@stdlib/math-base-special-gammaincinv/lib/higher_newton.js

Inverse 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.
*/

'use strict';

// MODULES //

var logger = require( 'debug' );
var gammainc = require( '@stdlib/math-base-special-gammainc' );
var abs = require( '@stdlib/math-base-special-abs' );
var exp = require( '@stdlib/math-base-special-exp' );
var ln = require( '@stdlib/math-base-special-ln' );
var MAX_FLOAT32 = require( '@stdlib/constants-float32-max' );


// VARIABLES //

var debug = logger( 'gammaincinv:higher_newton' );


// MAIN //

/**
* Implementation of the high order Newton-like method.
*
* @private
* @param {number} x0 - initial value
* @param {number} a - scale parameter
* @param {number} m - indicator
* @param {Probability} p - probability value
* @param {Probability} q - probability value
* @param {number} lgama - logarithm of scale parameter
* @param {number} invfp - one over `fp`
* @param {boolean} pcase - boolean indicating whether p < 0.5
* @returns {number} function value of the inverse
*/
function higherNewton( x0, a, m, p, q, lgama, invfp, pcase ) {
	var dlnr;
	var xini;
	var ck0;
	var ck1;
	var ck2;
	var a2;
	var x2;
	var px;
	var qx;
	var xr;
	var t;
	var n;
	var r;
	var x;

	x = x0;
	t = 1;
	n = 1;
	a2 = a * a;
	xini = x0;
	do {
		x = x0;
		x2 = x * x;
		if ( m === 0 ) {
			dlnr = ( ( 1.0-a ) * ln( x ) ) + x + lgama;
			if ( dlnr > ln( MAX_FLOAT32 ) ) {
				debug( 'Warning: overflow problems in one or more steps of the computation. The initial approximation to the root is returned.' );
				return xini;
			}
			r = exp( dlnr );
		} else {
			r = -invfp * x;
		}
		if ( pcase ) {
			// Call: gammainc( x, s[, regularized = true ][, upper = false ] )
			px = gammainc( x, a, true, false );
			ck0 = -r * ( px - p );
		} else {
			// Call: gammainc( x, s[, regularized = true ][, upper = true ] )
			qx = gammainc( x, a, true, true );
			ck0 = r * ( qx - q );
		}
		r = ck0;
		if ( ( p > 1e-120 ) || ( n > 1 ) ) {
			ck1 = 0.5 * ( x - a + 1.0 ) / x;
			ck2 = ( (2*x2) - (4*x*a) + (4*x) + (2*a2) - (3*a) + 1 ) / x2;
			ck2 /= 6.0;
			x0 = x + ( r * ( 1.0 + ( r * ( ck1 + (r*ck2) ) ) ) );
		} else {
			x0 = x + r;
		}
		t = abs( ( x/x0 ) - 1.0 );
		n += 1;
		x = x0;
		if ( x < 0 ) {
			x = xini;
			n = 100;
		}
	} while ( ( ( t > 2e-14 ) && ( n < 35 ) ) );
	if ( ( t > 2e-14 ) || ( n > 99 ) ) {
		debug( 'Warning: the number of iterations in the Newton method reached the upper limit N=35. The last value obtained for the root is given as output.' );
	}
	xr = x || 0;
	return xr;
}


// EXPORTS //

module.exports = higherNewton;