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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217
|
/**
* @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/factorials.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006, 2010.
*
* 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 isNonNegativeInteger = require( './../../../../base/assert/is-nonnegative-integer' );
var isInteger = require( './../../../../base/assert/is-integer' );
var isnan = require( './../../../../base/assert/is-nan' );
var gammaDeltaRatio = require( './../../../../base/special/gamma-delta-ratio' );
var floor = require( './../../../../base/special/floor' );
var abs = require( './../../../../base/special/abs' );
var FLOAT64_MAX = require( '@stdlib/constants/float64/max' );
var PINF = require( '@stdlib/constants/float64/pinf' );
// VARIABLES //
var MAX_FACTORIAL = 170; // TODO: consider packaging as constant
// FUNCTIONS //
/**
* Computes the rising factorial of `x` and `n`.
*
* ## Notes
*
* - The rising factorial is defined as
*
* ```tex
* \operatorname{risingFactorial}(x, n) = x (x-1) (x-2) (x-3) \ldots (x-n+1)
* ```
*
* or equivalently
*
* ```tex
* \operatorname{risingFactorial}(x, n) = \frac{ \Gamma(x + n) }{ \Gamma(x) };
* ```
*
* @param {number} x - first function parameter
* @param {integer} n - second function parameter
* @returns {number} function value
*
* @example
* var v = risingFactorial( 0.9, 5 );
* // returns ~94.766
*
* @example
* var v = risingFactorial( -9.0, 3 );
* // returns -504.0
*
* @example
* var v = risingFactorial( 0.0, 2 );
* // returns 0.0
*
* @example
* var v = risingFactorial( 3.0, -2 );
* // returns 0.5
*/
function risingFactorial( x, n ) {
var result;
var inv;
if ( isnan( x ) || !isInteger( n ) ) {
return NaN;
}
if ( x < 0.0 ) {
// For `x < 0`, we really have a falling factorial, modulo a possible change of sign. Note that the falling factorial isn't defined for negative `n`, so we'll get rid of that case first:
if ( n < 0.0 ) {
x += n;
n = -n;
inv = true;
}
result = ( (n&1) ? -1.0 : 1.0 ) * fallingFactorial( -x, n );
if ( inv ) {
result = 1.0 / result;
}
return result;
}
if ( n === 0 ) {
return 1.0;
}
if ( x === 0.0 ) {
if ( n < 0 ) {
return -gammaDeltaRatio( x+1.0, -n );
}
return 0.0;
}
if ( x < 1.0 && x+n < 0.0 ) {
result = gammaDeltaRatio( 1.0-x, -n );
return ( n&1 ) ? -result : result;
}
// We don't optimize this for small `n`, because `gammaDeltaRatio` is already optimized for that use case:
return 1.0 / gammaDeltaRatio( x, n );
}
// MAIN //
/**
* Computes the falling factorial of `x` and `n`.
*
* ## Notes
*
* - The falling factorial is defined as
*
* ```tex
* \operatorname{fallingFactorial}(x, n) = x (x-1) (x-2) (x-3) \ldots (x-n+1)
* ```
*
* @param {number} x - first function parameter
* @param {NonNegativeInteger} n - second function parameter
* @returns {number} function value
*
* @example
* var v = fallingFactorial( 0.9, 5 );
* // returns ~0.644
*
* @example
* var v = fallingFactorial( -9.0, 3 );
* // returns -990.0
*
* @example
* var v = fallingFactorial( 0.0, 2 );
* // returns 0.0
*
* @example
* var v = fallingFactorial( 3.0, -2 );
* // returns NaN
*/
function fallingFactorial( x, n ) {
var result;
var xp1;
var n2;
var t1;
var t2;
if ( isnan( x ) || !isNonNegativeInteger( n ) ) {
return NaN;
}
if ( x === 0.0 ) {
return 0.0;
}
if ( x < 0.0 ) {
// For `x < 0`, we really have a rising factorial modulo a possible change of sign:
return ( ( n&1 ) ? -1.0 : 1.0 ) * risingFactorial( -x, n );
}
if ( n === 0 ) {
return 1.0;
}
if ( x < 0.5 ) {
// Computing `1 + x` will throw away digits, so split up calculation...
if ( n > MAX_FACTORIAL-2 ) {
// Given a ratio of two very large numbers, we need to split the calculation up into two blocks:
t1 = x * fallingFactorial( x-1.0, MAX_FACTORIAL-2 );
t2 = fallingFactorial( x-MAX_FACTORIAL+1.0, n-MAX_FACTORIAL+1 );
if ( FLOAT64_MAX/abs(t1) < abs(t2) ) {
return PINF;
}
return t1 * t2;
}
return x * fallingFactorial( x-1.0, n-1.0 );
}
if ( x <= n-1.0 ) {
// `x+1-n` will be negative and computing the ratio of two gammas will not work, so split the product up into three parts:
xp1 = x + 1.0;
n2 = abs( floor( xp1 ) );
if ( n2 === xp1 ) {
return 0.0;
}
result = gammaDeltaRatio( xp1, -n2 );
x -= n2;
result *= x;
n2 += 1.0;
if ( n2 < n ) {
result *= fallingFactorial( x-1.0, n-n2 );
}
return result;
}
// Simple case: just the ratio of two (positive argument) gamma functions. Note that we don't optimize this for small `n`, because `gammaDeltaRatio` is already optimized for that use case:
return gammaDeltaRatio( x+1.0, -n );
}
// EXPORTS //
module.exports = fallingFactorial;
|