forked from tsai/mathnet-numerics
Christoph Ruegg
14 years ago
16 changed files with 1368 additions and 72 deletions
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// <copyright file="Evaluate.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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// http://mathnetnumerics.codeplex.com
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//
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// Copyright (c) 2009-2012 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
|
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// files (the "Software"), to deal in the Software without
|
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// restriction, including without limitation the rights to use,
|
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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// <contribution>
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// CERN - European Laboratory for Particle Physics
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// http://www.docjar.com/html/api/cern/jet/math/Bessel.java.html
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// Copyright 1999 CERN - European Laboratory for Particle Physics.
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// Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
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// is hereby granted without fee, provided that the above copyright notice appear in all copies and
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// that both that copyright notice and this permission notice appear in supporting documentation.
|
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// CERN makes no representations about the suitability of this software for any purpose.
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// It is provided "as is" without expressed or implied warranty.
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// TOMS757 - Uncommon Special Functions (Fortran77) by Allan McLeod
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// http://people.sc.fsu.edu/~jburkardt/f77_src/toms757/toms757.html
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// Wei Wu
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// Cephes Math Library, Stephen L. Moshier
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// ALGLIB 2.0.1, Sergey Bochkanov
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// </contribution>
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// ReSharper disable CheckNamespace
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namespace MathNet.Numerics |
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// ReSharper restore CheckNamespace
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{ |
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using System; |
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/// <summary>
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/// Evaluation functions, useful for function approximation.
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/// </summary>
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public static class Evaluate |
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{ |
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/// <summary>
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/// Evaluate polynomials.
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/// </summary>
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/// <param name="coefficients">The coefficients of the polynomial.</param>
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/// <param name="z">The location where to evaluate the polynomial at.</param>
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/// <returns>the evaluation of the polynomial.</returns>
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public static double Polynomial(double[] coefficients, double z) |
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{ |
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int count = coefficients.Length; |
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double sum = coefficients[count - 1]; |
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for (int i = count - 2; i >= 0; --i) |
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{ |
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sum *= z; |
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sum += coefficients[i]; |
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} |
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return sum; |
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} |
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/// <summary>
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/// Numerically stable series summation
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/// </summary>
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/// <param name="nextSummand">provides the summands sequentially</param>
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/// <returns>Sum</returns>
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internal static double Series(Func<double> nextSummand) |
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{ |
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double compensation = 0.0; |
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double current; |
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const double factor = 1 << 16; |
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double sum = nextSummand(); |
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do |
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{ |
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// Kahan Summation
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// NOTE (ruegg): do NOT optimize. Now, how to tell that the compiler?
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current = nextSummand(); |
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double y = current - compensation; |
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double t = sum + y; |
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compensation = t - sum; |
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compensation -= y; |
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sum = t; |
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} |
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while (Math.Abs(sum) < Math.Abs(factor * current)); |
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return sum; |
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} |
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/// <summary> Evaluates the series of Chebyshev polynomials Ti at argument x/2.
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/// The series is given by
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/// <pre>
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/// N-1
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/// - '
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/// y = > coef[i] T (x/2)
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/// - i
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/// i=0
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/// </pre>
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/// Coefficients are stored in reverse order, i.e. the zero
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/// order term is last in the array. Note N is the number of
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/// coefficients, not the order.
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/// <p/>
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/// If coefficients are for the interval a to b, x must
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/// have been transformed to x -> 2(2x - b - a)/(b-a) before
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/// entering the routine. This maps x from (a, b) to (-1, 1),
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/// over which the Chebyshev polynomials are defined.
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/// <p/>
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/// If the coefficients are for the inverted interval, in
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/// which (a, b) is mapped to (1/b, 1/a), the transformation
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/// required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity,
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/// this becomes x -> 4a/x - 1.
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/// <p/>
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/// SPEED:
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/// <p/>
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/// Taking advantage of the recurrence properties of the
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/// Chebyshev polynomials, the routine requires one more
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/// addition per loop than evaluating a nested polynomial of
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/// the same degree.
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/// </summary>
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/// <param name="coefficients">The coefficients of the polynomial.
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/// </param>
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/// <param name="x">Argument to the polynomial.
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/// </param>
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/// <param name="N">The number of coefficients.
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/// </param>
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/// <remarks>
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/// Reference: https://bpm2.svn.codeplex.com/svn/Common.Numeric/Arithmetic.cs
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/// <p/>
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/// Marked as Deprecated in
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/// http://people.apache.org/~isabel/mahout_site/mahout-matrix/apidocs/org/apache/mahout/jet/math/Arithmetic.html
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/// </remarks>
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internal static double ChebyshevA(double[] coefficients, double x) |
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{ |
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// TODO: Unify, normalize, then make public
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double b2; |
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int p = 0; |
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double b0 = coefficients[p++]; |
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double b1 = 0.0; |
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int i = coefficients.Length - 1; |
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do |
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{ |
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b2 = b1; |
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b1 = b0; |
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b0 = x * b1 - b2 + coefficients[p++]; |
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} |
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while (--i > 0); |
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return (0.5 * (b0 - b2)); |
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} |
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/// <summary>
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/// Summation of Chebyshev polynomials, using the Clenshaw method with Reinsch modification.
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/// </summary>
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/// <param name="n">The no. of terms in the sequence.</param>
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/// <param name="coefficients">The coefficients of the Chebyshev series, length n+1.</param>
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/// <param name="x">The value at which the series is to be evaluated.</param>
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/// <remarks>
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/// ORIGINAL AUTHOR:
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/// Dr. Allan J. MacLeod; Dept. of Mathematics and Statistics, University of Paisley; High St., PAISLEY, SCOTLAND
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/// REFERENCES:
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/// "An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series"
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/// J. Oliver, J.I.M.A., vol. 20, 1977, pp379-391
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/// </remarks>
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internal static double ChebyshevSum(int n, double[] coefficients, double x) |
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{ |
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// TODO: Unify, normalize, then make public
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// If |x| < 0.6 use the standard Clenshaw method
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if (Math.Abs(x) < 0.6) |
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{ |
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double u0 = 0.0; |
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double u1 = 0.0; |
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double u2 = 0.0; |
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double xx = x + x; |
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for (int i = n; i >= 0; i--) |
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{ |
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u2 = u1; |
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u1 = u0; |
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u0 = xx * u1 + coefficients[i] - u2; |
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} |
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return (u0 - u2) / 2.0; |
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} |
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// If ABS ( T ) > = 0.6 use the Reinsch modification
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// T > = 0.6 code
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if (x > 0.0) |
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{ |
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double u1 = 0.0; |
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double d1 = 0.0; |
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double d2 = 0.0; |
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double xx = (x - 0.5) - 0.5; |
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xx = xx + xx; |
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for (int i = n; i >= 0; i--) |
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{ |
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d2 = d1; |
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double u2 = u1; |
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d1 = xx * u2 + coefficients[i] + d2; |
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u1 = d1 + u2; |
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} |
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return (d1 + d2) / 2.0; |
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} |
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else |
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{ |
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// T < = -0.6 code
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double u1 = 0.0; |
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double d1 = 0.0; |
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double d2 = 0.0; |
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double xx = (x + 0.5) + 0.5; |
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xx = xx + xx; |
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for (int i = n; i >= 0; i--) |
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{ |
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d2 = d1; |
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double u2 = u1; |
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d1 = xx * u2 + coefficients[i] - d2; |
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u1 = d1 - u2; |
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} |
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return (d1 - d2) / 2.0; |
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} |
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} |
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} |
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} |
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@ -0,0 +1,287 @@ |
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// <copyright file="ModifiedBessel.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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// http://mathnetnumerics.codeplex.com
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//
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// Copyright (c) 2009-2012 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
|
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// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
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// restriction, including without limitation the rights to use,
|
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
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// copies of the Software, and to permit persons to whom the
|
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// Software is furnished to do so, subject to the following
|
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// conditions:
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//
|
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// The above copyright notice and this permission notice shall be
|
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// included in all copies or substantial portions of the Software.
|
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//
|
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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// <contribution>
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// CERN - European Laboratory for Particle Physics
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// http://www.docjar.com/html/api/cern/jet/math/Bessel.java.html
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// Copyright 1999 CERN - European Laboratory for Particle Physics.
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// Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
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// is hereby granted without fee, provided that the above copyright notice appear in all copies and
|
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// that both that copyright notice and this permission notice appear in supporting documentation.
|
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// CERN makes no representations about the suitability of this software for any purpose.
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// It is provided "as is" without expressed or implied warranty.
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// TOMS757 - Uncommon Special Functions (Fortran77) by Allan McLeod
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// http://people.sc.fsu.edu/~jburkardt/f77_src/toms757/toms757.html
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// Wei Wu
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// Cephes Math Library, Stephen L. Moshier
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// ALGLIB 2.0.1, Sergey Bochkanov
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// </contribution>
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// ReSharper disable CheckNamespace
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namespace MathNet.Numerics |
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// ReSharper restore CheckNamespace
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{ |
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using System; |
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/// <summary>
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/// This partial implementation of the SpecialFunctions class contains all methods related to the modified bessel function.
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/// </summary>
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public static partial class SpecialFunctions |
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{ |
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/// <summary>
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/// **************************************
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/// COEFFICIENTS FOR METHODS bessi0 *
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/// **************************************
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/// </summary>
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/// <summary> Chebyshev coefficients for exp(-x) I0(x)
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/// in the interval [0, 8].
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///
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/// lim(x->0){ exp(-x) I0(x) } = 1.
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/// </summary>
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private static readonly double[] BesselI0A = new[] { -4.41534164647933937950e-18, 3.33079451882223809783e-17, -2.43127984654795469359e-16, 1.71539128555513303061e-15, -1.16853328779934516808e-14, 7.67618549860493561688e-14, -4.85644678311192946090e-13, 2.95505266312963983461e-12, -1.72682629144155570723e-11, 9.67580903537323691224e-11, -5.18979560163526290666e-10, 2.65982372468238665035e-9, -1.30002500998624804212e-8, 6.04699502254191894932e-8, -2.67079385394061173391e-7, 1.11738753912010371815e-6, -4.41673835845875056359e-6, 1.64484480707288970893e-5, -5.75419501008210370398e-5, 1.88502885095841655729e-4, -5.76375574538582365885e-4, 1.63947561694133579842e-3, -4.32430999505057594430e-3, 1.05464603945949983183e-2, -2.37374148058994688156e-2, 4.93052842396707084878e-2, -9.49010970480476444210e-2, 1.71620901522208775349e-1, -3.04682672343198398683e-1, 6.76795274409476084995e-1 }; |
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/// <summary> Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
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/// in the inverted interval [8, infinity].
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///
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/// lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
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/// </summary>
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private static readonly double[] BesselI0B = new[] { -7.23318048787475395456e-18, -4.83050448594418207126e-18, 4.46562142029675999901e-17, 3.46122286769746109310e-17, -2.82762398051658348494e-16, -3.42548561967721913462e-16, 1.77256013305652638360e-15, 3.81168066935262242075e-15, -9.55484669882830764870e-15, -4.15056934728722208663e-14, 1.54008621752140982691e-14, 3.85277838274214270114e-13, 7.18012445138366623367e-13, -1.79417853150680611778e-12, -1.32158118404477131188e-11, -3.14991652796324136454e-11, 1.18891471078464383424e-11, 4.94060238822496958910e-10, 3.39623202570838634515e-9, 2.26666899049817806459e-8, 2.04891858946906374183e-7, 2.89137052083475648297e-6, 6.88975834691682398426e-5, 3.36911647825569408990e-3, 8.04490411014108831608e-1 }; |
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/// <summary>
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/// **************************************
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/// COEFFICIENTS FOR METHODS bessi1 *
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/// **************************************
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/// </summary>
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/// <summary> Chebyshev coefficients for exp(-x) I1(x) / x
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/// in the interval [0, 8].
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///
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/// lim(x->0){ exp(-x) I1(x) / x } = 1/2.
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/// </summary>
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private static readonly double[] BesselI1A = new[] { 2.77791411276104639959e-18, -2.11142121435816608115e-17, 1.55363195773620046921e-16, -1.10559694773538630805e-15, 7.60068429473540693410e-15, -5.04218550472791168711e-14, 3.22379336594557470981e-13, -1.98397439776494371520e-12, 1.17361862988909016308e-11, -6.66348972350202774223e-11, 3.62559028155211703701e-10, -1.88724975172282928790e-9, 9.38153738649577178388e-9, -4.44505912879632808065e-8, 2.00329475355213526229e-7, -8.56872026469545474066e-7, 3.47025130813767847674e-6, -1.32731636560394358279e-5, 4.78156510755005422638e-5, -1.61760815825896745588e-4, 5.12285956168575772895e-4, -1.51357245063125314899e-3, 4.15642294431288815669e-3, -1.05640848946261981558e-2, 2.47264490306265168283e-2, -5.29459812080949914269e-2, 1.02643658689847095384e-1, -1.76416518357834055153e-1, 2.52587186443633654823e-1 }; |
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/// <summary> Chebyshev coefficients for exp(-x) sqrt(x) I1(x)
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/// in the inverted interval [8, infinity].
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///
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/// lim(x->inf){ exp(-x) sqrt(x) I1(x) } = 1/sqrt(2pi).
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/// </summary>
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private static readonly double[] BesselI1B = new[] { 7.51729631084210481353e-18, 4.41434832307170791151e-18, -4.65030536848935832153e-17, -3.20952592199342395980e-17, 2.96262899764595013876e-16, 3.30820231092092828324e-16, -1.88035477551078244854e-15, -3.81440307243700780478e-15, 1.04202769841288027642e-14, 4.27244001671195135429e-14, -2.10154184277266431302e-14, -4.08355111109219731823e-13, -7.19855177624590851209e-13, 2.03562854414708950722e-12, 1.41258074366137813316e-11, 3.25260358301548823856e-11, -1.89749581235054123450e-11, -5.58974346219658380687e-10, -3.83538038596423702205e-9, -2.63146884688951950684e-8, -2.51223623787020892529e-7, -3.88256480887769039346e-6, -1.10588938762623716291e-4, -9.76109749136146840777e-3, 7.78576235018280120474e-1 }; |
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/// <summary>
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/// **************************************
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/// COEFFICIENTS FOR METHODS bessk0, bessk0e *
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/// **************************************
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/// </summary>
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/// <summary> Chebyshev coefficients for K0(x) + log(x/2) I0(x)
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/// in the interval [0, 2]. The odd order coefficients are all
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/// zero; only the even order coefficients are listed.
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///
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/// lim(x->0){ K0(x) + log(x/2) I0(x) } = -EUL.
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/// </summary>
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private static readonly double[] BesselK0A = new[] { 1.37446543561352307156e-16, 4.25981614279661018399e-14, 1.03496952576338420167e-11, 1.90451637722020886025e-9, 2.53479107902614945675e-7, 2.28621210311945178607e-5, 1.26461541144692592338e-3, 3.59799365153615016266e-2, 3.44289899924628486886e-1, -5.35327393233902768720e-1 }; |
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/// <summary> Chebyshev coefficients for exp(x) sqrt(x) K0(x)
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/// in the inverted interval [2, infinity].
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///
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/// lim(x->inf){ exp(x) sqrt(x) K0(x) } = sqrt(pi/2).
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/// </summary>
|
|||
private static readonly double[] BesselK0B = new[] { 5.30043377268626276149e-18, -1.64758043015242134646e-17, 5.21039150503902756861e-17, -1.67823109680541210385e-16, 5.51205597852431940784e-16, -1.84859337734377901440e-15, 6.34007647740507060557e-15, -2.22751332699166985548e-14, 8.03289077536357521100e-14, -2.98009692317273043925e-13, 1.14034058820847496303e-12, -4.51459788337394416547e-12, 1.85594911495471785253e-11, -7.95748924447710747776e-11, 3.57739728140030116597e-10, -1.69753450938905987466e-9, 8.57403401741422608519e-9, -4.66048989768794782956e-8, 2.76681363944501510342e-7, -1.83175552271911948767e-6, 1.39498137188764993662e-5, -1.28495495816278026384e-4, 1.56988388573005337491e-3, -3.14481013119645005427e-2, 2.44030308206595545468e0 }; |
|||
|
|||
/// <summary>
|
|||
/// **************************************
|
|||
/// COEFFICIENTS FOR METHODS bessk1, bessk1e *
|
|||
/// **************************************
|
|||
/// </summary>
|
|||
/// <summary> Chebyshev coefficients for x(K1(x) - log(x/2) I1(x))
|
|||
/// in the interval [0, 2].
|
|||
///
|
|||
/// lim(x->0){ x(K1(x) - log(x/2) I1(x)) } = 1.
|
|||
/// </summary>
|
|||
private static readonly double[] BesselK1A = new[] { -7.02386347938628759343e-18, -2.42744985051936593393e-15, -6.66690169419932900609e-13, -1.41148839263352776110e-10, -2.21338763073472585583e-8, -2.43340614156596823496e-6, -1.73028895751305206302e-4, -6.97572385963986435018e-3, -1.22611180822657148235e-1, -3.53155960776544875667e-1, 1.52530022733894777053e0 }; |
|||
|
|||
/// <summary> Chebyshev coefficients for exp(x) sqrt(x) K1(x)
|
|||
/// in the interval [2, infinity].
|
|||
///
|
|||
/// lim(x->inf){ exp(x) sqrt(x) K1(x) } = sqrt(pi/2).
|
|||
/// </summary>
|
|||
private static readonly double[] BesselK1B = new[] { -5.75674448366501715755e-18, 1.79405087314755922667e-17, -5.68946255844285935196e-17, 1.83809354436663880070e-16, -6.05704724837331885336e-16, 2.03870316562433424052e-15, -7.01983709041831346144e-15, 2.47715442448130437068e-14, -8.97670518232499435011e-14, 3.34841966607842919884e-13, -1.28917396095102890680e-12, 5.13963967348173025100e-12, -2.12996783842756842877e-11, 9.21831518760500529508e-11, -4.19035475934189648750e-10, 2.01504975519703286596e-9, -1.03457624656780970260e-8, 5.74108412545004946722e-8, -3.50196060308781257119e-7, 2.40648494783721712015e-6, -1.93619797416608296024e-5, 1.95215518471351631108e-4, -2.85781685962277938680e-3, 1.03923736576817238437e-1, 2.72062619048444266945e0 }; |
|||
|
|||
/// <summary>Returns the modified Bessel function of first kind, order 0 of the argument.
|
|||
/// <p/>
|
|||
/// The function is defined as <tt>i0(x) = j0( ix )</tt>.
|
|||
/// <p/>
|
|||
/// The range is partitioned into the two intervals [0, 8] and
|
|||
/// (8, infinity). Chebyshev polynomial expansions are employed
|
|||
/// in each interval.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the bessel function of.
|
|||
/// </param>
|
|||
public static double BesselI0(double x) |
|||
{ |
|||
if (x < 0) |
|||
{ |
|||
x = -x; |
|||
} |
|||
if (x <= 8.0) |
|||
{ |
|||
double y = (x / 2.0) - 2.0; |
|||
return (Math.Exp(x) * Evaluate.ChebyshevA(BesselI0A, y)); |
|||
} |
|||
|
|||
double x1 = 32.0 / x - 2.0; |
|||
return (Math.Exp(x) * Evaluate.ChebyshevA(BesselI0B, x1) / Math.Sqrt(x)); |
|||
} |
|||
|
|||
/// <summary>Returns the modified Bessel function of first kind,
|
|||
/// order 1 of the argument.
|
|||
/// <p/>
|
|||
/// The function is defined as <tt>i1(x) = -i j1( ix )</tt>.
|
|||
/// <p/>
|
|||
/// The range is partitioned into the two intervals [0, 8] and
|
|||
/// (8, infinity). Chebyshev polynomial expansions are employed
|
|||
/// in each interval.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the bessel function of.
|
|||
/// </param>
|
|||
public static double BesselI1(double x) |
|||
{ |
|||
double z = Math.Abs(x); |
|||
if (z <= 8.0) |
|||
{ |
|||
double y = (z / 2.0) - 2.0; |
|||
z = Evaluate.ChebyshevA(BesselI1A, y) * z * Math.Exp(z); |
|||
} |
|||
else |
|||
{ |
|||
double x1 = 32.0 / z - 2.0; |
|||
z = Math.Exp(z) * Evaluate.ChebyshevA(BesselI1B, x1) / Math.Sqrt(z); |
|||
} |
|||
if (x < 0.0) |
|||
{ |
|||
z = -z; |
|||
} |
|||
return z; |
|||
} |
|||
|
|||
/// <summary> Returns the modified Bessel function of the second kind
|
|||
/// of order 0 of the argument.
|
|||
/// <p/>
|
|||
/// The range is partitioned into the two intervals [0, 8] and
|
|||
/// (8, infinity). Chebyshev polynomial expansions are employed
|
|||
/// in each interval.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the bessel function of.
|
|||
/// </param>
|
|||
public static double BesselK0(double x) |
|||
{ |
|||
if (x <= 0.0) |
|||
{ |
|||
throw new ArithmeticException(); |
|||
} |
|||
if (x <= 2.0) |
|||
{ |
|||
double y = x * x - 2.0; |
|||
return Evaluate.ChebyshevA(BesselK0A, y) - Math.Log(0.5 * x) * BesselI0(x); |
|||
} |
|||
|
|||
double z = 8.0 / x - 2.0; |
|||
return Math.Exp(-x) * Evaluate.ChebyshevA(BesselK0B, z) / Math.Sqrt(x); |
|||
} |
|||
|
|||
/// <summary>Returns the exponentially scaled modified Bessel function
|
|||
/// of the second kind of order 0 of the argument.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the bessel function of.
|
|||
/// </param>
|
|||
public static double BesselK0e(double x) |
|||
{ |
|||
if (x <= 0.0) |
|||
{ |
|||
throw new ArithmeticException(); |
|||
} |
|||
if (x <= 2.0) |
|||
{ |
|||
double y = x * x - 2.0; |
|||
return Evaluate.ChebyshevA(BesselK0A, y) - Math.Log(0.5 * x) * BesselI0(x) * Math.Exp(x); |
|||
} |
|||
|
|||
double x1 = 8.0 / x - 2.0; |
|||
return Evaluate.ChebyshevA(BesselK0B, x1) / Math.Sqrt(x); |
|||
} |
|||
|
|||
/// <summary> Returns the modified Bessel function of the second kind
|
|||
/// of order 1 of the argument.
|
|||
/// <p/>
|
|||
/// The range is partitioned into the two intervals [0, 2] and
|
|||
/// (2, infinity). Chebyshev polynomial expansions are employed
|
|||
/// in each interval.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the bessel function of.
|
|||
/// </param>
|
|||
public static double BesselK1(double x) |
|||
{ |
|||
double z = 0.5 * x; |
|||
if (z <= 0.0) |
|||
{ |
|||
throw new ArithmeticException(); |
|||
} |
|||
if (x <= 2.0) |
|||
{ |
|||
double y = x * x - 2.0; |
|||
return Math.Log(z) * BesselI1(x) + Evaluate.ChebyshevA(BesselK1A, y) / x; |
|||
} |
|||
|
|||
double x1 = 8.0 / x - 2.0; |
|||
return Math.Exp(-x) * Evaluate.ChebyshevA(BesselK1B, x1) / Math.Sqrt(x); |
|||
} |
|||
|
|||
/// <summary> Returns the exponentially scaled modified Bessel function
|
|||
/// of the second kind of order 1 of the argument.
|
|||
/// <p/>
|
|||
/// <tt>k1e(x) = exp(x) * k1(x)</tt>.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the bessel function of.
|
|||
/// </param>
|
|||
public static double BesselK1e(double x) |
|||
{ |
|||
if (x <= 0.0) |
|||
{ |
|||
throw new ArithmeticException(); |
|||
} |
|||
if (x <= 2.0) |
|||
{ |
|||
double y = x * x - 2.0; |
|||
return Math.Log(0.5 * x) * BesselI1(x) + Evaluate.ChebyshevA(BesselK1A, y) / x * Math.Exp(x); |
|||
} |
|||
|
|||
double x1 = 8.0 / x - 2.0; |
|||
return Evaluate.ChebyshevA(BesselK1B, x1) / Math.Sqrt(x); |
|||
} |
|||
} |
|||
} |
|||
@ -0,0 +1,556 @@ |
|||
// <copyright file="ModifiedStruve.cs" company="Math.NET">
|
|||
// Math.NET Numerics, part of the Math.NET Project
|
|||
// http://numerics.mathdotnet.com
|
|||
// http://github.com/mathnet/mathnet-numerics
|
|||
// http://mathnetnumerics.codeplex.com
|
|||
//
|
|||
// Copyright (c) 2009-2012 Math.NET
|
|||
//
|
|||
// Permission is hereby granted, free of charge, to any person
|
|||
// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
|||
// restriction, including without limitation the rights to use,
|
|||
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
|||
// copies of the Software, and to permit persons to whom the
|
|||
// Software is furnished to do so, subject to the following
|
|||
// conditions:
|
|||
//
|
|||
// The above copyright notice and this permission notice shall be
|
|||
// included in all copies or substantial portions of the Software.
|
|||
//
|
|||
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
|||
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
|||
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
|||
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
|||
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
|||
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
|||
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
|||
// OTHER DEALINGS IN THE SOFTWARE.
|
|||
// </copyright>
|
|||
|
|||
// <contribution>
|
|||
// CERN - European Laboratory for Particle Physics
|
|||
// http://www.docjar.com/html/api/cern/jet/math/Bessel.java.html
|
|||
// Copyright 1999 CERN - European Laboratory for Particle Physics.
|
|||
// Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
|
|||
// is hereby granted without fee, provided that the above copyright notice appear in all copies and
|
|||
// that both that copyright notice and this permission notice appear in supporting documentation.
|
|||
// CERN makes no representations about the suitability of this software for any purpose.
|
|||
// It is provided "as is" without expressed or implied warranty.
|
|||
// TOMS757 - Uncommon Special Functions (Fortran77) by Allan McLeod
|
|||
// http://people.sc.fsu.edu/~jburkardt/f77_src/toms757/toms757.html
|
|||
// Wei Wu
|
|||
// Cephes Math Library, Stephen L. Moshier
|
|||
// ALGLIB 2.0.1, Sergey Bochkanov
|
|||
// </contribution>
|
|||
|
|||
// ReSharper disable CheckNamespace
|
|||
namespace MathNet.Numerics |
|||
// ReSharper restore CheckNamespace
|
|||
{ |
|||
using System; |
|||
|
|||
/// <summary>
|
|||
/// This partial implementation of the SpecialFunctions class contains all methods related to the modified bessel function.
|
|||
/// </summary>
|
|||
public static partial class SpecialFunctions |
|||
{ |
|||
/// <summary>
|
|||
/// Returns the modified Struve function of order 0.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the function of.</param>
|
|||
/// <returns></returns>
|
|||
public static double StruveL0(double x) |
|||
{ |
|||
//*********************************************************************72
|
|||
//
|
|||
//c STRVL0 calculates the modified Struve function of order 0.
|
|||
//
|
|||
// DESCRIPTION:
|
|||
//
|
|||
// This function calculates the modified Struve function of
|
|||
// order 0, denoted L0(x), defined as the solution of the
|
|||
// second-order equation
|
|||
//
|
|||
// x*D(Df) + Df - x*f = 2x/pi
|
|||
//
|
|||
// This subroutine is set up to work on IEEE machines.
|
|||
// For other machines, you should retrieve the code
|
|||
// from the general MISCFUN archive.
|
|||
//
|
|||
//
|
|||
// ERROR RETURNS:
|
|||
//
|
|||
// If the value of |XVALUE| is too large, the result
|
|||
// would cause an floating-pt overflow. An error message
|
|||
// is printed and the function returns the value of
|
|||
// sign(XVALUE)*XMAX where XMAX is the largest possible
|
|||
// floating-pt argument.
|
|||
//
|
|||
//
|
|||
// MACHINE-DEPENDENT PARAMETERS:
|
|||
//
|
|||
// NTERM1 - INTEGER - The no. of terms for the array ARL0.
|
|||
// The recommended value is such that
|
|||
// ABS(ARL0(NTERM1)) < EPS/100
|
|||
//
|
|||
// NTERM2 - INTEGER - The no. of terms for the array ARL0AS.
|
|||
// The recommended value is such that
|
|||
// ABS(ARL0AS(NTERM2)) < EPS/100
|
|||
//
|
|||
// NTERM3 - INTEGER - The no. of terms for the array AI0ML0.
|
|||
// The recommended value is such that
|
|||
// ABS(AI0ML0(NTERM3)) < EPS/100
|
|||
//
|
|||
// XLOW - DOUBLE PRECISION - The value of x below which L0(x) = 2*x/pi
|
|||
// to machine precision. The recommended value is
|
|||
// 3*SQRT(EPS)
|
|||
//
|
|||
// XHIGH1 - DOUBLE PRECISION - The value beyond which the Chebyshev series
|
|||
// in the asymptotic expansion of I0 - L0 gives
|
|||
// 1.0 to machine precision. The recommended value
|
|||
// is SQRT( 30/EPSNEG )
|
|||
//
|
|||
// XHIGH2 - DOUBLE PRECISION - The value beyond which the Chebyshev series
|
|||
// in the asymptotic expansion of I0 gives 1.0
|
|||
// to machine precision. The recommended value
|
|||
// is 28 / EPSNEG
|
|||
//
|
|||
// XMAX - DOUBLE PRECISION - The value of XMAX, where XMAX is the
|
|||
// largest possible floating-pt argument.
|
|||
// This is used to prevent overflow.
|
|||
//
|
|||
// For values of EPS, EPSNEG and XMAX the user should refer
|
|||
// to the file MACHCON.TXT
|
|||
//
|
|||
// The machine-arithmetic constants are given in DATA
|
|||
// statements.
|
|||
//
|
|||
//
|
|||
// INTRINSIC FUNCTIONS USED:
|
|||
//
|
|||
// EXP , LOG , SQRT
|
|||
//
|
|||
//
|
|||
// OTHER MISCFUN SUBROUTINES USED:
|
|||
//
|
|||
// CHEVAL , ERRPRN
|
|||
//
|
|||
//
|
|||
// AUTHOR:
|
|||
// DR. ALLAN J. MACLEOD
|
|||
// DEPT. OF MATHEMATICS AND STATISTICS
|
|||
// UNIVERSITY OF PAISLEY
|
|||
// HIGH ST.
|
|||
// PAISLEY
|
|||
// SCOTLAND
|
|||
// PA1 2BE
|
|||
//
|
|||
// (e-mail: macl_ms0@paisley.ac.uk )
|
|||
//
|
|||
//
|
|||
// LATEST REVISION:
|
|||
// 12 JANUARY, 1996
|
|||
//
|
|||
//
|
|||
|
|||
if (x < 0.0) |
|||
{ |
|||
return -StruveL0(-x); |
|||
} |
|||
|
|||
const double LNR2PI = 0.91893853320467274178; |
|||
const double TWOBPI = 0.63661977236758134308; |
|||
|
|||
double[] ARL0 = new double[28]; |
|||
ARL0[0] = 0.42127458349979924863; |
|||
ARL0[1] = -0.33859536391220612188; |
|||
ARL0[2] = 0.21898994812710716064; |
|||
ARL0[3] = -0.12349482820713185712; |
|||
ARL0[4] = 0.6214209793866958440e-1; |
|||
ARL0[5] = -0.2817806028109547545e-1; |
|||
ARL0[6] = 0.1157419676638091209e-1; |
|||
ARL0[7] = -0.431658574306921179e-2; |
|||
ARL0[8] = 0.146142349907298329e-2; |
|||
ARL0[9] = -0.44794211805461478e-3; |
|||
ARL0[10] = 0.12364746105943761e-3; |
|||
ARL0[11] = -0.3049028334797044e-4; |
|||
ARL0[12] = 0.663941401521146e-5; |
|||
ARL0[13] = -0.125538357703889e-5; |
|||
ARL0[14] = 0.20073446451228e-6; |
|||
ARL0[15] = -0.2588260170637e-7; |
|||
ARL0[16] = 0.241143742758e-8; |
|||
ARL0[17] = -0.10159674352e-9; |
|||
ARL0[18] = -0.1202430736e-10; |
|||
ARL0[19] = 0.262906137e-11; |
|||
ARL0[20] = -0.15313190e-12; |
|||
ARL0[21] = -0.1574760e-13; |
|||
ARL0[22] = 0.315635e-14; |
|||
ARL0[23] = -0.4096e-16; |
|||
ARL0[24] = -0.3620e-16; |
|||
ARL0[25] = 0.239e-17; |
|||
ARL0[26] = 0.36e-18; |
|||
ARL0[27] = -0.4e-19; |
|||
|
|||
double[] ARL0AS = new double[16]; |
|||
ARL0AS[0] = 2.00861308235605888600; |
|||
ARL0AS[1] = 0.403737966500438470e-2; |
|||
ARL0AS[2] = -0.25199480286580267e-3; |
|||
ARL0AS[3] = 0.1605736682811176e-4; |
|||
ARL0AS[4] = -0.103692182473444e-5; |
|||
ARL0AS[5] = 0.6765578876305e-7; |
|||
ARL0AS[6] = -0.444999906756e-8; |
|||
ARL0AS[7] = 0.29468889228e-9; |
|||
ARL0AS[8] = -0.1962180522e-10; |
|||
ARL0AS[9] = 0.131330306e-11; |
|||
ARL0AS[10] = -0.8819190e-13; |
|||
ARL0AS[11] = 0.595376e-14; |
|||
ARL0AS[12] = -0.40389e-15; |
|||
ARL0AS[13] = 0.2651e-16; |
|||
ARL0AS[14] = -0.208e-17; |
|||
ARL0AS[15] = 0.11e-18; |
|||
|
|||
double[] AI0ML0 = new double[24]; |
|||
AI0ML0[0] = 2.00326510241160643125; |
|||
AI0ML0[1] = 0.195206851576492081e-2; |
|||
AI0ML0[2] = 0.38239523569908328e-3; |
|||
AI0ML0[3] = 0.7534280817054436e-4; |
|||
AI0ML0[4] = 0.1495957655897078e-4; |
|||
AI0ML0[5] = 0.299940531210557e-5; |
|||
AI0ML0[6] = 0.60769604822459e-6; |
|||
AI0ML0[7] = 0.12399495544506e-6; |
|||
AI0ML0[8] = 0.2523262552649e-7; |
|||
AI0ML0[9] = 0.504634857332e-8; |
|||
AI0ML0[10] = 0.97913236230e-9; |
|||
AI0ML0[11] = 0.18389115241e-9; |
|||
AI0ML0[12] = 0.3376309278e-10; |
|||
AI0ML0[13] = 0.611179703e-11; |
|||
AI0ML0[14] = 0.108472972e-11; |
|||
AI0ML0[15] = 0.18861271e-12; |
|||
AI0ML0[16] = 0.3280345e-13; |
|||
AI0ML0[17] = 0.565647e-14; |
|||
AI0ML0[18] = 0.93300e-15; |
|||
AI0ML0[19] = 0.15881e-15; |
|||
AI0ML0[20] = 0.2791e-16; |
|||
AI0ML0[21] = 0.389e-17; |
|||
AI0ML0[22] = 0.70e-18; |
|||
AI0ML0[23] = 0.16e-18; |
|||
|
|||
// MACHINE-DEPENDENT VALUES (Suitable for IEEE-arithmetic machines)
|
|||
const int NTERM1 = 25; const int NTERM2 = 14; const int NTERM3 = 21; |
|||
const double XLOW = 4.4703484e-8; const double XMAX = 1.797693e308; |
|||
const double XHIGH1 = 5.1982303e8; const double XHIGH2 = 2.5220158e17; |
|||
|
|||
// Code for |xvalue| <= 16
|
|||
if (x <= 16.0) |
|||
{ |
|||
if (x < XLOW) |
|||
{ |
|||
return TWOBPI * x; |
|||
} |
|||
|
|||
double T = (4.0 * x - 24.0) / (x + 24.0); |
|||
return TWOBPI * x * Evaluate.ChebyshevSum(NTERM1, ARL0, T) * Math.Exp(x); |
|||
} |
|||
|
|||
// Code for |xvalue| > 16
|
|||
double ch1; |
|||
if (x > XHIGH2) |
|||
{ |
|||
ch1 = 1.0; |
|||
} |
|||
else |
|||
{ |
|||
double T = (x - 28.0) / (4.0 - x); |
|||
ch1 = Evaluate.ChebyshevSum(NTERM2, ARL0AS, T); |
|||
} |
|||
|
|||
double ch2; |
|||
if (x > XHIGH1) |
|||
{ |
|||
ch2 = 1.0; |
|||
} |
|||
else |
|||
{ |
|||
double xsq = x * x; |
|||
double T = (800.0 - xsq) / (288.0 + xsq); |
|||
ch2 = Evaluate.ChebyshevSum(NTERM3, AI0ML0, T); |
|||
} |
|||
|
|||
double test = Math.Log(ch1) - LNR2PI - Math.Log(x) / 2.0 + x; |
|||
if (test > Math.Log(XMAX)) |
|||
{ |
|||
throw new ArithmeticException("ERROR IN MISCFUN FUNCTION STRVL0: ARGUMENT CAUSES OVERFLOW"); |
|||
} |
|||
|
|||
return Math.Exp(test) - TWOBPI * ch2 / x; |
|||
} |
|||
|
|||
/// <summary>
|
|||
/// Returns the modified Struve function of order 1.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the function of.</param>
|
|||
/// <returns></returns>
|
|||
public static double StruveL1(double x) |
|||
{ |
|||
//*********************************************************************72
|
|||
//
|
|||
//c STRVL1 calculates the modified Struve function of order 1.
|
|||
//
|
|||
// DESCRIPTION:
|
|||
//
|
|||
// This function calculates the modified Struve function of
|
|||
// order 1, denoted L1(x), defined as the solution of
|
|||
//
|
|||
// x*x*D(Df) + x*Df - (x*x+1)f = 2*x*x/pi
|
|||
//
|
|||
// This subroutine is set up to work on IEEE machines.
|
|||
// For other machines, you should retrieve the code
|
|||
// from the general MISCFUN archive.
|
|||
//
|
|||
//
|
|||
// ERROR RETURNS:
|
|||
//
|
|||
// If the value of |XVALUE| is too large, the result
|
|||
// would cause an floating-pt overflow. An error message
|
|||
// is printed and the function returns the value of
|
|||
// sign(XVALUE)*XMAX where XMAX is the largest possible
|
|||
// floating-pt argument.
|
|||
//
|
|||
//
|
|||
// MACHINE-DEPENDENT PARAMETERS:
|
|||
//
|
|||
// NTERM1 - INTEGER - The no. of terms for the array ARL1.
|
|||
// The recommended value is such that
|
|||
// ABS(ARL1(NTERM1)) < EPS/100
|
|||
//
|
|||
// NTERM2 - INTEGER - The no. of terms for the array ARL1AS.
|
|||
// The recommended value is such that
|
|||
// ABS(ARL1AS(NTERM2)) < EPS/100
|
|||
//
|
|||
// NTERM3 - INTEGER - The no. of terms for the array AI1ML1.
|
|||
// The recommended value is such that
|
|||
// ABS(AI1ML1(NTERM3)) < EPS/100
|
|||
//
|
|||
// XLOW1 - DOUBLE PRECISION - The value of x below which
|
|||
// L1(x) = 2*x*x/(3*pi)
|
|||
// to machine precision. The recommended
|
|||
// value is SQRT(15*EPS)
|
|||
//
|
|||
// XLOW2 - DOUBLE PRECISION - The value of x below which L1(x) set to 0.0.
|
|||
// This is used to prevent underflow. The
|
|||
// recommended value is
|
|||
// SQRT(5*XMIN)
|
|||
//
|
|||
// XHIGH1 - DOUBLE PRECISION - The value of |x| above which the Chebyshev
|
|||
// series in the asymptotic expansion of I1
|
|||
// equals 1.0 to machine precision. The
|
|||
// recommended value is SQRT( 30 / EPSNEG ).
|
|||
//
|
|||
// XHIGH2 - DOUBLE PRECISION - The value of |x| above which the Chebyshev
|
|||
// series in the asymptotic expansion of I1 - L1
|
|||
// equals 1.0 to machine precision. The recommended
|
|||
// value is 30 / EPSNEG.
|
|||
//
|
|||
// XMAX - DOUBLE PRECISION - The value of XMAX, where XMAX is the
|
|||
// largest possible floating-pt argument.
|
|||
// This is used to prevent overflow.
|
|||
//
|
|||
// For values of EPS, EPSNEG, XMIN, and XMAX the user should refer
|
|||
// to the file MACHCON.TXT
|
|||
//
|
|||
// The machine-arithmetic constants are given in DATA
|
|||
// statements.
|
|||
//
|
|||
//
|
|||
// INTRINSIC FUNCTIONS USED:
|
|||
//
|
|||
// EXP , LOG , SQRT
|
|||
//
|
|||
//
|
|||
// OTHER MISCFUN SUBROUTINES USED:
|
|||
//
|
|||
// CHEVAL , ERRPRN
|
|||
//
|
|||
//
|
|||
// AUTHOR:
|
|||
// DR. ALLAN J. MACLEOD
|
|||
// DEPT. OF MATHEMATICS AND STATISTICS
|
|||
// UNIVERSITY OF PAISLEY
|
|||
// HIGH ST.
|
|||
// PAISLEY
|
|||
// SCOTLAND
|
|||
// PA1 2BE
|
|||
//
|
|||
// (e-mail: macl_ms0@paisley.ac.uk )
|
|||
//
|
|||
//
|
|||
// LATEST UPDATE:
|
|||
// 12 JANUARY, 1996
|
|||
//
|
|||
//
|
|||
|
|||
if (x < 0.0) |
|||
{ |
|||
return StruveL1(-x); |
|||
} |
|||
|
|||
const double LNR2PI = 0.91893853320467274178; |
|||
const double PI3BY2 = 4.71238898038468985769; |
|||
const double TWOBPI = 0.63661977236758134308; |
|||
|
|||
double[] ARL1 = new double[27]; |
|||
ARL1[0] = 0.38996027351229538208; |
|||
ARL1[1] = -0.33658096101975749366; |
|||
ARL1[2] = 0.23012467912501645616; |
|||
ARL1[3] = -0.13121594007960832327; |
|||
ARL1[4] = 0.6425922289912846518e-1; |
|||
ARL1[5] = -0.2750032950616635833e-1; |
|||
ARL1[6] = 0.1040234148637208871e-1; |
|||
ARL1[7] = -0.350532294936388080e-2; |
|||
ARL1[8] = 0.105748498421439717e-2; |
|||
ARL1[9] = -0.28609426403666558e-3; |
|||
ARL1[10] = 0.6925708785942208e-4; |
|||
ARL1[11] = -0.1489693951122717e-4; |
|||
ARL1[12] = 0.281035582597128e-5; |
|||
ARL1[13] = -0.45503879297776e-6; |
|||
ARL1[14] = 0.6090171561770e-7; |
|||
ARL1[15] = -0.623543724808e-8; |
|||
ARL1[16] = 0.38430012067e-9; |
|||
ARL1[17] = 0.790543916e-11; |
|||
ARL1[18] = -0.489824083e-11; |
|||
ARL1[19] = 0.46356884e-12; |
|||
ARL1[20] = 0.684205e-14; |
|||
ARL1[21] = -0.569748e-14; |
|||
ARL1[22] = 0.35324e-15; |
|||
ARL1[23] = 0.4244e-16; |
|||
ARL1[24] = -0.644e-17; |
|||
ARL1[25] = -0.21e-18; |
|||
ARL1[26] = 0.9e-19; |
|||
|
|||
double[] ARL1AS = new double[17]; |
|||
ARL1AS[0] = 1.97540378441652356868; |
|||
ARL1AS[1] = -0.1195130555088294181e-1; |
|||
ARL1AS[2] = 0.33639485269196046e-3; |
|||
ARL1AS[3] = -0.1009115655481549e-4; |
|||
ARL1AS[4] = 0.30638951321998e-6; |
|||
ARL1AS[5] = -0.953704370396e-8; |
|||
ARL1AS[6] = 0.29524735558e-9; |
|||
ARL1AS[7] = -0.951078318e-11; |
|||
ARL1AS[8] = 0.28203667e-12; |
|||
ARL1AS[9] = -0.1134175e-13; |
|||
ARL1AS[10] = 0.147e-17; |
|||
ARL1AS[11] = -0.6232e-16; |
|||
ARL1AS[12] = -0.751e-17; |
|||
ARL1AS[13] = -0.17e-18; |
|||
ARL1AS[14] = 0.51e-18; |
|||
ARL1AS[15] = 0.23e-18; |
|||
ARL1AS[16] = 0.5e-19; |
|||
|
|||
double[] AI1ML1 = new double[26]; |
|||
AI1ML1[0] = 1.99679361896789136501; |
|||
AI1ML1[1] = -0.190663261409686132e-2; |
|||
AI1ML1[2] = -0.36094622410174481e-3; |
|||
AI1ML1[3] = -0.6841847304599820e-4; |
|||
AI1ML1[4] = -0.1299008228509426e-4; |
|||
AI1ML1[5] = -0.247152188705765e-5; |
|||
AI1ML1[6] = -0.47147839691972e-6; |
|||
AI1ML1[7] = -0.9020819982592e-7; |
|||
AI1ML1[8] = -0.1730458637504e-7; |
|||
AI1ML1[9] = -0.332323670159e-8; |
|||
AI1ML1[10] = -0.63736421735e-9; |
|||
AI1ML1[11] = -0.12180239756e-9; |
|||
AI1ML1[12] = -0.2317346832e-10; |
|||
AI1ML1[13] = -0.439068833e-11; |
|||
AI1ML1[14] = -0.82847110e-12; |
|||
AI1ML1[15] = -0.15562249e-12; |
|||
AI1ML1[16] = -0.2913112e-13; |
|||
AI1ML1[17] = -0.543965e-14; |
|||
AI1ML1[18] = -0.101177e-14; |
|||
AI1ML1[19] = -0.18767e-15; |
|||
AI1ML1[20] = -0.3484e-16; |
|||
AI1ML1[21] = -0.643e-17; |
|||
AI1ML1[22] = -0.118e-17; |
|||
AI1ML1[23] = -0.22e-18; |
|||
AI1ML1[24] = -0.4e-19; |
|||
AI1ML1[25] = -0.1e-19; |
|||
|
|||
// MACHINE-DEPENDENT VALUES (Suitable for IEEE-arithmetic machines)
|
|||
const int NTERM1 = 24; const int NTERM2 = 13; const int NTERM3 = 22; |
|||
const double XLOW1 = 5.7711949e-8; const double XLOW2 = 3.3354714e-154; const double XMAX = 1.797693e308; |
|||
const double XHIGH1 = 5.19823025e8; const double XHIGH2 = 2.7021597e17; |
|||
|
|||
// CODE FOR |x| <= 16
|
|||
if (x <= 16.0) |
|||
{ |
|||
if (x <= XLOW2) |
|||
{ |
|||
return 0.0; |
|||
} |
|||
|
|||
double xsq = x * x; |
|||
if (x < XLOW1) |
|||
{ |
|||
return xsq / PI3BY2; |
|||
} |
|||
|
|||
double t = (4.0 * x - 24.0) / (x + 24.0); |
|||
return xsq * Evaluate.ChebyshevSum(NTERM1, ARL1, t) * Math.Exp(x) / PI3BY2; |
|||
} |
|||
|
|||
// CODE FOR |x| > 16
|
|||
double ch1; |
|||
if (x > XHIGH2) |
|||
{ |
|||
ch1 = 1.0; |
|||
} |
|||
else |
|||
{ |
|||
double t = (x - 30.0) / (2.0 - x); |
|||
ch1 = Evaluate.ChebyshevSum(NTERM2, ARL1AS, t); |
|||
} |
|||
|
|||
double ch2; |
|||
if (x > XHIGH1) |
|||
{ |
|||
ch2 = 1.0; |
|||
} |
|||
else |
|||
{ |
|||
double xsq = x * x; |
|||
double t = (800.0 - xsq) / (288.0 + xsq); |
|||
ch2 = Evaluate.ChebyshevSum(NTERM3, AI1ML1, t); |
|||
} |
|||
|
|||
double test = Math.Log(ch1) - LNR2PI - Math.Log(x) / 2.0 + x; |
|||
if (test > Math.Log(XMAX)) |
|||
{ |
|||
throw new ArithmeticException("ERROR IN MISCFUN FUNCTION STRVL1: ARGUMENT CAUSES OVERFLOW"); |
|||
} |
|||
|
|||
return Math.Exp(test) - TWOBPI * ch2; |
|||
} |
|||
|
|||
/// <summary>
|
|||
/// Returns the difference between the Bessel I0 and Struve L0 functions.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the function of.</param>
|
|||
/// <returns></returns>
|
|||
public static double BesselI0MStruveL0(double x) |
|||
{ |
|||
// TODO: way off for large x (e.g. 100) - needs direct approximation
|
|||
return BesselI0(x) - StruveL0(x); |
|||
} |
|||
|
|||
/// <summary>
|
|||
/// Returns the difference between the Bessel I1 and Struve L1 functions.
|
|||
/// </summary>
|
|||
/// <param name="x">The value to compute the function of.</param>
|
|||
/// <returns></returns>
|
|||
public static double BesselI1MStruveL1(double x) |
|||
{ |
|||
// TODO: way off for large x (e.g. 100) - needs direct approximation
|
|||
return BesselI1(x) - StruveL1(x); |
|||
} |
|||
} |
|||
} |
|||
@ -0,0 +1,120 @@ |
|||
// <copyright file="ModifiedBesselTests.cs" company="Math.NET">
|
|||
// Math.NET Numerics, part of the Math.NET Project
|
|||
// http://numerics.mathdotnet.com
|
|||
// http://github.com/mathnet/mathnet-numerics
|
|||
// http://mathnetnumerics.codeplex.com
|
|||
//
|
|||
// Copyright (c) 2009-2012 Math.NET
|
|||
//
|
|||
// Permission is hereby granted, free of charge, to any person
|
|||
// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
|||
// restriction, including without limitation the rights to use,
|
|||
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
|||
// copies of the Software, and to permit persons to whom the
|
|||
// Software is furnished to do so, subject to the following
|
|||
// conditions:
|
|||
//
|
|||
// The above copyright notice and this permission notice shall be
|
|||
// included in all copies or substantial portions of the Software.
|
|||
//
|
|||
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
|||
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
|||
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
|||
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
|||
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
|||
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
|||
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
|||
// OTHER DEALINGS IN THE SOFTWARE.
|
|||
// </copyright>
|
|||
|
|||
namespace MathNet.Numerics.UnitTests.SpecialFunctionsTests |
|||
{ |
|||
using System; |
|||
using NUnit.Framework; |
|||
|
|||
/// <summary>
|
|||
/// Modified Bessel functions tests.
|
|||
/// </summary>
|
|||
[TestFixture] |
|||
public class BodifiedBesselTests |
|||
{ |
|||
[Test] |
|||
public void BesselI0Approx([Range(-3.75, 3.75, 0.25)] double x) |
|||
{ |
|||
// Approx by Abramowitz/Stegun 9.8.1
|
|||
Assert.AreEqual(Evaluate.Polynomial(new[] { 1.0, 0.0, 3.5156229, 0.0, 3.0899424, 0.0, 1.2067492, 0.0, 0.2659732, 0.0, 0.0360768, 0.0, 0.0045813 }, x / 3.75), SpecialFunctions.BesselI0(x), 1e-7); |
|||
} |
|||
|
|||
[TestCase(0.0, 1.0)] |
|||
[TestCase(0.005, 1.000006250009766)] |
|||
[TestCase(0.5, 1.063483370741324)] |
|||
[TestCase(1.5, 1.646723189772891)] |
|||
[TestCase(10.0, 2815.716628466254)] |
|||
[TestCase(100.0, 1.073751707131074e+42)] |
|||
[TestCase(-0.005, 1.000006250009766)] |
|||
[TestCase(-10.0, 2815.716628466254)] |
|||
public void BesselI0Exact(double x, double expected) |
|||
{ |
|||
AssertHelpers.AlmostEqual(expected, SpecialFunctions.BesselI0(x), 14); |
|||
} |
|||
|
|||
[Test] |
|||
public void BesselI1Approx([Range(-3.75, 3.75, 0.25)] double x) |
|||
{ |
|||
// Approx by Abramowitz/Stegun 9.8.3
|
|||
Assert.AreEqual(Evaluate.Polynomial(new[] { 0.5, 0.0, 0.87890594, 0.0, 0.51498869, 0.0, 0.15084934, 0.0, 0.02658733, 0.0, 0.00301532, 0.0, 0.00032411 }, x / 3.75) * x, SpecialFunctions.BesselI1(x), 1e-8); |
|||
} |
|||
|
|||
[TestCase(0.0, 0.0)] |
|||
[TestCase(0.005, 0.002500007812508138)] |
|||
[TestCase(0.5, 0.2578943053908963)] |
|||
[TestCase(1.5, 0.9816664285779076)] |
|||
[TestCase(10.0, 2670.988303701255)] |
|||
[TestCase(100.0, 1.068369390338162e+42)] |
|||
[TestCase(-0.005, -0.002500007812508138)] |
|||
[TestCase(-10.0, -2670.988303701255)] |
|||
public void BesselI1Exact(double x, double expected) |
|||
{ |
|||
AssertHelpers.AlmostEqual(expected, SpecialFunctions.BesselI1(x), 14); |
|||
} |
|||
|
|||
[Test] |
|||
public void BesselK0Approx([Range(0.20, 2.0, 0.20)] double x) |
|||
{ |
|||
// Approx by Abramowitz/Stegun 9.8.5
|
|||
Assert.AreEqual(Evaluate.Polynomial(new[] { -Math.Log(x/2.0)*SpecialFunctions.BesselI0(x)-0.57721566, 0.0, 0.42278420, 0.0, 0.23069756, 0.0, 0.03488590, 0.0, 0.00262698, 0.0, 0.00010750, 0.0, 0.00000740 }, x / 2.0), SpecialFunctions.BesselK0(x), 1e-8); |
|||
} |
|||
|
|||
[TestCase(1e-10, 23.14178244559887)] |
|||
[TestCase(1e-5, 11.62885698094436)] |
|||
[TestCase(0.005, 5.414288971329485)] |
|||
[TestCase(0.5, 0.9244190712276659)] |
|||
[TestCase(1.5, 0.2138055626475257)] |
|||
[TestCase(10.0, 0.00001778006231616765)] |
|||
[TestCase(100.0, 4.656628229175902e-45)] |
|||
public void BesselK0Exact(double x, double expected) |
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{ |
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AssertHelpers.AlmostEqual(expected, SpecialFunctions.BesselK0(x), 14); |
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} |
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|
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[Test] |
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public void BesselK1Approx([Range(0.20, 2.0, 0.20)] double x) |
|||
{ |
|||
// Approx by Abramowitz/Stegun 9.8.7
|
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Assert.AreEqual(Evaluate.Polynomial(new[] { x * Math.Log(x / 2.0) * SpecialFunctions.BesselI1(x) + 1.0, 0.0, 0.15443144, 0.0, -0.67278579, 0.0, -0.18156897, 0.0, -0.01919402, 0.0, -0.00110404, 0.0, -0.00004686 }, x / 2.0), SpecialFunctions.BesselK1(x) * x, 1e-8); |
|||
} |
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|
|||
[TestCase(1e-10, 1.0e+10)] |
|||
[TestCase(1e-5, 99999.99993935572)] |
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[TestCase(0.005, 199.9852143257300)] |
|||
[TestCase(0.5, 1.656441120003301)] |
|||
[TestCase(1.5, 0.2773878004568438)] |
|||
[TestCase(10.0, 0.00001864877345382558)] |
|||
[TestCase(100.0, 4.679853735636909e-45)] |
|||
public void BesselK1Exact(double x, double expected) |
|||
{ |
|||
AssertHelpers.AlmostEqual(expected, SpecialFunctions.BesselK1(x), 14); |
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} |
|||
} |
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} |
|||
@ -0,0 +1,99 @@ |
|||
// <copyright file="ModifiedBesselTests.cs" company="Math.NET">
|
|||
// Math.NET Numerics, part of the Math.NET Project
|
|||
// http://numerics.mathdotnet.com
|
|||
// http://github.com/mathnet/mathnet-numerics
|
|||
// http://mathnetnumerics.codeplex.com
|
|||
//
|
|||
// Copyright (c) 2009-2012 Math.NET
|
|||
//
|
|||
// Permission is hereby granted, free of charge, to any person
|
|||
// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
|||
// restriction, including without limitation the rights to use,
|
|||
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
|||
// copies of the Software, and to permit persons to whom the
|
|||
// Software is furnished to do so, subject to the following
|
|||
// conditions:
|
|||
//
|
|||
// The above copyright notice and this permission notice shall be
|
|||
// included in all copies or substantial portions of the Software.
|
|||
//
|
|||
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
|||
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
|||
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
|||
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
|||
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
|||
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
|||
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
|||
// OTHER DEALINGS IN THE SOFTWARE.
|
|||
// </copyright>
|
|||
|
|||
namespace MathNet.Numerics.UnitTests.SpecialFunctionsTests |
|||
{ |
|||
using NUnit.Framework; |
|||
|
|||
/// <summary>
|
|||
/// Modified Struve functions tests.
|
|||
/// </summary>
|
|||
[TestFixture] |
|||
public class ModifiedStruveTests |
|||
{ |
|||
[TestCase(0.0, 0.0)] |
|||
[TestCase(0.005, 0.003183107703788032)] |
|||
[TestCase(0.5, 0.3272406993941808)] |
|||
[TestCase(1.5, 1.216162510717182)] |
|||
[TestCase(10.0, 2815.652249374595)] |
|||
[TestCase(100.0, 1.073751707131074e+42)] |
|||
[TestCase(-0.005, -0.003183107703788032)] |
|||
[TestCase(-10.0, -2815.652249374595)] |
|||
public void StruveL0Exact(double x, double expected) |
|||
{ |
|||
AssertHelpers.AlmostEqual(expected, SpecialFunctions.StruveL0(x), 14); |
|||
} |
|||
|
|||
[TestCase(0.0, 0.0)] |
|||
[TestCase(0.005, 5.305173611677443e-6)] |
|||
[TestCase(0.5, 0.05394218262352266)] |
|||
[TestCase(1.5, 0.5538569084469910)] |
|||
[TestCase(10.0, 2670.358285208483)] |
|||
[TestCase(100.0, 1.068369390338162e+42)] |
|||
[TestCase(-0.005, 5.305173611677443e-6)] |
|||
[TestCase(-10.0, 2670.358285208483)] |
|||
public void StruveL1Exact(double x, double expected) |
|||
{ |
|||
AssertHelpers.AlmostEqual(expected, SpecialFunctions.StruveL1(x), 14); |
|||
} |
|||
|
|||
[TestCase(0.0, 1.0)] |
|||
[TestCase(0.1, 0.938769)] |
|||
[TestCase(0.4, 0.781198)] |
|||
[TestCase(2.0, 0.342152)] |
|||
[TestCase(2.4, 0.289765)] |
|||
[TestCase(4.5, 0.150279)] |
|||
[TestCase(4.9, 0.136938)] |
|||
[TestCase(10.0, 0.064379)] |
|||
[TestCase(20.0, 0.031912)] |
|||
//[TestCase(100.0, 0.006367)] Needs direct approximation
|
|||
public void BesselI0MStruveL0Exact(double x, double expected) |
|||
{ |
|||
// Abramowitz/Stegun Table 12.1, 12.2
|
|||
AssertHelpers.AlmostEqual(expected, SpecialFunctions.BesselI0MStruveL0(x), 5); |
|||
} |
|||
|
|||
[TestCase(0.0, 0.0)] |
|||
[TestCase(0.1, 0.047939)] |
|||
[TestCase(0.4, 0.169710)] |
|||
[TestCase(2.0, 0.487877)] |
|||
[TestCase(2.4, 0.521712)] |
|||
[TestCase(4.5, 0.600147)] |
|||
[TestCase(4.9, 0.606142)] |
|||
[TestCase(10.0, 0.630018)] |
|||
[TestCase(20.0, 0.635016)] |
|||
//[TestCase(100.0, 0.636556)] Needs direct approximation
|
|||
public void BesselI1MStruveL1Exact(double x, double expected) |
|||
{ |
|||
// Abramowitz/Stegun Table 12.1, 12.2
|
|||
AssertHelpers.AlmostEqual(expected, SpecialFunctions.BesselI1MStruveL1(x), 5); |
|||
} |
|||
} |
|||
} |
|||
Loading…
Reference in new issue