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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_61_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 gammaDeltaRatio = require( './../../../../base/special/gamma-delta-ratio' );
var factorial = require( './../../../../base/special/factorial' );
var gammainc = require( './../../../../base/special/gammainc' );
var log1p = require( './../../../../base/special/log1p' );
var abs = require( './../../../../base/special/abs' );
var pow = require( './../../../../base/special/pow' );
var ln = require( './../../../../base/special/ln' );
var MIN_VALUE = require( '@stdlib/constants/float64/smallest-normal' );
var EPSILON = require( '@stdlib/constants/float64/eps' );
var fullIGammaPrefix = require( './full_igamma_prefix.js' );
var regularizedGammaPrefix = require( './regularized_gamma_prefix.js' );
// VARIABLES //
var p = new Array( 30 );
// MAIN //
/**
* This is DiDonato and Morris's BGRAT routine, see Eq's 9 through 9.6.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {Probability} y - probability equal to `1-x`
* @param {NonNegativeInteger} s0 - initial value
* @param {number} mult - initial value
* @param {boolean} normalized - boolean indicating whether to evaluate the regularized or non-regularized incomplete beta function
* @returns {number} function value
*/
function betaSmallBLargeASeries( a, b, x, y, s0, mult, normalized ) {
var prefix;
var tmp1;
var tnp1;
var sum;
var b2n;
var bm1;
var lx2;
var lxp;
var mbn;
var lx;
var t4;
var h;
var j;
var m;
var n;
var r;
var t;
var u;
// Some values we'll need later, these are Eq 9.1:
bm1 = b - 1.0;
t = a + ( bm1 / 2.0 );
if ( y < 0.35 ) {
lx = log1p( -y );
} else {
lx = ln( x );
}
u = -t * lx;
// And from from 9.2:
h = regularizedGammaPrefix( b, u );
if ( h <= MIN_VALUE ) {
return s0;
}
if ( normalized ) {
prefix = h / gammaDeltaRatio( a, b );
prefix /= pow( t, b );
} else {
prefix = fullIGammaPrefix( b, u ) / pow( t, b );
}
prefix *= mult;
// We need the quantity Pn. Unfortunately, this is computed recursively and requires a full history of all the previous values. No choice but to declare a big table and hope it's big enough...
p[ 0 ] = 1; // see 9.3.
// Now an initial value for J, see 9.6: gammainc( u, b, regularized, upper )
j = gammainc( u, b, true, true );
j /= h;
// Now we can start to pull things together and evaluate the sum in Eq 9:
sum = s0 + ( prefix * j ); // Value at N = 0
// Some variables we'll need...
tnp1 = 1.0; // 2*N+1
lx2 = lx / 2.0;
lx2 *= lx2;
lxp = 1.0;
t4 = 4.0 * t * t;
b2n = b;
for ( n = 1; n < p.length; ++n ) {
// Begin by evaluating the next Pn from Eq 9.4:
tnp1 += 2.0;
p[ n ] = 0.0;
mbn = b - n;
tmp1 = 3;
for ( m = 1; m < n; ++m ) {
mbn = ( m * b ) - n;
p[ n ] += mbn * p[ n-m ] / factorial( tmp1 );
tmp1 += 2;
}
p[ n ] /= n;
p[ n ] += bm1 / factorial( tnp1 );
// Now we want Jn from Jn-1 using Eq 9.6:
j = ( ( b2n * ( b2n+1.0 ) * j ) + ( ( u+b2n+1.0 ) * lxp ) ) / t4;
lxp *= lx2;
b2n += 2.0;
// Pull it together with Eq 9:
r = prefix * p[ n ] * j;
sum += r;
if ( r > 1.0 ) {
if ( abs( r ) < abs( EPSILON * sum ) ) {
break;
}
} else if ( abs( r / EPSILON ) < abs( sum ) ) {
break;
}
}
return sum;
}
// EXPORTS //
module.exports = betaSmallBLargeASeries;
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