auto cit0=a.seq.begin(); cit0!=last; ++cit0) {
const ex & r = cit0->rest;
const ex & c = cit0->coeff;
if (c.is_equal(_ex1)) {
if (is_exactly_a<mul>(r)) {
- sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
- _ex1));
+ result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
+ _ex1));
} else {
- sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
- _ex1));
+ result.push_back(expair(dynallocate<power>(r, _ex2),
+ _ex1));
}
} else {
if (is_exactly_a<mul>(r)) {
- sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
- ex_to<numeric>(c).power_dyn(*_num2_p)));
+ result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
+ ex_to<numeric>(c).power_dyn(*_num2_p)));
} else {
- sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
- ex_to<numeric>(c).power_dyn(*_num2_p)));
+ result.push_back(expair(dynallocate<power>(r, _ex2),
+ ex_to<numeric>(c).power_dyn(*_num2_p)));
}
}
- for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
+ for (auto cit1=cit0+1; cit1!=last; ++cit1) {
const ex & r1 = cit1->rest;
const ex & c1 = cit1->coeff;
- sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
- _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
+ result.push_back(expair(mul(r,r1).expand(options),
+ _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
}
}
- GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
-
- // second part: add terms coming from overall_factor (if != 0)
+ // second part: add terms coming from overall_coeff (if != 0)
if (!a.overall_coeff.is_zero()) {
- epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
- while (i != end) {
- sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
- ++i;
- }
- sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
+ for (auto & i : a.seq)
+ result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
+ }
+
+ GINAC_ASSERT(result.size() == result_size);
+
+ if (a.overall_coeff.is_zero()) {
+ return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
+ } else {
+ return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
}
-
- GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
-
- return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
}
/** Expand factors of m in m^n where m is a mul and n is an integer.
* @see power::expand */
-ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
+ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
{
GINAC_ASSERT(n.is_integer());
}
// do not bother to rename indices if there are no any.
- if ((!(options & expand_options::expand_rename_idx))
- && m.info(info_flags::has_indices))
+ if (!(options & expand_options::expand_rename_idx) &&
+ m.info(info_flags::has_indices))
options |= expand_options::expand_rename_idx;
// Leave it to multiplication since dummy indices have to be renamed
if ((options & expand_options::expand_rename_idx) &&
- (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
+ (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
ex result = m;
exvector va = get_all_dummy_indices(m);
sort(va.begin(), va.end(), ex_is_less());
distrseq.reserve(m.seq.size());
bool need_reexpand = false;
- epvector::const_iterator last = m.seq.end();
- epvector::const_iterator cit = m.seq.begin();
- while (cit!=last) {
- expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
- if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
+ for (auto & cit : m.seq) {
+ expair p = m.combine_pair_with_coeff_to_pair(cit, n);
+ if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
// this happens when e.g. (a+b)^(1/2) gets squared and
// the resulting product needs to be reexpanded
need_reexpand = true;
}
distrseq.push_back(p);
- ++cit;
}
- const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
+ const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
if (need_reexpand)
return ex(result).expand(options);
if (from_expand)
git://www.ginac.de / ginac.git / blobdiff