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* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* 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.
*
*
* ## Notice
*
* The original C++ code and copyright notice are from the [Boost library]{@link http://www.boost.org/doc/libs/1_85_0/boost/math/special_functions/gamma.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* Copyright John Maddock 2006-7, 2013-14.
* Copyright Paul A. Bristow 2007, 2013-14.
* Copyright Nikhar Agrawal 2013-14.
* Copyright Christopher Kormanyos 2013-14.
*
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
* ```
*/
'use strict';
// MODULES //
var lanczosSum = require( '@stdlib/math/base/special/gamma-lanczos-sum' );
var gamma = require( '@stdlib/math/base/special/gamma' );
var log1p = require( '@stdlib/math/base/special/log1p' );
var abs = require( '@stdlib/math/base/special/abs' );
var exp = require( '@stdlib/math/base/special/exp' );
var pow = require( '@stdlib/math/base/special/pow' );
var EPSILON = require( '@stdlib/constants/float64/eps' );
var E = require( '@stdlib/constants/float64/e' );
var G = require( '@stdlib/constants/float64/gamma-lanczos-g' );
var FLOAT64_MAX_NTH_FACTORIAL = require( '@stdlib/constants/float64/max-nth-factorial' );
// VARIABLES //
var FACTORIAL_169 = 4.269068009004705e+304;
// MAIN //
/**
* Calculates the ratio of two gamma functions via Lanczos approximation.
*
* ## Notes
*
* - When \\( z < \epsilon \\), we get spurious numeric overflow unless we're very careful. This can occur either inside `lanczosSum(z)` or in the final combination of terms. To avoid this, split the product up into 2 (or 3) parts:
*
* ```tex
* \begin{align*}
* G(z) / G(L) &= 1 / (z \cdot G(L)) ; z < \eps, L = z + \delta = \delta \\
* z * G(L) &= z * G(lim) \cdot (G(L)/G(lim)) ; lim = \text{largest factorial}
* \end{align*}
* ```
*
* @private
* @param {number} z - first gamma parameter
* @param {number} delta - difference
* @returns {number} gamma ratio
*/
function gammaDeltaRatioLanczos( z, delta ) {
var result;
var ratio;
var zgh;
if ( z < EPSILON ) {
if ( delta >= FLOAT64_MAX_NTH_FACTORIAL ) {
ratio = gammaDeltaRatioLanczos( delta, FLOAT64_MAX_NTH_FACTORIAL-delta );
ratio *= z;
ratio *= FACTORIAL_169;
return 1.0 / ratio;
}
return 1.0 / ( z * gamma( z+delta ) );
}
zgh = z + G - 0.5;
if ( z + delta === z ) {
if ( abs( delta / zgh ) < EPSILON ) {
result = exp( -delta );
} else {
result = 1.0;
}
} else {
if ( abs(delta) < 10.0 ) {
result = exp( ( 0.5-z ) * log1p( delta/zgh ) );
} else {
result = pow( zgh / (zgh+delta), z-0.5 );
}
// Split up the calculation to avoid spurious overflow:
result *= lanczosSum( z ) / lanczosSum( z + delta );
}
result *= pow( E / ( zgh+delta ), delta );
return result;
}
// EXPORTS //
module.exports = gammaDeltaRatioLanczos;
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