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* @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_65_0/boost/math/special_functions/detail/polygamma.hpp}. The implementation follows the original but has been modified for JavaScript.
*
* ```text
* (C) Copyright Nikhar Agrawal 2013.
* (C) Copyright Christopher Kormanyos 2013.
* (C) Copyright John Maddock 2014.
* (C) Copyright Paul Bristow 2013.
*
* 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 logger = require( 'debug' );
var bernoulli = require( '@stdlib/math/base/special/bernoulli' );
var factorial = require( '@stdlib/math/base/special/factorial' );
var gammaln = require( '@stdlib/math/base/special/gammaln' );
var abs = require( '@stdlib/math/base/special/abs' );
var exp = require( '@stdlib/math/base/special/exp' );
var pow = require( '@stdlib/math/base/special/pow' );
var ln = require( '@stdlib/math/base/special/ln' );
var MAX_LN = require( '@stdlib/constants/float64/max-ln' );
var LN_TWO = require( '@stdlib/constants/float64/ln-two' );
var EPS = require( '@stdlib/constants/float64/eps' );
// VARIABLES //
var debug = logger( 'polygamma' );
var MAX_SERIES_ITERATIONS = 1000000;
var MAX_FACTORIAL = 172;
// MAIN //
/**
* Evaluates the polygamma function for large values of `x` such as for `x > 400`.
*
* @private
* @param {PositiveInteger} n - derivative to evaluate
* @param {number} x - input
* @returns {number} (n+1)'th derivative
* @see {@link http://functions.wolfram.com/GammaBetaErf/PolyGamma2/06/02/0001/}
*/
function atinfinityplus( n, x ) {
var partTerm; // Value of current term excluding the Bernoulli number part
var xsquared;
var term; // Value of current term to be added to sum
var sum; // Current value of accumulated sum
var nlx;
var k2;
var k;
if ( n+x === x ) {
// If `x` is very large, just concentrate on the first part of the expression and use logs:
if ( n === 1 ) {
return 1.0 / x;
}
nlx = n * ln( x );
if ( nlx < MAX_LN && n < MAX_FACTORIAL ) {
return ( (n & 1) ? 1.0 : -1.0 ) * factorial( n-1 ) * pow( x, -n );
}
return ( (n & 1) ? 1.0 : -1.0 ) * exp( gammaln( n ) - ( n*ln(x) ) );
}
xsquared = x * x;
// 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:
if ( n > MAX_FACTORIAL && n*n > MAX_LN ) {
partTerm = 0.0;
} else {
partTerm = factorial( n-1 ) * pow( x, -n-1 );
}
if ( partTerm === 0.0 ) {
// Either `n` is very large, or the power term underflows. Set the initial values of `partTerm`, `term`, and `sum` via logs:
partTerm = gammaln(n) - ( (n+1) * ln(x) );
sum = exp( partTerm + ln( n + (2.0*x) ) - LN_TWO );
partTerm += ln( n*(n+1) ) - LN_TWO - ln(x);
partTerm = exp( partTerm );
} else {
sum = partTerm * ( n+(2.0*x) ) / 2.0;
partTerm *= ( n*(n+1) ) / 2.0;
partTerm /= x;
}
// If the leading term is 0, so is the result:
if ( sum === 0.0 ) {
return sum;
}
for ( k = 1; ; ) {
term = partTerm * bernoulli( k*2 );
sum += term;
// Normal termination condition:
if ( abs( term/sum ) < EPS ) {
break;
}
// Increment our counter, and move `partTerm` on to the next value:
k += 1;
k2 = 2 * k;
partTerm *= ( n+k2-2 ) * ( n-1+k2 );
partTerm /= ( k2-1 ) * k2;
partTerm /= xsquared;
if ( k > MAX_SERIES_ITERATIONS ) {
debug( 'Series did not converge, closest value was: %d.', sum );
return NaN;
}
}
if ( ( n-1 ) & 1 ) {
sum = -sum;
}
return sum;
}
// EXPORTS //
module.exports = atinfinityplus;
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