/** @file power.cpp
*
* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
* GiNaC Copyright (C) 1999-2007 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
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* 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 "mul.h"
#include "ncmul.h"
#include "numeric.h"
#include "constant.h"
#include "operators.h"
#include "inifcns.h" // for log() in power::derivative()
#include "matrix.h"
#include "indexed.h"
#include "symbol.h"
#include "lst.h"
#include "archive.h"
#include "utils.h"
#include "relational.h"
#include "compiler.h"
namespace GiNaC {
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
print_func<print_dflt>(&power::do_print_dflt).
print_func<print_latex>(&power::do_print_latex).
print_func<print_csrc>(&power::do_print_csrc).
print_func<print_python>(&power::do_print_python).
print_func<print_python_repr>(&power::do_print_python_repr))
typedef std::vector<int> intvector;
//////////
// default constructor
//////////
power::power() : inherited(TINFO_power) { }
//////////
// other constructors
//////////
// all inlined
//////////
// archiving
//////////
power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
{
n.find_ex("basis", basis, sym_lst);
n.find_ex("exponent", exponent, sym_lst);
}
void power::archive(archive_node &n) const
{
inherited::archive(n);
n.add_ex("basis", basis);
n.add_ex("exponent", exponent);
}
DEFAULT_UNARCHIVE(power)
//////////
// functions overriding virtual functions from base classes
//////////
// public
void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
{
// Ordinary output of powers using '^' or '**'
if (precedence() <= level)
c.s << openbrace << '(';
basis.print(c, precedence());
c.s << powersymbol;
c.s << openbrace;
exponent.print(c, precedence());
c.s << closebrace;
if (precedence() <= level)
c.s << ')' << closebrace;
}
void power::do_print_dflt(const print_dflt & c, unsigned level) const
{
if (exponent.is_equal(_ex1_2)) {
// Square roots are printed in a special way
c.s << "sqrt(";
basis.print(c);
c.s << ')';
} else
print_power(c, "^", "", "", level);
}
void power::do_print_latex(const print_latex & c, unsigned level) const
{
if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
// Powers with negative numeric exponents are printed as fractions
c.s << "\\frac{1}{";
power(basis, -exponent).eval().print(c);
c.s << '}';
} else if (exponent.is_equal(_ex1_2)) {
// Square roots are printed in a special way
c.s << "\\sqrt{";
basis.print(c);
c.s << '}';
} else
print_power(c, "^", "{", "}", level);
}
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
// to learn why such a parenthesation is really necessary.
if (exp == 1) {
x.print(c);
} else if (exp == 2) {
x.print(c);
c.s << "*";
x.print(c);
} else if (exp & 1) {
x.print(c);
c.s << "*";
print_sym_pow(c, x, exp-1);
} else {
c.s << "(";
print_sym_pow(c, x, exp >> 1);
c.s << ")*(";
print_sym_pow(c, x, exp >> 1);
c.s << ")";
}
}
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))) {
int exp = ex_to<numeric>(exponent).to_int();
if (exp > 0)
c.s << '(';
else {
exp = -exp;
if (is_a<print_csrc_cl_N>(c))
c.s << "recip(";
else
c.s << "1.0/(";
}
print_sym_pow(c, ex_to<symbol>(basis), exp);
c.s << ')';
// <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
} else if (exponent.is_equal(_ex_1)) {
if (is_a<print_csrc_cl_N>(c))
c.s << "recip(";
else
c.s << "1.0/(";
basis.print(c);
c.s << ')';
// Otherwise, use the pow() or expt() (CLN) functions
} else {
if (is_a<print_csrc_cl_N>(c))
c.s << "expt(";
else
c.s << "pow(";
basis.print(c);
c.s << ',';
exponent.print(c);
c.s << ')';
}
}
void power::do_print_python(const print_python & c, unsigned level) const
{
print_power(c, "**", "", "", level);
}
void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
{
c.s << class_name() << '(';
basis.print(c);
c.s << ',';
exponent.print(c);
c.s << ')';
}
bool power::info(unsigned inf) const
{
switch (inf) {
case info_flags::polynomial:
case info_flags::integer_polynomial:
case info_flags::cinteger_polynomial:
case info_flags::rational_polynomial:
case info_flags::crational_polynomial:
return exponent.info(info_flags::nonnegint) &&
basis.info(inf);
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);
case info_flags::expanded:
return (flags & status_flags::expanded);
}
return inherited::info(inf);
}
size_t power::nops() const
{
return 2;
}
ex power::op(size_t i) const
{
GINAC_ASSERT(i<2);
return i==0 ? basis : exponent;
}
ex power::map(map_function & f) const
{
const ex &mapped_basis = f(basis);
const ex &mapped_exponent = f(exponent);
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);
else
return *this;
}
int power::degree(const ex & s) const
{
if (is_equal(ex_to<basic>(s)))
return 1;
else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
if (basis.is_equal(s))
return ex_to<numeric>(exponent).to_int();
else
return basis.degree(s) * ex_to<numeric>(exponent).to_int();
} else if (basis.has(s))
throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
else
return 0;
}
int power::ldegree(const ex & s) const
{
if (is_equal(ex_to<basic>(s)))
return 1;
else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
if (basis.is_equal(s))
return ex_to<numeric>(exponent).to_int();
else
return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
} else if (basis.has(s))
throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
else
return 0;
}
ex power::coeff(const ex & s, int n) const
{
if (is_equal(ex_to<basic>(s)))
return n==1 ? _ex1 : _ex0;
else if (!basis.is_equal(s)) {
// basis not equal to s
if (n == 0)
return *this;
else
return _ex0;
} else {
// basis equal to s
if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
// integer exponent
int int_exp = ex_to<numeric>(exponent).to_int();
if (n == int_exp)
return _ex1;
else
return _ex0;
} else {
// non-integer exponents are treated as zero
if (n == 0)
return *this;
else
return _ex0;
}
}
}
/** Perform automatic term rewriting rules in this class. In the following
* x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
* stand for such expressions that contain a plain number.
* - ^(x,0) -> 1 (also handles ^(0,0))
* - ^(x,1) -> x
* - ^(0,c) -> 0 or exception (depending on the real part of c)
* - ^(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,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
{
if ((level==1) && (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);
}
if (is_exactly_a<numeric>(eexponent)) {
exponent_is_numerical = true;
num_exponent = &ex_to<numeric>(eexponent);
}
// ^(x,0) -> 1 (0^0 also handled here)
if (eexponent.is_zero()) {
if (ebasis.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;
// ^(0,c1) -> 0 or exception (depending on real value of c1)
if (ebasis.is_zero() && exponent_is_numerical) {
if ((num_exponent->real()).is_zero())
throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
else if ((num_exponent->real()).is_negative())
throw (pole_error("power::eval(): division by zero",1));
else
return _ex0;
}
// ^(1,x) -> 1
if (ebasis.is_equal(_ex1))
return _ex1;
// 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 (exponent_is_numerical) {
// ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
// except if c1,c2 are rational, but c1^c2 is not)
if (basis_is_numerical) {
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);
}
const numeric res = num_basis->power(*num_exponent);
if (res.is_crational()) {
return res;
}
GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
// ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
if (basis_is_crational && exponent_is_crational
&& num_exponent->is_real()
&& !num_exponent->is_integer()) {
const numeric n = num_exponent->numer();
const numeric m = num_exponent->denom();
numeric r;
numeric q = iquo(n, m, r);
if (r.is_negative()) {
r += m;
--q;
}
if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
if (num_basis->is_rational() && !num_basis->is_integer()) {
// try it for numerator and denominator separately, in order to
// partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
const numeric bnum = num_basis->numer();
const numeric bden = num_basis->denom();
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);
if (res_bden.is_integer())
return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
}
return this->hold();
} else {
// assemble resulting product, but allowing for a re-evaluation,
// 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);
}
}
}
// ^(^(x,c1),c2) -> ^(x,c1*c2)
// (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
// case c1==1 should not happen, see below!)
if (is_exactly_a<power>(ebasis)) {
const power & sub_power = ex_to<power>(ebasis);
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));
}
}
// ^(*(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);
}
// (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();
const numeric& lead_coeff =
ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div_dyn(icont);
const bool canonicalizable = lead_coeff.is_integer();
const bool unit_normal = lead_coeff.is_pos_integer();
if (canonicalizable && (! unit_normal))
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 numeric c = icont.power(*num_exponent);
if (likely(c != *_num1_p))
return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
else
return 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)) {
GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
const mul & mulref = ex_to<mul>(ebasis);
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);
} 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);
}
}
}
}
}
// ^(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);
}
}
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);
}
ex power::evalf(int level) const
{
ex ebasis;
ex eexponent;
if (level==1) {
ebasis = basis;
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);
}
ex power::evalm() const
{
const ex ebasis = basis.evalm();
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 (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
}
// from mul.cpp
extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
ex power::subs(const exmap & m, unsigned options) const
{
const ex &subsed_basis = basis.subs(m, options);
const ex &subsed_exponent = exponent.subs(m, options);
if (!are_ex_trivially_equal(basis, subsed_basis)
|| !are_ex_trivially_equal(exponent, subsed_exponent))
return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
if (!(options & subs_options::algebraic))
return subs_one_level(m, options);
for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
int nummatches = std::numeric_limits<int>::max();
lst repls;
if (tryfactsubs(*this, it->first, nummatches, repls))
return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
}
return subs_one_level(m, options);
}
ex power::eval_ncmul(const exvector & v) const
{
return inherited::eval_ncmul(v);
}
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;
}
return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
}
// protected
/** Implementation of ex::diff() for a power.
* @see ex::diff */
ex power::derivative(const symbol & s) const
{
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);
} 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))));
}
}
int power::compare_same_type(const basic & other) const
{
GINAC_ASSERT(is_exactly_a<power>(other));
const power &o = static_cast<const power &>(other);
int cmpval = basis.compare(o.basis);
if (cmpval)
return cmpval;
else
return exponent.compare(o.exponent);
}
unsigned power::return_type() const
{
return basis.return_type();
}
unsigned power::return_type_tinfo() const
{
return basis.return_type_tinfo();
}
ex power::expand(unsigned options) const
{
if (options == 0 && (flags & status_flags::expanded))
return *this;
const ex expanded_basis = basis.expand(options);
const ex expanded_exponent = exponent.expand(options);
// x^(a+b) -> x^a * x^b
if (is_exactly_a<add>(expanded_exponent)) {
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;
}
// 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();
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));
} else
distrseq.push_back(power(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);
return r.expand(options);
}
if (!is_exactly_a<numeric>(expanded_exponent) ||
!ex_to<numeric>(expanded_exponent).is_integer()) {
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));
}
}
// integer numeric exponent
const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
int int_exponent = num_exponent.to_int();
// (x+y)^n, n>0
if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
// (x*y)^n -> x^n * y^n
if (is_exactly_a<mul>(expanded_basis))
return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
// cannot expand further
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));
}
//////////
// new virtual functions which can be overridden by derived classes
//////////
// none
//////////
// non-virtual functions in this class
//////////
/** 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
{
if (n==2)
return expand_add_2(a, options);
const size_t m = a.nops();
exvector 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);
int l;
for (size_t l=0; l<m-1; ++l) {
k[l] = 0;
k_cum[l] = 0;
upper_limit[l] = n;
}
while (true) {
exvector term;
term.reserve(m+1);
for (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(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(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 (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[]
l = m-2;
while ((l>=0) && ((++k[l])>upper_limit[l])) {
k[l] = 0;
--l;
}
if (l<0) 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];
}
return (new add(result))->setflag(status_flags::dynallocated |
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
{
epvector sum;
size_t a_nops = a.nops();
sum.reserve((a_nops*(a_nops+1))/2);
epvector::const_iterator 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) {
const ex & r = cit0->rest;
const ex & c = cit0->coeff;
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));
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));
} else {
sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
_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)));
} 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)));
}
}
for (epvector::const_iterator 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))));
}
}
GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
// second part: add terms coming from overall_factor (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));
}
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
{
GINAC_ASSERT(n.is_integer());
if (n.is_zero()) {
return _ex1;
}
// Leave it to multiplication since dummy indices have to be renamed
if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
ex result = m;
for (int i=1; i < n.to_int(); i++)
result *= rename_dummy_indices_uniquely(m,m);
return result;
}
epvector distrseq;
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()) {
// 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));
if (need_reexpand)
return ex(result).expand(options);
if (from_expand)
return result.setflag(status_flags::expanded);
return result;
}
} // namespace GiNaC
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