lmgamma.hpp

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/*################################################################################
  ##
  ##   Copyright (C) 2016-2022 Keith O'Hara
  ##
  ##   This file is part of the GCE-Math C++ library.
  ##
  ##   Licensed under the Apache License, Version 2.0 (the "License");
  ##   you may not use this file except in compliance with the License.
  ##   You may obtain a copy of the License at
  ##
  ##       http://www.apache.org/licenses/LICENSE-2.0
  ##
  ##   Unless required by applicable law or agreed to in writing, software
  ##   distributed under the License is distributed on an "AS IS" BASIS,
  ##   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  ##   See the License for the specific language governing permissions and
  ##   limitations under the License.
  ##
  ################################################################################*/
 
/* 
 * log multivariate gamma function
 */
 
#ifndef _gcem_lmgamma_HPP
#define _gcem_lmgamma_HPP
 
namespace internal
{
 
// see https://en.wikipedia.org/wiki/Multivariate_gamma_function
 
template<typename T1, typename T2>
constexpr
T1
lmgamma_recur(const T1 a, const T2 p)
noexcept
{
    return( // NaN check
            is_nan(a) ? \
                GCLIM<T1>::quiet_NaN() :
            //
            p == T2(1) ? \
                lgamma(a) :
            p <  T2(1) ? \
                GCLIM<T1>::quiet_NaN() :
            // else
                T1(GCEM_LOG_PI) * (p - T1(1))/T1(2) \
                    + lgamma(a) + lmgamma_recur(a - T1(0.5),p-T2(1)) );
}
 
}
 
/**
 * Compile-time log multivariate gamma function
 *
 * @param a a real-valued input.
 * @param p integral-valued input.
 * @return computes log-multivariate gamma function via recursion
 * \f[ \Gamma_p(a) = \pi^{(p-1)/2} \Gamma(a) \Gamma_{p-1}(a-0.5) \f]
 * where \f$ \Gamma_1(a) = \Gamma(a) \f$.
 */
 
template<typename T1, typename T2>
constexpr
return_t<T1>
lmgamma(const T1 a, const T2 p)
noexcept
{
    return internal::lmgamma_recur(static_cast<return_t<T1>>(a),p);
}
 
#endif