Press n or j to go to the next uncovered block, b, p or k for the previous block.
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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.
*/
'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;
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