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// (C) 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)
#define L22
#include "../tools/ntl_rr_lanczos.hpp"
#include "../tools/ntl_rr_digamma.hpp"
#include <boost/math/bindings/rr.hpp>
#include <boost/math/tools/polynomial.hpp>
#include <boost/math/special_functions.hpp>
#include <cmath>
boost::math::ntl::RR f(const boost::math::ntl::RR& x, int variant)
{
static const boost::math::ntl::RR tiny = boost::math::tools::min_value<boost::math::ntl::RR>() * 64;
switch(variant)
{
case 0:
return boost::math::erfc(x) * x / exp(-x * x);
case 1:
return boost::math::erf(x);
case 2:
{
boost::math::ntl::RR x_ = x == 0 ? 1e-80 : x;
return boost::math::erf(x_) / x_;
}
case 3:
{
boost::math::ntl::RR y(x);
if(y == 0)
y += tiny;
return boost::math::lgamma(y+2) / y - 0.5;
}
case 4:
//
// lgamma in the range [2,3], use:
//
// lgamma(x) = (x-2) * (x + 1) * (c + R(x - 2))
//
// Works well at 80-bit long double precision, but doesn't
// stretch to 128-bit precision.
//
if(x == 0)
{
return boost::lexical_cast<boost::math::ntl::RR>("0.42278433509846713939348790991759756895784066406008") / 3;
}
return boost::math::lgamma(x+2) / (x * (x+3));
case 5:
{
//
// lgamma in the range [1,2], use:
//
// lgamma(x) = (x - 1) * (x - 2) * (c + R(x - 1))
//
// works well over [1, 1.5] but not near 2 :-(
//
boost::math::ntl::RR r1 = boost::lexical_cast<boost::math::ntl::RR>("0.57721566490153286060651209008240243104215933593992");
boost::math::ntl::RR r2 = boost::lexical_cast<boost::math::ntl::RR>("0.42278433509846713939348790991759756895784066406008");
if(x == 0)
{
return r1;
}
if(x == 1)
{
return r2;
}
return boost::math::lgamma(x+1) / (x * (x - 1));
}
case 6:
{
//
// lgamma in the range [1.5,2], use:
//
// lgamma(x) = (2 - x) * (1 - x) * (c + R(2 - x))
//
// works well over [1.5, 2] but not near 1 :-(
//
boost::math::ntl::RR r1 = boost::lexical_cast<boost::math::ntl::RR>("0.57721566490153286060651209008240243104215933593992");
boost::math::ntl::RR r2 = boost::lexical_cast<boost::math::ntl::RR>("0.42278433509846713939348790991759756895784066406008");
if(x == 0)
{
return r2;
}
if(x == 1)
{
return r1;
}
return boost::math::lgamma(2-x) / (x * (x - 1));
}
case 7:
{
//
// erf_inv in range [0, 0.5]
//
boost::math::ntl::RR y = x;
if(y == 0)
y = boost::math::tools::epsilon<boost::math::ntl::RR>() / 64;
return boost::math::erf_inv(y) / (y * (y+10));
}
case 8:
{
//
// erfc_inv in range [0.25, 0.5]
// Use an y-offset of 0.25, and range [0, 0.25]
// abs error, auto y-offset.
//
boost::math::ntl::RR y = x;
if(y == 0)
y = boost::lexical_cast<boost::math::ntl::RR>("1e-5000");
return sqrt(-2 * log(y)) / boost::math::erfc_inv(y);
}
case 9:
{
boost::math::ntl::RR x2 = x;
if(x2 == 0)
x2 = boost::lexical_cast<boost::math::ntl::RR>("1e-5000");
boost::math::ntl::RR y = exp(-x2*x2); // sqrt(-log(x2)) - 5;
return boost::math::erfc_inv(y) / x2;
}
case 10:
{
//
// Digamma over the interval [1,2], set x-offset to 1
// and optimise for absolute error over [0,1].
//
int current_precision = boost::math::ntl::RR::precision();
if(current_precision < 1000)
boost::math::ntl::RR::SetPrecision(1000);
//
// This value for the root of digamma is calculated using our
// differentiated lanczos approximation. It agrees with Cody
// to ~ 25 digits and to Morris to 35 digits. See:
// TOMS ALGORITHM 708 (Didonato and Morris).
// and Math. Comp. 27, 123-127 (1973) by Cody, Strecok and Thacher.
//
//boost::math::ntl::RR root = boost::lexical_cast<boost::math::ntl::RR>("1.4616321449683623412626595423257213234331845807102825466429633351908372838889871");
//
// Actually better to calculate the root on the fly, it appears to be more
// accurate: convergence is easier with the 1000-bit value, the approximation
// produced agrees with functions.mathworld.com values to 35 digits even quite
// near the root.
//
static boost::math::tools::eps_tolerance<boost::math::ntl::RR> tol(1000);
static boost::uintmax_t max_iter = 1000;
static const boost::math::ntl::RR root = boost::math::tools::bracket_and_solve_root(&boost::math::digamma, boost::math::ntl::RR(1.4), boost::math::ntl::RR(1.5), true, tol, max_iter).first;
boost::math::ntl::RR x2 = x;
double lim = 1e-65;
if(fabs(x2 - root) < lim)
{
//
// This is a problem area:
// x2-root suffers cancellation error, so does digamma.
// That gets compounded again when Remez calculates the error
// function. This cludge seems to stop the worst of the problems:
//
static const boost::math::ntl::RR a = boost::math::digamma(root - lim) / -lim;
static const boost::math::ntl::RR b = boost::math::digamma(root + lim) / lim;
boost::math::ntl::RR fract = (x2 - root + lim) / (2*lim);
boost::math::ntl::RR r = (1-fract) * a + fract * b;
std::cout << "In root area: " << r;
return r;
}
boost::math::ntl::RR result = boost::math::digamma(x2) / (x2 - root);
if(current_precision < 1000)
boost::math::ntl::RR::SetPrecision(current_precision);
return result;
}
case 11:
// expm1:
if(x == 0)
{
static boost::math::ntl::RR lim = 1e-80;
static boost::math::ntl::RR a = boost::math::expm1(-lim);
static boost::math::ntl::RR b = boost::math::expm1(lim);
static boost::math::ntl::RR l = (b-a) / (2 * lim);
return l;
}
return boost::math::expm1(x) / x;
case 12:
// demo, and test case:
return exp(x);
case 13:
// K(k):
{
// x = k^2
boost::math::ntl::RR k2 = x;
if(k2 > boost::math::ntl::RR(1) - 1e-40)
k2 = boost::math::ntl::RR(1) - 1e-40;
/*if(k < 1e-40)
k = 1e-40;*/
boost::math::ntl::RR p2 = boost::math::constants::pi<boost::math::ntl::RR>() / 2;
return (boost::math::ellint_1(sqrt(k2))) / (p2 - boost::math::log1p(-k2));
}
case 14:
// K(k)
{
// x = 1 - k^2
boost::math::ntl::RR mp = x;
if(mp < 1e-20)
mp = 1e-20;
boost::math::ntl::RR k = sqrt(1 - mp);
static const boost::math::ntl::RR l4 = log(boost::math::ntl::RR(4));
boost::math::ntl::RR p2 = boost::math::constants::pi<boost::math::ntl::RR>() / 2;
return (boost::math::ellint_1(k) + 1) / (1 + l4 - log(mp));
}
case 15:
// E(k)
{
// x = 1-k^2
boost::math::ntl::RR z = 1 - x * log(x);
return boost::math::ellint_2(sqrt(1-x)) / z;
}
case 16:
// Bessel I0(x) over [0,16]:
{
return boost::math::cyl_bessel_i(0, sqrt(x));
}
case 17:
// Bessel I0(x) over [16,INF]
{
boost::math::ntl::RR z = 1 / (boost::math::ntl::RR(1)/16 - x);
return boost::math::cyl_bessel_i(0, z) * sqrt(z) / exp(z);
}
}
return 0;
}
void show_extra(
const boost::math::tools::polynomial<boost::math::ntl::RR>& n,
const boost::math::tools::polynomial<boost::math::ntl::RR>& d,
const boost::math::ntl::RR& x_offset,
const boost::math::ntl::RR& y_offset,
int variant)
{
switch(variant)
{
default:
// do nothing here...
;
}
}
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