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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_62_0/boost/math/special_functions/gamma.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006-7, 2013-14.
* (C) Copyright Paul A. Bristow 2007, 2013-14.
* (C) Copyright Nikhar Agrawal 2013-14.
* (C) 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 gammaln = require( '@stdlib/math/base/special/gammaln' );
var floor = require( '@stdlib/math/base/special/floor' );
var gamma = require( '@stdlib/math/base/special/gamma' );
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 ln = require( '@stdlib/math/base/special/ln' );
var SQRT_EPSILON = require( '@stdlib/constants/float64/sqrt-eps' );
var FLOAT64_MAX = require( '@stdlib/constants/float64/max' );
var SQRT_TWO_PI = require( '@stdlib/constants/float64/sqrt-two-pi' );
var MAX_LN = require( '@stdlib/constants/float64/max-ln' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var finiteGammaQ = require( './finite_gamma_q.js' );
var finiteHalfGammaQ = require( './finite_half_gamma_q.js' );
var fullIGammaPrefix = require( './full_igamma_prefix.js' );
var igammaTemmeLarge = require( './igamma_temme_large.js' );
var lowerGammaSeries = require( './lower_gamma_series.js' );
var regularisedGammaPrefix = require( './regularised_gamma_prefix.js' );
var tgammaSmallUpperPart = require( './tgamma_small_upper_part.js' );
var upperGammaFraction = require( './upper_gamma_fraction.js' );
// VARIABLES //
var MAX_FACTORIAL = 170; // TODO: consider extracting as a constant
// MAIN //
/**
* Computes the regularized incomplete gamma function. The upper tail is calculated via the modified Lentz's method for computing continued fractions, the lower tail using a power expansion.
*
* ## Notes
*
* - When `a >= MAX_FACTORIAL` and computing the non-normalized incomplete gamma, result is rather hard to compute unless we use logs. There are really two options a) if `x` is a long way from `a` in value then we can reliably use methods 2 and 4 below in logarithmic form and go straight to the result. Otherwise we let the regularized gamma take the strain (the result is unlikely to underflow in the central region anyway) and combine with `lgamma` in the hopes that we get a finite result.
*
* @param {NonNegativeNumber} x - function parameter
* @param {PositiveNumber} a - function parameter
* @param {boolean} [regularized=true] - boolean indicating if the function should evaluate the regularized or non-regularized incomplete gamma functions
* @param {boolean} [upper=false] - boolean indicating if the function should return the upper tail of the incomplete gamma function
* @returns {number} function value
*/
function gammainc( x, a, regularized, upper ) {
var optimisedInvert;
var normalized;
var evalMethod;
var initValue;
var isHalfInt;
var useTemme;
var isSmallA;
var invert;
var result;
var isInt;
var sigma;
var gam;
var res;
var fa;
var g;
if ( x < 0.0 || a <= 0.0 ) {
return NaN;
}
normalized = ( regularized === void 0 ) ? true : regularized;
invert = upper;
result = 0.0;
if ( a >= MAX_FACTORIAL && !normalized ) {
if ( invert && ( a * 4.0 < x ) ) {
// This is method 4 below, done in logs:
result = ( a * ln(x) ) - x;
result += ln( upperGammaFraction( a, x ) );
}
else if ( !invert && ( a > 4.0 * x ) ) {
// This is method 2 below, done in logs:
result = ( a * ln(x) ) - x;
initValue = 0;
result += ln( lowerGammaSeries( a, x, initValue ) / a );
}
else {
result = gammainc( a, x, true, invert );
if ( result === 0.0 ) {
if ( invert ) {
// Try http://functions.wolfram.com/06.06.06.0039.01
result = 1.0 + ( 1.0 / (12.0*a) ) + ( 1.0 / (288.0*a*a) );
result = ln( result ) - a + ( ( a-0.5 ) * ln(a) );
result += ln( SQRT_TWO_PI );
} else {
// This is method 2 below, done in logs, we're really outside the range of this method, but since the result is almost certainly infinite, we should probably be OK:
result = ( a * ln( x ) ) - x;
initValue = 0.0;
result += ln( lowerGammaSeries( a, x, initValue ) / a );
}
}
else {
result = ln( result ) + gammaln( a );
}
}
if ( result > MAX_LN ) {
return PINF;
}
return exp( result );
}
isSmallA = ( a < 30 ) && ( a <= x + 1.0 ) && ( x < MAX_LN );
if ( isSmallA ) {
fa = floor( a );
isInt = ( fa === a );
isHalfInt = ( isInt ) ? false : ( abs( fa - a ) === 0.5 );
} else {
isInt = isHalfInt = false;
}
if ( isInt && x > 0.6 ) {
// Calculate Q via finite sum:
invert = !invert;
evalMethod = 0;
}
else if ( isHalfInt && x > 0.2 ) {
// Calculate Q via finite sum for half integer a:
invert = !invert;
evalMethod = 1;
}
else if ( x < SQRT_EPSILON && a > 1.0 ) {
evalMethod = 6;
}
else if ( x < 0.5 ) {
// Changeover criterion chosen to give a changeover at Q ~ 0.33:
if ( -0.4 / ln( x ) < a ) {
evalMethod = 2;
} else {
evalMethod = 3;
}
}
else if ( x < 1.1 ) {
// Changeover here occurs when P ~ 0.75 or Q ~ 0.25:
if ( x * 0.75 < a ) {
evalMethod = 2;
} else {
evalMethod = 3;
}
}
else {
// Begin by testing whether we're in the "bad" zone where the result will be near 0.5 and the usual series and continued fractions are slow to converge:
useTemme = false;
if ( normalized && a > 20 ) {
sigma = abs( (x-a)/a );
if ( a > 200 ) {
// Limit chosen so that we use Temme's expansion only if the result would be larger than about 10^-6. Below that the regular series and continued fractions converge OK, and if we use Temme's method we get increasing errors from the dominant erfc term as it's (inexact) argument increases in magnitude.
if ( 20 / a > sigma * sigma ) {
useTemme = true;
}
} else if ( sigma < 0.4 ) {
useTemme = true;
}
}
if ( useTemme ) {
evalMethod = 5;
}
// Regular case where the result will not be too close to 0.5: Changeover occurs at P ~ Q ~ 0.5. Note that series computation of P is about x2 faster than continued fraction calculation of Q, so try and use the CF only when really necessary, especially for small x.
else if ( x - ( 1.0 / (3.0 * x) ) < a ) {
evalMethod = 2;
} else {
evalMethod = 4;
invert = !invert;
}
}
/* eslint-disable default-case */
switch ( evalMethod ) {
case 0:
result = finiteGammaQ( a, x );
if (normalized === false ) {
result *= gamma( a );
}
break;
case 1:
result = finiteHalfGammaQ( a, x );
if ( normalized === false ) {
result *= gamma( a );
}
break;
case 2:
// Compute P:
result = ( normalized ) ?
regularisedGammaPrefix( a, x ) :
fullIGammaPrefix( a, x );
if ( result !== 0.0 ) {
initValue = 0.0;
optimisedInvert = false;
if ( invert ) {
initValue = ( normalized ) ? 1.0 : gamma( a );
if (
normalized ||
result >= 1.0 ||
FLOAT64_MAX * result > initValue
) {
initValue /= result;
if (
normalized ||
a < 1.0 ||
( FLOAT64_MAX / a > initValue )
) {
initValue *= -a;
optimisedInvert = true;
}
else {
initValue = 0.0;
}
}
else {
initValue = 0.0;
}
}
}
result *= lowerGammaSeries( a, x, initValue ) / a;
if ( optimisedInvert ) {
invert = false;
result = -result;
}
break;
case 3:
// Compute Q:
invert = !invert;
res = tgammaSmallUpperPart( a, x, invert );
result = res[ 0 ];
g = res[ 1 ];
invert = false;
if ( normalized ) {
result /= g;
}
break;
case 4:
// Compute Q:
result = ( normalized ) ?
regularisedGammaPrefix( a, x ) :
fullIGammaPrefix( a, x );
if ( result !== 0 ) {
result *= upperGammaFraction( a, x );
}
break;
case 5:
result = igammaTemmeLarge( a, x );
if ( x >= a ) {
invert = !invert;
}
break;
case 6:
// Since x is so small that P is necessarily very small too, use http://functions.wolfram.com/GammaBetaErf/GammaRegularized/06/01/05/01/01/
result = ( normalized ) ?
pow(x, a) / gamma( a + 1.0 ) :
pow( x, a ) / a;
result *= 1.0 - ( a * x / ( a + 1.0 ) );
break;
}
if ( normalized && result > 1.0 ) {
result = 1.0;
}
if ( invert ) {
gam = ( normalized ) ? 1.0 : gamma( a );
result = gam - result;
}
return result;
}
// EXPORTS //
module.exports = gammainc;
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