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Merge pull request #267 from glyphard/gh_#228_exponential_integral
Gh #228 exponential integral
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// <copyright file="ExponentialIntegral.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) 2014 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> | ||
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// <contribution> | ||
// Ashley Messer | ||
// </contribution> | ||
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// ReSharper disable CheckNamespace | ||
namespace MathNet.Numerics | ||
// ReSharper restore CheckNamespace | ||
{ | ||
using System; | ||
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public static partial class SpecialFunctions | ||
{ | ||
/// <summary> | ||
/// Computes the Exponential Integral function. | ||
/// </summary> | ||
/// <param name="x">The argument of the Exponential Integral function.</param> | ||
/// <returns>The value of the Exponential Integral function.</returns> | ||
/// <remarks> | ||
/// <para>This implementation of the computation of the Exponential Integral function follows the derivation in | ||
/// "Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55", Abramowitz, M., and Stegun, I.A. 1964, reprinted 1968 by | ||
/// Dover Publications, New York), Chapters 6, 7, and 26. | ||
/// AND | ||
/// "Advanced mathematical methods for scientists and engineers", Bender, Carl M.; Steven A. Orszag (1978). page 253 | ||
/// </para> | ||
/// <para> | ||
/// for x > 1 uses continued fraction approach that is often used to compute incomplete gamma. | ||
/// for 0 < x <= 1 uses taylor series expansion | ||
/// </para> | ||
/// <para>Our unit tests suggest that the accuracy of the Exponential Integral function is correct up to 13 floating point digits.</para> | ||
/// </remarks> | ||
public static double ExponentialIntegral(double x, int n) | ||
{ | ||
//parameter validation | ||
if (n < 0 || x < 0.0 ) { | ||
throw new ArgumentOutOfRangeException(string.Format("x and n must be positive: x={0}, n={1}", x, n)); | ||
} | ||
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const double epsilon = 0.00000000000000001; | ||
int maxIterations = 100; | ||
int i, ii; | ||
double ndbl = (double)n; | ||
double result = double.NaN; | ||
double nearDoubleMin = 1e-100; //needs a very small value that is not quite as small as the lowest value double can take | ||
double factorial = 1.0d; | ||
double del; | ||
double psi; | ||
double a, b, c, d, h; //variables for continued fraction | ||
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//special cases | ||
if (n == 0) | ||
{ | ||
result = Math.Exp( -1.0d * x ) / x; | ||
return result; | ||
} | ||
else if (x == 0.0d) | ||
{ | ||
result = 1.0d / (ndbl - 1.0d); | ||
return result; | ||
} | ||
//general cases | ||
//continued fraction for large x | ||
if (x > 1.0d) | ||
{ | ||
b = x + ((double)n); | ||
c = 1.0d / nearDoubleMin; | ||
d = 1.0d / b; | ||
h = d; | ||
for (i = 1; i <= maxIterations; i++) | ||
{ | ||
a = -1.0d * ((double)i) * ((ndbl - 1.0d) + (double)i); | ||
b += 2.0d; | ||
d = 1.0d / (a * d + b); | ||
c = b + a / c; | ||
del = c * d; | ||
h = h * del; | ||
if (Math.Abs(del - 1.0d) < epsilon) | ||
{ | ||
result = h * Math.Exp( -x ); | ||
return result; | ||
} | ||
} | ||
throw new ArithmeticException(string.Format("continued fraction failed to converge for x={0}, n={1})", x, n)); | ||
} | ||
//series computation for small x | ||
else | ||
{ | ||
result = ((ndbl - 1.0d) != 0 ? 1.0 / (ndbl - 1.0d) : (-1.0d * Math.Log(x) - Constants.EulerMascheroni)); //Set first term. | ||
for (i = 1; i <= maxIterations; i++) | ||
{ | ||
factorial *= (-1.0d * x / ((double)i)); | ||
if (i != (ndbl - 1.0d)) { del = -factorial / (i - (ndbl - 1.0d)); } | ||
else | ||
{ | ||
psi = -1.0d * Constants.EulerMascheroni; | ||
for (ii = 1; ii <= (ndbl - 1.0d); ii++) | ||
{ | ||
psi += (1.0d / ((double)ii)); | ||
} | ||
del = factorial * (-1.0d * Math.Log(x) + psi); | ||
} | ||
result += del; | ||
if (Math.Abs(del) < Math.Abs(result) * epsilon) | ||
{ | ||
return result; | ||
} | ||
} | ||
throw new ArithmeticException(string.Format("series failed to converge for x={0}, n={1})", x, n)); | ||
} | ||
} | ||
} | ||
} |
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src/UnitTests/SpecialFunctionsTests/ExponentialIntegralTests.cs
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namespace MathNet.Numerics.UnitTests.SpecialFunctionsTests | ||
{ | ||
using System; | ||
using NUnit.Framework; | ||
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/// <summary> | ||
/// Exponential Integral tests. | ||
/// </summary> | ||
[TestFixture, Category("Distributions")] | ||
public class ExponentialIntegralTests | ||
{ | ||
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[TestCase(0.001d, 6.33153936413614904)] | ||
[TestCase(0.1d, 1.82292395841939059)] | ||
[TestCase(1.0d, 0.219383934395520286d)] | ||
[TestCase(2.0d, 0.0489005107080611248d)] | ||
[TestCase(2.5d, 0.0249149178702697399)] | ||
[TestCase(10.0d, 4.15696892968532464e-06)] | ||
public void ExponentialIntegral_Matches_MATLAB_and_R_expint_E1(double x, double result) | ||
{ | ||
double actual = SpecialFunctions.ExponentialIntegral( x, 1 ); | ||
double delta = Math.Abs( result - actual ); | ||
AssertHelpers.AlmostEqualRelative( result, actual, 13 ); | ||
} | ||
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[TestCase(0.001d, 2, 0.992668960469238915)] | ||
[TestCase(0.1d, 2, 0.722545022194020392)] | ||
[TestCase(1.0d, 2, 0.148495506775922048)] | ||
[TestCase(2.0d, 2, 0.0375342618204904527)] | ||
[TestCase(10.0d, 2, 3.830240465631608e-06)] | ||
public void ExponentialIntegral_Matches_R_expint_En(double x, int n, double result) | ||
{ | ||
double actual = SpecialFunctions.ExponentialIntegral(x, n); | ||
double delta = Math.Abs(result - actual); | ||
AssertHelpers.AlmostEqualRelative(result, actual, 13); | ||
} | ||
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[TestCase(0.001d, 0, 999.000499833375)] | ||
[TestCase(0.1d, 0, 9.048374180359595)] | ||
[TestCase(1.0d, 0, 0.3678794411714423)] | ||
[TestCase(2.0d, 0, 0.06766764161830635)] | ||
[TestCase(10.0d, 0, 4.539992976248485e-06)] | ||
public void ExponentialIntegral_SpecialCase_EXP_Matches_from_R_expint_En(double x, int n, double result) | ||
{ | ||
double actual = SpecialFunctions.ExponentialIntegral(x, n); | ||
double delta = Math.Abs(result - actual); | ||
AssertHelpers.AlmostEqualRelative(result, actual, 13); | ||
} | ||
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} | ||
} |
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