forked from tsai/mathnet-numerics
Ashley Messer
12 years ago
4 changed files with 163 additions and 0 deletions
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using System; |
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using System.Collections.Generic; |
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using System.Linq; |
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using System.Text; |
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namespace MathNet.Numerics |
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{ |
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public static partial class SpecialFunctions |
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{ |
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/// <summary>
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/// Computes the Exponential Integral function.
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/// </summary>
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/// <param name="x">The argument of the Exponential Integral function.</param>
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/// <returns>The value of the Exponential Integral function.</returns>
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/// <remarks>
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/// <para>This implementation of the computation of the Exponential Integral function follows the derivation in
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/// "Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55", Abramowitz, M., and Stegun, I.A. 1964, reprinted 1968 by
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/// Dover Publications, New York), Chapters 6, 7, and 26.
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/// AND
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/// "Advanced mathematical methods for scientists and engineers", Bender, Carl M.; Steven A. Orszag (1978). page 253
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/// </para>
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/// <para>
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/// for x > 1 uses continued fraction approach that is often used to compute incomplete gamma.
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/// for 0 < x <= 1 uses taylor series expansion
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/// </para>
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/// <para>Our unit tests suggest that the accuracy of the Exponential Integral function is correct up to 13 floating point digits.</para>
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/// </remarks>
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public static double ExponentialIntegral(double x, int n) |
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{ |
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//parameter validation
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if (n < 0 || x < 0.0 ) { |
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throw new ArgumentOutOfRangeException(string.Format("x and n must be positive: x={0}, n={1}", x, n)); |
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} |
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const double epsilon = 0.00000000000000001; |
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int maxIterations = 100; |
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int i, ii; |
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double ndbl = (double)n; |
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double result = double.NaN; |
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double nearDoubleMin = 1e-100; //needs a very small value that is not quite as small as the lowest value double can take
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double factorial = 1.0d; |
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double del; |
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double psi; |
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double a, b, c, d, h; //variables for continued fraction
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//special cases
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if (n == 0) |
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{ |
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result = Math.Exp( -1.0d * x ) / x; |
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return result; |
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} |
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else if (x == 0.0d) |
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{ |
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result = 1.0d / (ndbl - 1.0d); |
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return result; |
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} |
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//general cases
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//continued fraction for large x
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if (x > 1.0d) |
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{ |
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b = x + ((double)n); |
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c = 1.0d / nearDoubleMin; |
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d = 1.0d / b; |
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h = d; |
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for (i = 1; i <= maxIterations; i++) |
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{ |
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a = -1.0d * ((double)i) * ((ndbl - 1.0d) + (double)i); |
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b += 2.0d; |
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d = 1.0d / (a * d + b); |
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c = b + a / c; |
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del = c * d; |
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h = h * del; |
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if (Math.Abs(del - 1.0d) < epsilon) |
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{ |
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result = h * Math.Exp( -x ); |
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return result; |
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} |
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} |
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throw new ArithmeticException(string.Format("continued fraction failed to converge for x={0}, n={1})", x, n)); |
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} |
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//series computation for small x
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else |
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{ |
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result = ((ndbl - 1.0d) != 0 ? 1.0 / (ndbl - 1.0d) : (-1.0d * Math.Log(x) - Constants.EulerMascheroni)); //Set first term.
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for (i = 1; i <= maxIterations; i++) |
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{ |
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factorial *= (-1.0d * x / ((double)i)); |
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if (i != (ndbl - 1.0d)) { del = -factorial / (i - (ndbl - 1.0d)); } |
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else |
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{ |
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psi = -1.0d * Constants.EulerMascheroni; |
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for (ii = 1; ii <= (ndbl - 1.0d); ii++) |
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{ |
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psi += (1.0d / ((double)ii)); |
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} |
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del = factorial * (-1.0d * Math.Log(x) + psi); |
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} |
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result += del; |
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if (Math.Abs(del) < Math.Abs(result) * epsilon) |
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{ |
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return result; |
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} |
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} |
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throw new ArithmeticException(string.Format("series failed to converge for x={0}, n={1})", x, n)); |
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} |
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} |
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} |
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} |
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@ -0,0 +1,53 @@ |
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namespace MathNet.Numerics.UnitTests.SpecialFunctionsTests |
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{ |
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using System; |
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using NUnit.Framework; |
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/// <summary>
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/// Factorial tests.
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/// </summary>
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[TestFixture, Category("Distributions")] |
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public class ExponentialIntegralTests |
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{ |
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[TestCase(0.001d, 6.33153936413614904)] |
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[TestCase(0.1d, 1.82292395841939059)] |
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[TestCase(1.0d, 0.219383934395520286d)] |
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[TestCase(2.0d, 0.0489005107080611248d)] |
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[TestCase(2.5d, 0.0249149178702697399)] |
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[TestCase(10.0d, 4.15696892968532464e-06)] |
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public void ExponentialIntegral_Matches_MATLAB_and_R_expint_E1(double x, double result) |
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{ |
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double actual = SpecialFunctions.ExponentialIntegral( x, 1 ); |
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double delta = Math.Abs( result - actual ); |
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AssertHelpers.AlmostEqualRelative( result, actual, 13 ); |
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} |
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[TestCase(0.001d, 2, 0.992668960469238915)] |
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[TestCase(0.1d, 2, 0.722545022194020392)] |
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[TestCase(1.0d, 2, 0.148495506775922048)] |
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[TestCase(2.0d, 2, 0.0375342618204904527)] |
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[TestCase(10.0d, 2, 3.830240465631608e-06)] |
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public void ExponentialIntegral_Matches_R_expint_En(double x, int n, double result) |
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{ |
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double actual = SpecialFunctions.ExponentialIntegral(x, n); |
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double delta = Math.Abs(result - actual); |
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AssertHelpers.AlmostEqualRelative(result, actual, 13); |
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} |
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[TestCase(0.001d, 0, 999.000499833375)] |
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[TestCase(0.1d, 0, 9.048374180359595)] |
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[TestCase(1.0d, 0, 0.3678794411714423)] |
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[TestCase(2.0d, 0, 0.06766764161830635)] |
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[TestCase(10.0d, 0, 4.539992976248485e-06)] |
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public void ExponentialIntegral_SpecialCase_EXP_Matches_from_R_expint_En(double x, int n, double result) |
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{ |
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double actual = SpecialFunctions.ExponentialIntegral(x, n); |
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double delta = Math.Abs(result - actual); |
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AssertHelpers.AlmostEqualRelative(result, actual, 13); |
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} |
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} |
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} |
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