time-to-botec

Benchmark sampling in different programming languages
Log | Files | Refs | README

atinfinityplus.js (4646B)

      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 bernoulli = require( './../../../../base/special/bernoulli' );
     41 var factorial = require( './../../../../base/special/factorial' );
     42 var gammaln = require( './../../../../base/special/gammaln' );
     43 var abs = require( './../../../../base/special/abs' );
     44 var exp = require( './../../../../base/special/exp' );
     45 var pow = require( './../../../../base/special/pow' );
     46 var ln = require( './../../../../base/special/ln' );
     47 var MAX_LN = require( '@stdlib/constants/float64/max-ln' );
     48 var LN_TWO = require( '@stdlib/constants/float64/ln-two' );
     49 var EPS = require( '@stdlib/constants/float64/eps' );
     50 
     51 
     52 // VARIABLES //
     53 
     54 var debug = logger( 'polygamma' );
     55 var MAX_SERIES_ITERATIONS = 1000000;
     56 var MAX_FACTORIAL = 172;
     57 
     58 
     59 // MAIN //
     60 
     61 /**
     62 * Evaluates the polygamma function for large values of `x` such as for `x > 400`.
     63 *
     64 * @private
     65 * @param {PositiveInteger} n - derivative to evaluate
     66 * @param {number} x - input
     67 * @returns {number} (n+1)'th derivative
     68 * @see {@link http://functions.wolfram.com/GammaBetaErf/PolyGamma2/06/02/0001/}
     69 */
     70 function atinfinityplus( n, x ) {
     71 	var partTerm; // Value of current term excluding the Bernoulli number part
     72 	var xsquared;
     73 	var term; // Value of current term to be added to sum
     74 	var sum; // Current value of accumulated sum
     75 	var nlx;
     76 	var k2;
     77 	var k;
     78 
     79 	if ( n+x === x ) {
     80 		// If `x` is very large, just concentrate on the first part of the expression and use logs:
     81 		if ( n === 1 ) {
     82 			return 1.0 / x;
     83 		}
     84 		nlx = n * ln( x );
     85 		if ( nlx < MAX_LN && n < MAX_FACTORIAL ) {
     86 			return ( (n & 1) ? 1.0 : -1.0 ) * factorial( n-1 ) * pow( x, -n );
     87 		}
     88 		return ( (n & 1) ? 1.0 : -1.0 ) * exp( gammaln( n ) - ( n*ln(x) ) );
     89 	}
     90 	xsquared = x * x;
     91 
     92 	// Start by setting `partTerm` to `(n-1)! / x^(n+1)`, which is common to both the first term of the series (with k = 1) and to the leading part. We can then get to the leading term by: `partTerm * (n + 2 * x) / 2` and to the first term in the series (excluding the Bernoulli number) by: `partTerm n * (n + 1) / (2x)`. If either the factorial would over- or the power term underflow, set `partTerm` to 0 and then we know that we have to use logs for the initial terms:
     93 	if ( n > MAX_FACTORIAL && n*n > MAX_LN ) {
     94 		partTerm = 0.0;
     95 	} else {
     96 		partTerm = factorial( n-1 ) * pow( x, -n-1 );
     97 	}
     98 	if ( partTerm === 0.0 ) {
     99 		// Either `n` is very large, or the power term underflows. Set the initial values of `partTerm`, `term`, and `sum` via logs:
    100 		partTerm = gammaln(n) - ( (n+1) * ln(x) );
    101 		sum = exp( partTerm + ln( n + (2.0*x) ) - LN_TWO );
    102 		partTerm += ln( n*(n+1) ) - LN_TWO - ln(x);
    103 		partTerm = exp( partTerm );
    104 	} else {
    105 		sum = partTerm * ( n+(2.0*x) ) / 2.0;
    106 		partTerm *= ( n*(n+1) ) / 2.0;
    107 		partTerm /= x;
    108 	}
    109 	// If the leading term is 0, so is the result:
    110 	if ( sum === 0.0 ) {
    111 		return sum;
    112 	}
    113 	for ( k = 1; ; ) {
    114 		term = partTerm * bernoulli( k*2 );
    115 		sum += term;
    116 
    117 		// Normal termination condition:
    118 		if ( abs( term/sum ) < EPS ) {
    119 			break;
    120 		}
    121 
    122 		// Increment our counter, and move `partTerm` on to the next value:
    123 		k += 1;
    124 		k2 = 2 * k;
    125 		partTerm *= ( n+k2-2 ) * ( n-1+k2 );
    126 		partTerm /= ( k2-1 ) * k2;
    127 		partTerm /= xsquared;
    128 		if ( k > MAX_SERIES_ITERATIONS ) {
    129 			debug( 'Series did not converge, closest value was: %d.', sum );
    130 			return NaN;
    131 		}
    132 	}
    133 	if ( ( n-1 ) & 1 ) {
    134 		sum = -sum;
    135 	}
    136 	return sum;
    137 }
    138 
    139 
    140 // EXPORTS //
    141 
    142 module.exports = atinfinityplus;