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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2014 Klaus Spanderen
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
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 license for more details.
*/
/*! \file modifiedbessel.cpp
\brief modified Bessel functions of first and second kind
*/
#include <ql/math/modifiedbessel.hpp>
#include <ql/math/distributions/gammadistribution.hpp>
#include <cmath>
namespace QuantLib {
namespace {
template <class T> struct I {};
template <> struct I<Real> { Real value() { return 0.0;} };
template <> struct I<std::complex<Real> > {
std::complex<Real> value() { return std::complex<Real>(0.0,1.0);}
};
template <class T> struct Unweighted {
T weightSmallX(const T& x) { return 1.0; }
T weight1LargeX(const T& x) { return std::exp(x); }
T weight2LargeX(const T& x) { return std::exp(-x); }
};
template <class T> struct ExponentiallyWeighted {
T weightSmallX(const T& x) { return std::exp(-x); }
T weight1LargeX(const T& x) { return 1.0; }
T weight2LargeX(const T& x) { return std::exp(-2.0*x); }
};
template <class T, template <class> class W>
T modifiedBesselFunction_i_impl(Real nu, const T& x) {
if (std::abs(x) < 13.0) {
const T alpha = std::pow(0.5*x, nu)
/GammaFunction().value(1.0+nu);
const T Y = 0.25*x*x;
Size k=1;
T sum=alpha, B_k=alpha;
while (std::abs(B_k*=Y/(k*(k+nu)))>std::abs(sum)*QL_EPSILON) {
sum += B_k;
QL_REQUIRE(++k < 1000, "max iterations exceeded");
}
return sum * W<T>().weightSmallX(x);
}
else {
Real na_k=1.0, sign=1.0;
T da_k=T(1.0);
T s1=T(1.0), s2=T(1.0);
for (Size k=1; k < 30; ++k) {
sign*=-1;
na_k *= (4.0 * nu * nu -
(2.0 * static_cast<Real>(k) - 1.0) *
(2.0 * static_cast<Real>(k) - 1.0));
da_k *= (8.0 * k) * x;
const T a_k = na_k/da_k;
s2+=a_k;
s1+=sign*a_k;
}
const T i = I<T>().value();
return 1.0 / std::sqrt(2 * M_PI * x) *
(W<T>().weight1LargeX(x) * s1 +
i * std::exp(i * nu * M_PI) * W<T>().weight2LargeX(x) * s2);
}
}
template <class T, template <class> class W>
T modifiedBesselFunction_k_impl(Real nu, const T& x) {
return M_PI_2 * (modifiedBesselFunction_i_impl<T,W>(-nu, x) -
modifiedBesselFunction_i_impl<T,W>(nu, x)) /
std::sin(M_PI * nu);
}
}
Real modifiedBesselFunction_i(Real nu, Real x) {
QL_REQUIRE(x >= 0.0, "negative argument requires complex version of "
"modifiedBesselFunction");
return modifiedBesselFunction_i_impl<Real, Unweighted>(nu, x);
}
std::complex<Real> modifiedBesselFunction_i(Real nu,
const std::complex<Real> &z) {
if (z.imag() == 0.0 && z.real() >= 0.0)
return std::complex<Real>(modifiedBesselFunction_i(nu, z.real()));
return modifiedBesselFunction_i_impl<
std::complex<Real>, Unweighted>(nu, z);
}
Real modifiedBesselFunction_k(Real nu, Real x) {
return modifiedBesselFunction_k_impl<Real, Unweighted>(nu, x);
}
std::complex<Real> modifiedBesselFunction_k(Real nu,
const std::complex<Real> &z) {
if (z.imag() == 0.0 && z.real() >= 0.0)
return std::complex<Real>(modifiedBesselFunction_k(nu, z.real()));
return modifiedBesselFunction_k_impl<
std::complex<Real>, Unweighted>(nu, z);
}
Real modifiedBesselFunction_i_exponentiallyWeighted(Real nu, Real x) {
QL_REQUIRE(x >= 0.0, "negative argument requires complex version of "
"modifiedBesselFunction");
return modifiedBesselFunction_i_impl<Real, ExponentiallyWeighted>(
nu, x);
}
std::complex<Real> modifiedBesselFunction_i_exponentiallyWeighted(
Real nu, const std::complex<Real> &z) {
if (z.imag() == 0.0 && z.real() >= 0.0)
return std::complex<Real>(
modifiedBesselFunction_i_exponentiallyWeighted(nu, z.real()));
return modifiedBesselFunction_i_impl<
std::complex<Real>, ExponentiallyWeighted>(nu, z);
}
Real modifiedBesselFunction_k_exponentiallyWeighted(Real nu, Real x) {
return modifiedBesselFunction_k_impl<Real, ExponentiallyWeighted>(
nu, x);
}
std::complex<Real> modifiedBesselFunction_k_exponentiallyWeighted(
Real nu, const std::complex<Real> &z) {
if (z.imag() == 0.0 && z.real() >= 0.0)
return std::complex<Real>(
modifiedBesselFunction_k_exponentiallyWeighted(nu, z.real()));
return modifiedBesselFunction_k_impl<
std::complex<Real>, ExponentiallyWeighted>(nu, z);
}
}
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