1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
|
/**
* @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_64_0/boost/math/special_functions/detail/ibeta_inverse.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* Copyright John Maddock 2006.
* Copyright Paul A. Bristow 2007.
*
* 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 evalpoly = require( './../../../../base/tools/evalpoly' );
var erfcinv = require( './../../../../base/special/erfcinv' );
var sqrt = require( './../../../../base/special/sqrt' );
var exp = require( './../../../../base/special/exp' );
var SQRT2 = require( '@stdlib/constants/float64/sqrt-two' );
// VARIABLES //
// Workspaces for the polynomial coefficients:
var workspace = [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ];
var terms = [ 0.0, 0.0, 0.0, 0.0 ];
// MAIN //
/**
* Carries out the first method by Temme (described in section 2).
*
* ## References
*
* - Temme, N. M. 1992. "Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function." _Journal of Computational and Applied Mathematics_ 41 (1–2): 1638–63. doi:[10.1016/0377-0427(92)90244-R](https://doi.org/10.1016/0377-0427(92)90244-R).
*
* @private
* @param {PositiveNumber} a - function parameter
* @param {PositiveNumber} b - function parameter
* @param {Probability} z - function parameter
* @returns {number} function value
*/
function temme1( a, b, z ) {
var eta0;
var eta2;
var eta;
var B2;
var B3;
var B;
var c;
// Get the first approximation for eta from the inverse error function (Eq: 2.9 and 2.10):
eta0 = erfcinv( 2.0 * z );
eta0 /= -sqrt( a / 2.0 );
terms[ 0 ] = eta0;
// Calculate powers:
B = b - a;
B2 = B * B;
B3 = B2 * B;
// Calculate correction terms:
// See eq following 2.15:
workspace[ 0 ] = -B * SQRT2 / 2;
workspace[ 1 ] = ( 1 - (2.0*B) ) / 8.0;
workspace[ 2 ] = -(B * SQRT2 / 48.0);
workspace[ 3 ] = -1.0 / 192.0;
workspace[ 4 ] = -B * SQRT2 / 3840.0;
workspace[ 5 ] = 0.0;
workspace[ 6 ] = 0.0;
terms[ 1 ] = evalpoly( workspace, eta0 );
// Eq Following 2.17:
workspace[ 0 ] = B * SQRT2 * ( (3.0*B) - 2.0) / 12.0;
workspace[ 1 ] = ( (20.0*B2) - (12.0*B) + 1.0 ) / 128.0;
workspace[ 2 ] = B * SQRT2 * ( (20.0*B) - 1.0) / 960.0;
workspace[ 3 ] = ( (16.0*B2) + (30.0*B) - 15.0) / 4608.0;
workspace[ 4 ] = B * SQRT2 * ( (21.0*B) + 32) / 53760.0;
workspace[ 5 ] = (-(32.0*B2) + 63.0) / 368640.0;
workspace[ 6 ] = -B * SQRT2 * ( (120.0*B) + 17.0) / 25804480.0;
terms[ 2 ] = evalpoly( workspace, eta0 );
// Eq Following 2.17:
workspace[ 0 ] = B * SQRT2 * ( (-75*B2) + (80.0*B) - 16.0) / 480.0;
workspace[ 1 ] = ( (-1080.0*B3) + (868.0*B2) - (90.0*B) - 45.0) / 9216.0;
workspace[ 2 ] = B * SQRT2 * ( (-1190.0*B2) + (84.0*B) + 373.0) / 53760.0;
workspace[ 3 ] = ( (-2240.0*B3)-(2508.0*B2)+(2100.0*B)-165.0 ) / 368640.0;
workspace[ 4 ] = 0.0;
workspace[ 5 ] = 0.0;
workspace[ 6 ] = 0.0;
terms[ 3 ] = evalpoly( workspace, eta0 );
// Bring them together to get a final estimate for eta:
eta = evalpoly( terms, 1.0/a );
// Now we need to convert eta to the return value `x`, by solving the appropriate quadratic equation:
eta2 = eta * eta;
c = -exp( -eta2 / 2.0 );
if ( eta2 === 0.0 ) {
return 0.5;
}
return ( 1.0 + ( eta * sqrt( ( 1.0+c ) / eta2 ) ) ) / 2.0;
}
// EXPORTS //
module.exports = temme1;
|
