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// Copyright John Maddock 2006.
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
#include "mp_t.hpp"
#include <boost/math/special_functions/log1p.hpp>
#include <boost/math/special_functions/erf.hpp>
#include <boost/math/constants/constants.hpp>
#include <map>
#include <iostream>
#include <iomanip>
using namespace std;
using namespace boost::math;
//
// This program calculates the coefficients of the polynomials
// used for the regularized incomplete gamma functions gamma_p
// and gamma_q when parameter a is large, and sigma is small
// (where sigma = fabs(1 - x/a) ).
//
// See "The Asymptotic Expansion of the Incomplete Gamma Functions"
// N. M. Temme.
// Siam J. Math Anal. Vol 10 No 4, July 1979, p757.
// Coefficient calculation is described from Eq 3.8 (p762) onwards.
//
//
// Alpha:
//
mp_t alpha(unsigned k)
{
static map<unsigned, mp_t> data;
if(data.empty())
{
data[1] = 1;
}
map<unsigned, mp_t>::const_iterator pos = data.find(k);
if(pos != data.end())
return (*pos).second;
//
// OK try and calculate the value:
//
mp_t result = alpha(k-1);
for(unsigned j = 2; j <= k-1; ++j)
{
result -= j * alpha(j) * alpha(k-j+1);
}
result /= (k+1);
data[k] = result;
return result;
}
mp_t gamma(unsigned k)
{
static map<unsigned, mp_t> data;
map<unsigned, mp_t>::const_iterator pos = data.find(k);
if(pos != data.end())
return (*pos).second;
mp_t result = (k&1) ? -1 : 1;
for(unsigned i = 1; i <= (2 * k + 1); i += 2)
result *= i;
result *= alpha(2 * k + 1);
data[k] = result;
return result;
}
mp_t Coeff(unsigned n, unsigned k)
{
map<unsigned, map<unsigned, mp_t> > data;
if(data.empty())
data[0][0] = mp_t(-1) / 3;
map<unsigned, map<unsigned, mp_t> >::const_iterator p1 = data.find(n);
if(p1 != data.end())
{
map<unsigned, mp_t>::const_iterator p2 = p1->second.find(k);
if(p2 != p1->second.end())
{
return p2->second;
}
}
//
// If we don't have the value, calculate it:
//
if(k == 0)
{
// special case:
mp_t result = (n+2) * alpha(n+2);
data[n][k] = result;
return result;
}
// general case:
mp_t result = gamma(k) * Coeff(n, 0) + (n+2) * Coeff(n+2, k-1);
data[n][k] = result;
return result;
}
void calculate_terms(double sigma, double a, unsigned bits)
{
cout << endl << endl;
cout << "Sigma: " << sigma << endl;
cout << "A: " << a << endl;
double lambda = 1 - sigma;
cout << "Lambda: " << lambda << endl;
double y = a * (-sigma - log1p(-sigma));
cout << "Y: " << y << endl;
double z = -sqrt(2 * (-sigma - log1p(-sigma)));
cout << "Z: " << z << endl;
double dom = erfc(sqrt(y)) / 2;
cout << "Erfc term: " << dom << endl;
double lead = exp(-y) / sqrt(2 * constants::pi<double>() * a);
cout << "Remainder factor: " << lead << endl;
double eps = ldexp(1.0, 1 - static_cast<int>(bits));
double target = dom * eps / lead;
cout << "Target smallest term: " << target << endl;
unsigned max_n = 0;
for(unsigned n = 0; n < 10000; ++n)
{
double term = tools::real_cast<double>(Coeff(n, 0) * pow(z, (double)n));
if(fabs(term) < target)
{
max_n = n-1;
break;
}
}
cout << "Max n required: " << max_n << endl;
unsigned max_k = 0;
for(unsigned k = 1; k < 10000; ++k)
{
double term = tools::real_cast<double>(Coeff(0, k) * pow(a, -((double)k)));
if(fabs(term) < target)
{
max_k = k-1;
break;
}
}
cout << "Max k required: " << max_k << endl << endl;
bool code = false;
cout << "Print code [0|1]? ";
cin >> code;
std::cout << std::scientific << std::setprecision(40);
if(code)
{
cout << " T workspace[" << max_k+1 << "];\n\n";
for(unsigned k = 0; k <= max_k; ++k)
{
cout <<
" static const T C" << k << "[] = {\n";
for(unsigned n = 0; n < 10000; ++n)
{
double term = tools::real_cast<double>(Coeff(n, k) * pow(a, -((double)k)) * pow(z, (double)n));
if(fabs(term) < target)
{
break;
}
cout << " " << Coeff(n, k) << "L,\n";
}
cout <<
" };\n"
" workspace[" << k << "] = tools::evaluate_polynomial(C" << k << ", z);\n\n";
}
cout << " T result = tools::evaluate_polynomial(workspace, 1/a);\n\n";
}
}
int main()
{
bool cont;
do{
cont = false;
double sigma;
cout << "Enter max value for sigma (sigma = |1 - x/a|): ";
cin >> sigma;
double a;
cout << "Enter min value for a: ";
cin >> a;
unsigned precision;
cout << "Enter number of bits precision required: ";
cin >> precision;
calculate_terms(sigma, a, precision);
cout << "Try again[0|1]: ";
cin >> cont;
}while(cont);
return 0;
}
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