@stdlib/math-base-special-gammainc/lib/igamma_temme_large.js

Incomplete gamma function.

/**
* @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 https://www.boost.org/doc/libs/1_88_0/boost/math/special_functions/gamma.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
* (C) Copyright Paul A. Bristow 2007.
*
* 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 evalpoly = require( '@stdlib/math-base-tools-evalpoly' );
var log1pmx = require( '@stdlib/math-base-special-log1pmx' );
var erfc = require( '@stdlib/math-base-special-erfc' );
var sqrt = require( '@stdlib/math-base-special-sqrt' );
var exp = require( '@stdlib/math-base-special-exp' );
var TWO_PI = require( '@stdlib/constants-float64-two-pi' );
var polyvalC0 = require( './polyval_c0.js' );
var polyvalC1 = require( './polyval_c1.js' );
var polyvalC2 = require( './polyval_c2.js' );
var polyvalC3 = require( './polyval_c3.js' );
var polyvalC4 = require( './polyval_c4.js' );
var polyvalC5 = require( './polyval_c5.js' );
var polyvalC6 = require( './polyval_c6.js' );
var polyvalC7 = require( './polyval_c7.js' );
var polyvalC8 = require( './polyval_c8.js' );


// VARIABLES //

// Pre-allocate workspace array:
var workspace = [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ]; // WARNING: not thread safe


// MAIN //

/**
* Asymptotic expansions of the incomplete gamma functions when `a` is large and `x ~ a` (IEEE double precision or 10^-17).
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @returns {number} value of asymptotic expansion
*/
function igammaTemmeLarge( a, x ) {
	var result;
	var sigma;
	var phi;
	var y;
	var z;

	sigma = ( x-a ) / a;
	phi = -log1pmx( sigma );
	y = a * phi;
	z = sqrt( 2.0 * phi );
	if ( x < a ) {
		z = -z;
	}
	workspace[ 0 ] = polyvalC0( z );
	workspace[ 1 ] = polyvalC1( z );
	workspace[ 2 ] = polyvalC2( z );
	workspace[ 3 ] = polyvalC3( z );
	workspace[ 4 ] = polyvalC4( z );
	workspace[ 5 ] = polyvalC5( z );
	workspace[ 6 ] = polyvalC6( z );
	workspace[ 7 ] = polyvalC7( z );
	workspace[ 8 ] = polyvalC8( z );
	workspace[ 9 ] = -0.00059676129019274625;
	result = evalpoly( workspace, 1.0/a );
	result *= exp( -y ) / sqrt( TWO_PI * a );
	if ( x < a ) {
		result = -result;
	}
	result += erfc( sqrt(y) ) / 2.0;
	return result;
}


// EXPORTS //

module.exports = igammaTemmeLarge;