main.js (3506B)
1 /** 2 * @license Apache-2.0 3 * 4 * Copyright (c) 2018 The Stdlib Authors. 5 * 6 * Licensed under the Apache License, Version 2.0 (the "License"); 7 * you may not use this file except in compliance with the License. 8 * You may obtain a copy of the License at 9 * 10 * http://www.apache.org/licenses/LICENSE-2.0 11 * 12 * Unless required by applicable law or agreed to in writing, software 13 * distributed under the License is distributed on an "AS IS" BASIS, 14 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 15 * See the License for the specific language governing permissions and 16 * limitations under the License. 17 */ 18 19 'use strict'; 20 21 // MODULES // 22 23 var factory = require( './factory.js' ); 24 25 26 // MAIN // 27 28 /** 29 * Generates a standard normally distributed random number. 30 * 31 * ## Method 32 * 33 * - Given two independent uniformly distributed random variables \\( U_1 \\) and \\( U_2 \\) in the interval \\( [0,1) \\), let 34 * 35 * ``` tex 36 * \begin{align*} 37 * Z_1 &= R \cos(\theta) = \sqrt{-2 \ln(U_1)} \cos(2\pi U_2) \\ 38 * Z_2 &= R \sin(\theta) = \sqrt{-2 \ln(U_1)} \sin(2\pi U_2) 39 * \end{align*} 40 * ``` 41 * 42 * where \\( Z_1 \\) and \\( Z_2 \\) are independent random variables with a standard normal distribution. 43 * 44 * - As two uniform random variates are mapped to two standard normal random variates, one of the random variates is cached and returned upon the following invocation. 45 * 46 * 47 * ## Notes 48 * 49 * - The minimum and maximum pseudorandom numbers which can be generated are dependent on the number of bits an underlying uniform pseudorandom number generator (PRNG) uses. For instance, if a PRNG uses \\( 32 \\) bits, the smallest non-zero number that can be generated is \\( 2^{-32}). When \\( U_1 \\) equals this value and \\( U_2 \\) equals \\( 0 \\), 50 * 51 * ``` tex 52 * r = \sqrt{-2\ln(2^{-32})} \cos(2\pi) \approx 6.66 53 * ``` 54 * 55 * which means that the algorithm cannot produce random variates more than \\( 6.66 \\) standard deviations from the mean. 56 * 57 * <!-- <note> --> 58 * 59 * This corresponds to a \\( 2.74 \times 10^{-11} \\) loss due to tail truncation. 60 * 61 * <!-- </note> --> 62 * 63 * 64 * ## References 65 * 66 * - Box, G. E. P., and Mervin E. Muller. 1958. "A Note on the Generation of Random Normal Deviates." _The Annals of Mathematical Statistics_ 29 (2). The Institute of Mathematical Statistics: 610–11. doi:[10.1214/aoms/1177706645](http://dx.doi.org/10.1214/aoms/1177706645). 67 * - Bell, James R. 1968. "Algorithm 334: Normal Random Deviates." _Communications of the ACM_ 11 (7). New York, NY, USA: ACM: 498. doi:[10.1145/363397.363547](http://dx.doi.org/10.1145/363397.363547). 68 * - Knop, R. 1969. "Remark on Algorithm 334 \[G5]: Normal Random Deviates." _Communications of the ACM_ 12 (5). New York, NY, USA: ACM: 281. doi:[10.1145/362946.362996](http://dx.doi.org/10.1145/362946.362996). 69 * - Marsaglia, G., and T. A. Bray. 1964. "A Convenient Method for Generating Normal Variables." _SIAM Review_ 6 (3). Society for Industrial; Applied Mathematics: 260–64. doi:[10.1137/1006063](http://dx.doi.org/10.1137/1006063). 70 * - Thomas, David B., Wayne Luk, Philip H.W. Leong, and John D. Villasenor. 2007. "Gaussian Random Number Generators." _ACM Computing Surveys_ 39 (4). New York, NY, USA: ACM. doi:[10.1145/1287620.1287622](http://dx.doi.org/10.1145/1287620.1287622). 71 * 72 * 73 * @name randn 74 * @type {PRNG} 75 * @returns {number} pseudorandom number 76 * 77 * @example 78 * var r = randn(); 79 * // returns <number> 80 */ 81 var randn = factory(); 82 83 84 // EXPORTS // 85 86 module.exports = randn;