SpecialFunctions (FsODE)

FsODE

Documentation for 0.0.2

SpecialFunctions Type

Namespace: Altaxo.Calc
Assembly: AltaxoCore.Redist.dll
Base Type: obj

Static members

Static member	Description
`SpecialFunctions.DiGamma(x)` Full Usage: `SpecialFunctions.DiGamma(x)` Parameters: x : `float` - The argument of the digamma function. Returns: `float` The value of the DiGamma function at x.	Computes the Digamma function which is mathematically defined as the derivative of the logarithm of the gamma function. This implementation is based on Jose Bernardo Algorithm AS 103: Psi ( Digamma ) Function, Applied Statistics, Volume 25, Number 3, 1976, pages 315-317. Using the modifications as in Tom Minka's lightspeed toolbox. x : `float` The argument of the digamma function. Returns: `float` The value of the DiGamma function at x.
`SpecialFunctions.DiGammaInv(p)` Full Usage: `SpecialFunctions.DiGammaInv(p)` Parameters: p : `float` - The argument of the inverse digamma function. Returns: `float` The positive solution to the inverse DiGamma function at p.	Computes the inverse Digamma function: this is the inverse of the logarithm of the gamma function. This function will only return solutions that are positive. This implementation is based on the bisection method. p : `float` The argument of the inverse digamma function. Returns: `float` The positive solution to the inverse DiGamma function at p.
`SpecialFunctions.Gamma(z)` Full Usage: `SpecialFunctions.Gamma(z)` Parameters: z : `float` - The argument of the gamma function. Returns: `float` The logarithm of the gamma function.	Computes the Gamma function. This implementation of the computation of the gamma and logarithm of the gamma function follows the derivation in "An Analysis Of The Lanczos Gamma Approximation", Glendon Ralph Pugh, 2004. We use the implementation listed on p. 116 which should achieve an accuracy of 16 floating point digits. Although 16 digit accuracy should be sufficient for double values, improving accuracy is possible (see p. 126 in Pugh). Our unit tests suggest that the accuracy of the Gamma function is correct up to 13 floating point digits. z : `float` The argument of the gamma function. Returns: `float` The logarithm of the gamma function.
`SpecialFunctions.GammaLn(z)` Full Usage: `SpecialFunctions.GammaLn(z)` Parameters: z : `float` - The argument of the gamma function. Returns: `float` The logarithm of the gamma function.	Computes the logarithm of the Gamma function. This implementation of the computation of the gamma and logarithm of the gamma function follows the derivation in "An Analysis Of The Lanczos Gamma Approximation", Glendon Ralph Pugh, 2004. We use the implementation listed on p. 116 which achieves an accuracy of 16 floating point digits. Although 16 digit accuracy should be sufficient for double values, improving accuracy is possible (see p. 126 in Pugh). Our unit tests suggest that the accuracy of the Gamma function is correct up to 14 floating point digits. z : `float` The argument of the gamma function. Returns: `float` The logarithm of the gamma function.
`SpecialFunctions.GammaLowerIncomplete(a, x)` Full Usage: `SpecialFunctions.GammaLowerIncomplete(a, x)` Parameters: a : `float` - The argument for the gamma function. x : `float` - The upper integral limit. Returns: `float` The lower incomplete gamma function.	Returns the lower incomplete gamma function gamma(a,x) = int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0. a : `float` The argument for the gamma function. x : `float` The upper integral limit. Returns: `float` The lower incomplete gamma function.
`SpecialFunctions.GammaLowerRegularized(a, x)` Full Usage: `SpecialFunctions.GammaLowerRegularized(a, x)` Parameters: a : `float` - The argument for the gamma function. x : `float` - The upper integral limit. Returns: `float` The lower incomplete gamma function.	Returns the lower incomplete regularized gamma function P(a,x) = 1/Gamma(a) * int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0. a : `float` The argument for the gamma function. x : `float` The upper integral limit. Returns: `float` The lower incomplete gamma function.
`SpecialFunctions.GammaLowerRegularizedInv(a, y0)` Full Usage: `SpecialFunctions.GammaLowerRegularizedInv(a, y0)` Parameters: a : `float` y0 : `float` Returns: `float`	Returns the inverse P^(-1) of the regularized lower incomplete gamma function P(a,x) = 1/Gamma(a) * int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0, such that P^(-1)(a,P(a,x)) == x. a : `float` y0 : `float` Returns: `float`
`SpecialFunctions.GammaUpperIncomplete(a, x)` Full Usage: `SpecialFunctions.GammaUpperIncomplete(a, x)` Parameters: a : `float` - The argument for the gamma function. x : `float` - The lower integral limit. Returns: `float` The upper incomplete gamma function.	Returns the upper incomplete gamma function Gamma(a,x) = int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0. a : `float` The argument for the gamma function. x : `float` The lower integral limit. Returns: `float` The upper incomplete gamma function.
`SpecialFunctions.GammaUpperRegularized(a, x)` Full Usage: `SpecialFunctions.GammaUpperRegularized(a, x)` Parameters: a : `float` - The argument for the gamma function. x : `float` - The lower integral limit. Returns: `float` The upper incomplete regularized gamma function.	Returns the upper incomplete regularized gamma function Q(a,x) = 1/Gamma(a) * int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0. a : `float` The argument for the gamma function. x : `float` The lower integral limit. Returns: `float` The upper incomplete regularized gamma function.