time-to-botec

Benchmark sampling in different programming languages
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nearzero.js (3987B)

      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 * ## Notice
     20 *
     21 * The original C++ code and copyright notice are from the [Boost library]{@link http://www.boost.org/doc/libs/1_65_0/boost/math/special_functions/detail/polygamma.hpp}. The implementation follows the original but has been modified for JavaScript.
     22 *
     23 * ```text
     24 * (C) Copyright Nikhar Agrawal 2013.
     25 * (C) Copyright Christopher Kormanyos 2013.
     26 * (C) Copyright John Maddock 2014.
     27 * (C) Copyright Paul Bristow 2013.
     28 *
     29 * Use, modification and distribution are subject to the
     30 * Boost Software License, Version 1.0. (See accompanying file
     31 * LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
     32 * ```
     33 */
     34 
     35 'use strict';
     36 
     37 // MODULES //
     38 
     39 var logger = require( 'debug' );
     40 var factorial = require( './../../../../base/special/factorial' );
     41 var zeta = require( './../../../../base/special/riemann-zeta' );
     42 var abs = require( './../../../../base/special/abs' );
     43 var pow = require( './../../../../base/special/pow' );
     44 var NINF = require( '@stdlib/constants/float64/ninf' );
     45 var PINF = require( '@stdlib/constants/float64/pinf' );
     46 var EPS = require( '@stdlib/constants/float64/eps' );
     47 var MAX = require( '@stdlib/constants/float64/max' );
     48 
     49 
     50 // VARIABLES //
     51 
     52 var debug = logger( 'polygamma' );
     53 var MAX_SERIES_ITERATIONS = 1000000;
     54 
     55 
     56 // MAIN //
     57 
     58 /**
     59 * Evaluates the polygamma function near zero.
     60 *
     61 * ## Notes
     62 *
     63 * -   If we take this [expansion][1] for `polygamma` and substitute in this [expression][2] for `polygamma(n, 1)`, we get an alternating series for polygamma when `x` is small in terms of zeta functions of integer arguments (which are easy to evaluate, at least when the integer is even).
     64 *
     65 * [1]: http://functions.wolfram.com/06.15.06.0003.02
     66 * [2]: http://functions.wolfram.com/06.15.03.0009.01
     67 *
     68 *
     69 * @private
     70 * @param {PositiveInteger} n - derivative to evaluate
     71 * @param {number} x - input value
     72 * @returns {number} (n+1)'th derivative
     73 */
     74 function nearzero( n, x ) {
     75 	var factorialPart;
     76 	var prefix;
     77 	var scale;
     78 	var term;
     79 	var sum;
     80 	var AX;
     81 	var k;
     82 
     83 	// In order to avoid spurious overflow, save the `n!` term for later, and rescale at the end:
     84 	scale = factorial( n );
     85 
     86 	// "factorialPart" contains everything except the zeta function evaluations in each term:
     87 	factorialPart = 1;
     88 
     89 	// "prefix" is what we'll be adding the accumulated sum to, it will be `n! / z^(n+1)`, but since we're scaling by `n!` it is just `1 / z^(n+1)` for now:
     90 	prefix = pow( x, n+1 );
     91 	if ( prefix === 0.0 ) {
     92 		return PINF;
     93 	}
     94 	prefix = 1.0 / prefix;
     95 
     96 	// First term in the series is necessarily `< zeta(2) < 2`, so ignore the sum if it will have no effect on the result:
     97 	if ( prefix > 2.0/EPS ) {
     98 		if ( n & 1 ) {
     99 			return ( AX/prefix < scale ) ? PINF : prefix * scale;
    100 		}
    101 		return ( AX/prefix < scale ) ? NINF : -prefix * scale;
    102 	}
    103 	sum = prefix;
    104 	for ( k = 0; ; ) {
    105 		// Get the k'th term:
    106 		term = factorialPart * zeta( k+n+1 );
    107 		sum += term;
    108 
    109 		// Termination condition:
    110 		if ( abs( term ) < abs(sum * EPS ) ) {
    111 			break;
    112 		}
    113 		// Move on `k` and `factorialPart`:
    114 		k += 1;
    115 		factorialPart *= (-x * (n+k)) / k;
    116 
    117 		// Last chance exit:
    118 		if ( k > MAX_SERIES_ITERATIONS ) {
    119 			debug( 'Series did not converge, best value is %d.', sum );
    120 			return NaN;
    121 		}
    122 	}
    123 	// We need to multiply by the scale, at each stage checking for overflow:
    124 	if ( MAX/scale < sum ) {
    125 		return PINF;
    126 	}
    127 	sum *= scale;
    128 	return ( n & 1 ) ? sum : -sum;
    129 }
    130 
    131 
    132 // EXPORTS //
    133 
    134 module.exports = nearzero;