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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 factorial = require( '@stdlib/math/base/special/factorial' );
var gammaln = require( '@stdlib/math/base/special/gammaln' );
var trunc = require( '@stdlib/math/base/special/trunc' );
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 atinfinityplus = require( './atinfinityplus.js' );
// VARIABLES //
var debug = logger( 'polygamma' );
var MAX_SERIES_ITERATIONS = 1000000;
var DIGITS_BASE10 = 19;
// MAIN //
/**
* Evaluates the polygamma function.
*
* @private
* @param {PositiveInteger} n - derivative to evaluate
* @param {number} x - input
* @returns {number} (n+1)'th derivative
* @see {@link http://functions.wolfram.com/GammaBetaErf/PolyGamma2/16/01/01/0017/}
*/
function attransitionplus( n, x ) {
var minusMminus1;
var lnterm;
var zpows;
var iter;
var sum0;
var d4d;
var N;
var m;
var k;
var z;
// Use N = (0.4 * digits) + (4 * n) for target value for x:
d4d = 0.4 * DIGITS_BASE10;
N = d4d + ( 4*n );
m = n;
iter = N - trunc( x );
if ( iter > MAX_SERIES_ITERATIONS ) {
debug( 'Exceeded maximum series evaluations when evaluated at n = %d and x = %d', n, x );
return NaN;
}
minusMminus1 = -m - 1;
z = x;
sum0 = 0.0;
zpows = 0.0;
// Forward recursion to larger `x`, need to check for overflow first though:
if ( ln( z+iter ) * minusMminus1 > -MAX_LN ) {
for ( k = 1; k <= iter; k++ ) {
zpows = pow( z, minusMminus1 );
sum0 += zpows;
z += 1;
}
sum0 *= factorial( n );
} else {
for ( k = 1; k <= iter; k++ ) {
lnterm = ( ln( z ) * minusMminus1 ) + gammaln( n+1 );
sum0 += exp( lnterm );
z += 1;
}
}
if ( ( n-1 ) & 1 ) {
sum0 = -sum0;
}
return sum0 + atinfinityplus( n, z );
}
// EXPORTS //
module.exports = attransitionplus;
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