@stdlib/math-base-special-gammainc/lib/tgamma_small_upper_part.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 http://www.boost.org/doc/libs/1_37_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 powm1 = require( '@stdlib/math-base-special-powm1' );
var sumSeries = require( '@stdlib/math-base-tools-sum-series' );
var gamma1pm1 = require( '@stdlib/math-base-special-gamma1pm1' );
var smallGamma2Series = require( './small_gamma2_series.js' );


// MAIN //

/**
* Compute the full upper fraction (Q) when `a` is very small.
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @param {boolean} invert - boolean indicating if the upper tail of the incomplete gamma function should be evaluated
* @returns {Array} full upper fraction (Q) and pgam
*/
function tgammaSmallUpperPart( a, x, invert ) {
	var initialValue;
	var result;
	var pgam;
	var p;
	var s;

	result = gamma1pm1( a );
	pgam = ( result + 1.0 ) / a;
	p = powm1( x, a );
	result -= p;
	result /= a;
	s = smallGamma2Series( a, x );
	p += 1.0;
	initialValue = ( invert ) ? pgam : 0.0;
	result = -p * sumSeries( s, {
		'initialValue': (initialValue - result) / p
	});
	if ( invert ) {
		result = -result;
	}
	return [ result, pgam ];
}


// EXPORTS //

module.exports = tgammaSmallUpperPart;