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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_64_0/boost/math/special_functions/beta.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
*
* 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 lanczosSumExpGScaled = require( './../../../../base/special/gamma-lanczos-sum-expg-scaled' );
var sumSeries = require( './../../../../base/tools/sum-series' );
var log1p = require( './../../../../base/special/log1p' );
var sqrt = require( './../../../../base/special/sqrt' );
var exp = require( './../../../../base/special/exp' );
var pow = require( './../../../../base/special/pow' );
var ln = require( './../../../../base/special/ln' );
var MIN_VALUE = require( '@stdlib/constants/float64/smallest-normal' );
var MAX_LN = require( '@stdlib/constants/float64/max-ln' );
var MIN_LN = require( '@stdlib/constants/float64/min-ln' );
var G = require( '@stdlib/constants/float64/gamma-lanczos-g' );
var E = require( '@stdlib/constants/float64/e' );
// VARIABLES //
var opts = {
'maxTerms': 100
};
// FUNCTIONS //
/**
* Series approximation to the incomplete beta.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {number} result - initial result value
* @returns {Function} series function
*/
function ibetaSeriesT( a, b, x, result ) {
var poch = 1.0 - b;
var n = 1;
return next;
/**
* Calculate the next term of the series.
*
* @private
* @returns {number} series expansion term
*/
function next() {
var r = result / a;
a += 1.0;
result *= poch * x / n;
n += 1;
poch += 1.0;
return r;
}
}
// MAIN //
/**
* Incomplete beta series.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {NonNegativeInteger} s0 - initial value
* @param {boolean} normalized - boolean indicating whether to evaluate the power terms of the regularized or non-regularized incomplete beta function
* @param {(Array|TypedArray|Object)} out - output array holding the derivative as the second element
* @param {Probability} y - probability equal to `1-x`
* @returns {number} function value
*/
function ibetaSeries( a, b, x, s0, normalized, out, y ) {
var result;
var agh;
var bgh;
var cgh;
var l1;
var l2;
var c;
var s;
if ( normalized ) {
c = a + b;
// Incomplete beta power term, combined with the Lanczos approximation:
agh = a + G - 0.5;
bgh = b + G - 0.5;
cgh = c + G - 0.5;
result = lanczosSumExpGScaled( c ) / ( lanczosSumExpGScaled( a ) * lanczosSumExpGScaled( b ) ); // eslint-disable-line max-len
l1 = ln( cgh / bgh ) * ( b - 0.5 );
l2 = ln( x * cgh / agh ) * a;
// Check for over/underflow in the power terms:
if (
l1 > MIN_LN &&
l1 < MAX_LN &&
l2 > MIN_LN &&
l2 < MAX_LN
) {
if ( a * b < bgh * 10.0 ) {
result *= exp( ( b-0.5 ) * log1p( a / bgh ) );
} else {
result *= pow( cgh / bgh, b - 0.5 );
}
result *= pow( x * cgh / agh, a );
result *= sqrt( agh / E );
if ( out ) {
out[ 1 ] = result * pow( y, b );
}
}
else {
// We need logs, and this *will* cancel:
result = ln( result ) + l1 + l2 + ( ( ln( agh ) - 1.0 ) / 2.0 );
if ( out ) {
out[ 1 ] = exp( result + ( b * ln( y ) ) );
}
result = exp( result );
}
}
else {
// Non-normalized, just compute the power:
result = pow( x, a );
}
if ( result < MIN_VALUE ) {
return s0; // Safeguard: series can't cope with denorms.
}
s = ibetaSeriesT( a, b, x, result );
opts.initialValue = s0;
return sumSeries( s, opts );
}
// EXPORTS //
module.exports = ibetaSeries;
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