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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 https://github.com/boostorg/math/blob/develop/include/boost/math/special_functions/detail/bessel_y1.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* Copyright Xiaogang Zhang, 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 ln = require( '@stdlib/math/base/special/ln' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var PI = require( '@stdlib/constants/float64/pi' );
var SQRT_PI = require( '@stdlib/constants/float64/sqrt-pi' );
var NINF = require( '@stdlib/constants/float64/ninf' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var sincos = require( '@stdlib/math/base/special/sincos' ).assign;
var besselj1 = require( '@stdlib/math/base/special/besselj1' );
var poly1 = require( './rational_p1q1.js' );
var poly2 = require( './rational_p2q2.js' );
var polyC = require( './rational_pcqc.js' );
var polyS = require( './rational_psqs.js' );
// VARIABLES //
var ONE_DIV_SQRT_PI = 1.0 / SQRT_PI;
var TWO_DIV_PI = 2.0 / PI;
var x1 = 2.1971413260310170351e+00;
var x2 = 5.4296810407941351328e+00;
var x11 = 5.620e+02;
var x12 = 1.8288260310170351490e-03;
var x21 = 1.3900e+03;
var x22 = -6.4592058648672279948e-06;
// `sincos` workspace:
var sc = [ 0.0, 0.0 ];
// MAIN //
/**
* Computes the Bessel function of the second kind of order one.
*
* ## Notes
*
* - Accuracy for subnormal `x` is very poor. Full accuracy is achieved at `1.0e-308` but trends progressively to zero at `5e-324`. This suggests that underflow (or overflow, perhaps due to a reciprocal) is effectively cutting off digits of precision until the computation loses all accuracy at `5e-324`.
*
* @param {number} x - input value
* @returns {number} evaluated Bessel function
*
* @example
* var v = y1( 0.0 );
* // returns -Infinity
*
* v = y1( 1.0 );
* // returns ~-0.781
*
* v = y1( -1.0 );
* // returns NaN
*
* v = y1( Infinity );
* // returns 0.0
*
* v = y1( -Infinity );
* // returns NaN
*
* v = y1( NaN );
* // returns NaN
*/
function y1( x ) {
var rc;
var rs;
var y2;
var r;
var y;
var z;
var f;
if ( x < 0.0 ) {
return NaN;
}
if ( x === 0.0 ) {
return NINF;
}
if ( x === PINF ) {
return 0.0;
}
if ( x <= 4.0 ) {
y = x * x;
z = ( ln( x/x1 ) * besselj1( x ) ) * TWO_DIV_PI;
r = poly1( y );
f = ( ( x+x1 ) * ( (x - (x11/256.0)) - x12 ) ) / x;
return z + ( f*r );
}
if ( x <= 8.0 ) {
y = x * x;
z = ( ln( x/x2 ) * besselj1( x ) ) * TWO_DIV_PI;
r = poly2( y );
f = ( ( x+x2 ) * ( (x - (x21/256.0)) - x22 ) ) / x;
return z + ( f*r );
}
y = 8.0 / x;
y2 = y * y;
rc = polyC( y2 );
rs = polyS( y2 );
f = ONE_DIV_SQRT_PI / sqrt( x );
/*
* This code is really just:
*
* ```
* z = x - 0.75 * PI;
* return f * (rc * sin(z) + y * rs * cos(z));
* ```
*
* But using the sin/cos addition rules, plus constants for sin/cos of `3π/4` which then cancel out with corresponding terms in "f".
*/
sincos( x, sc, 1, 0 );
return f * ( ( ( (y*rs) * (sc[0]-sc[1]) ) - ( rc * (sc[0]+sc[1]) ) ) );
}
// EXPORTS //
module.exports = y1;
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