C# Class NSoft.NFramework.Numerics.SpecialFunctions

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Public Methods

Method	Description
Beta ( this xs, IEnumerable ys ) : IEnumerable	Beta Function
Beta ( double x, double y ) : double	Beta Function
BetaIncomplete ( double a, double b, double x ) : double	Returns the lower incomplete (unregularized) beta function I_x(a,b) = int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.
BetaLn ( this xs, IEnumerable ys ) : IEnumerable	Log Beta Function
BetaLn ( double x, double y ) : double	Log Beta Function
BetaRegularized ( double a, double b, double x ) : double	Returns the regularized lower incomplete beta function I_x(a,b) = 1/Beta(a,b) * int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.
BetaRegularized ( double a, double b, double x, int maxIteration ) : double	Returns the regularized lower incomplete beta function I_x(a,b) = 1/Beta(a,b) * int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.
Binomial ( int n, int k ) : double	Computes the binomial coefficient: n choose k.
BinomialLn ( int n, int k ) : double	Computes the natural logarithm of the binomial coefficient: ln(n choose k).
DiGamma ( double x ) : double	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.
DiGammaInv ( double p ) : double	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.
Erf ( double x ) : double	Calculates the error function. returns 1 if `x == Double.PositiveInfinity`. returns -1 if `x == Double.NegativeInfinity`.
ErfInv ( double z ) : double	Calculates the inverse error function evaluated at z. Calculates the inverse error function evaluated at z. returns Double.PositiveInfinity if `z >= 1.0`. returns Double.NegativeInfinity if `z <= -1.0`.
Erfc ( double x ) : double	Calculates the complementary error function. returns 0 if `x == Double.PositiveInfinity`. returns 2 if `x == Double.NegativeInfinity`.
ErfcInv ( double z ) : double	Calculates the complementary inverse error function evaluated at z. calculates the complementary inverse error function evaluated at z. We have tested this implementation against the arbitrary precision mpmath library and found cases where we can only guarantee 9 significant figures correct. returns Double.PositiveInfinity if `z <= 0.0`. returns Double.NegativeInfinity if `z >= 2.0`.
ExponentialMinusOne ( double power ) : double	Numerically stable exponential minus one, i.e. `x -> exp(x)-1`
Factorial ( int x ) : double	Computes the factorial function x -> x! of an integer number > 0. The function can represent all number up to 22! exactly, all numbers up to 170! using a double representation. All larger values will overflow. If you need to multiply or divide various such factorials, consider using the logarithmic version FactorialLn instead so you can add instead of multiply and subtract instead of divide, and then exponentiate the result using System.Math.Exp. This will also circumvent the problem that factorials become very large even for small parameters.
FactorialLn ( int x ) : double	Computes the logarithmic factorial function x -> ln(x!) of an integer number > 0.
GeneralHarmonic ( int n, double m ) : double	Compute the generalized harmonic number of order n of m. (1 + 1/2^m + 1/3^m + ... + 1/n^m)
Harmonic ( int t ) : double	Computes the t'th Harmonic number.
Hypotenuse ( System.Complex a, System.Complex b ) : System.Complex	Numerically stable hypotenuse of a right angle triangle, i.e. `(a,b) -> sqrt(a^2 + b^2)`
Hypotenuse ( double a, double b ) : double	Numerically stable hypotenuse of a right angle triangle, i.e. `(a,b) -> sqrt(a^2 + b^2)`
Hypotenuse ( float a, float b ) : float	Numerically stable hypotenuse of a right angle triangle, i.e. `(a,b) -> sqrt(a^2 + b^2)`
Logistic ( double p ) : double	Computes the logistic function. see: http://en.wikipedia.org/wiki/Logistic
Logit ( double p ) : double	Computes the logit function. see: http://en.wikipedia.org/wiki/Logit
Multinomial ( int n, int ni ) : double	Computes the multinomial coefficient: n choose n1, n2, n3, ...

Private Methods

Method	Description
ErfImp ( double z, bool invert ) : double	Implementation of the error function.
ErfInvImpl ( double p, double q, double s ) : double	The implementation of the inverse error function.
EvaluatePolynomial ( double poly, double z ) : double	A helper function to evaluate polynomials fast.
IntializeFactorial ( ) : void
Series ( Func nextSummand ) : double	Numerically stable series summation
SpecialFunctions ( )

Method Details

Beta() public static method

Beta Function

public static Beta ( this xs, IEnumerable ys ) : IEnumerable
xs	this
ys	IEnumerable
return	IEnumerable

Beta() public static method

Beta Function

public static Beta ( double x, double y ) : double
x	double
y	double
return	double

BetaIncomplete() public static method

Returns the lower incomplete (unregularized) beta function I_x(a,b) = int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.

public static BetaIncomplete ( double a, double b, double x ) : double
a	double	The first Beta parameter, a positive real number.
b	double	The second Beta parameter, a positive real number.
x	double	The upper limit of the integral.
return	double

BetaLn() public static method

Log Beta Function

public static BetaLn ( this xs, IEnumerable ys ) : IEnumerable
xs	this
ys	IEnumerable
return	IEnumerable

BetaLn() public static method

Log Beta Function

public static BetaLn ( double x, double y ) : double
x	double
y	double
return	double

BetaRegularized() public static method

Returns the regularized lower incomplete beta function I_x(a,b) = 1/Beta(a,b) * int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.

public static BetaRegularized ( double a, double b, double x ) : double
a	double	The first Beta parameter, a positive real number.
b	double	The second Beta parameter, a positive real number.
x	double	The upper limit of the integral.
return	double

BetaRegularized() public static method

Returns the regularized lower incomplete beta function I_x(a,b) = 1/Beta(a,b) * int(t^(a-1)*(1-t)^(b-1),t=0..x) for real a > 0, b > 0, 1 >= x >= 0.

public static BetaRegularized ( double a, double b, double x, int maxIteration ) : double
a	double	The first Beta parameter, a positive real number.
b	double	The second Beta parameter, a positive real number.
x	double	The upper limit of the integral.
maxIteration	int	최대 반복 횟수
return	double

Binomial() public static method

Computes the binomial coefficient: n choose k.

public static Binomial ( int n, int k ) : double
n	int	A nonnegative value n.
k	int	A nonnegative value h.
return	double

BinomialLn() public static method

Computes the natural logarithm of the binomial coefficient: ln(n choose k).

public static BinomialLn ( int n, int k ) : double
n	int	A nonnegative value n.
k	int	A nonnegative value h.
return	double

DiGamma() public static method

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.

public static DiGamma ( double x ) : double
x	double	The argument of the digamma function.
return	double

DiGammaInv() public static method

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.

public static DiGammaInv ( double p ) : double
p	double	The argument of the inverse digamma function.
return	double

Erf() public static method

Calculates the error function.

returns 1 if x == Double.PositiveInfinity. returns -1 if x == Double.NegativeInfinity.

public static Erf ( double x ) : double
x	double	The value to evaluate.
return	double

ErfInv() public static method

Calculates the inverse error function evaluated at z. Calculates the inverse error function evaluated at z.

returns Double.PositiveInfinity if z >= 1.0. returns Double.NegativeInfinity if z <= -1.0.

public static ErfInv ( double z ) : double
z	double	value to evaluate.
return	double

Erfc() public static method

Calculates the complementary error function.

returns 0 if x == Double.PositiveInfinity. returns 2 if x == Double.NegativeInfinity.

public static Erfc ( double x ) : double
x	double	The value to evaluate.
return	double

ErfcInv() public static method

Calculates the complementary inverse error function evaluated at z. calculates the complementary inverse error function evaluated at z.

We have tested this implementation against the arbitrary precision mpmath library and found cases where we can only guarantee 9 significant figures correct. returns Double.PositiveInfinity if z <= 0.0. returns Double.NegativeInfinity if z >= 2.0.

public static ErfcInv ( double z ) : double
z	double	value to evaluate.
return	double

ExponentialMinusOne() public static method

Numerically stable exponential minus one, i.e. x -> exp(x)-1

public static ExponentialMinusOne ( double power ) : double
power	double	A number specifying a power.
return	double

Factorial() public static method

Computes the factorial function x -> x! of an integer number > 0. The function can represent all number up to 22! exactly, all numbers up to 170! using a double representation. All larger values will overflow.

If you need to multiply or divide various such factorials, consider using the logarithmic version FactorialLn instead so you can add instead of multiply and subtract instead of divide, and then exponentiate the result using System.Math.Exp. This will also circumvent the problem that factorials become very large even for small parameters.

public static Factorial ( int x ) : double
x	int
return	double

FactorialLn() public static method

Computes the logarithmic factorial function x -> ln(x!) of an integer number > 0.

public static FactorialLn ( int x ) : double
x	int
return	double

GeneralHarmonic() public static method

Compute the generalized harmonic number of order n of m. (1 + 1/2^m + 1/3^m + ... + 1/n^m)

public static GeneralHarmonic ( int n, double m ) : double
n	int	The order parameter.
m	double	The power parameter.
return	double

Harmonic() public static method

Computes the t'th Harmonic number.

public static Harmonic ( int t ) : double
t	int	The Harmonic number which needs to be computed.
return	double

Hypotenuse() public static method

Numerically stable hypotenuse of a right angle triangle, i.e. (a,b) -> sqrt(a^2 + b^2)

public static Hypotenuse ( System.Complex a, System.Complex b ) : System.Complex
a	System.Complex	The length of side a of the triangle.
b	System.Complex	The length of side b of the triangle.
return	System.Complex

Hypotenuse() public static method

Numerically stable hypotenuse of a right angle triangle, i.e. (a,b) -> sqrt(a^2 + b^2)

public static Hypotenuse ( double a, double b ) : double
a	double	The length of side a of the triangle.
b	double	The length of side b of the triangle.
return	double

Hypotenuse() public static method

Numerically stable hypotenuse of a right angle triangle, i.e. (a,b) -> sqrt(a^2 + b^2)

public static Hypotenuse ( float a, float b ) : float
a	float	The length of side a of the triangle.
b	float	The length of side b of the triangle.
return	float

Logistic() public static method

Computes the logistic function. see: http://en.wikipedia.org/wiki/Logistic

public static Logistic ( double p ) : double
p	double	The parameter for which to compute the logistic function.
return	double

Logit() public static method

Computes the logit function. see: http://en.wikipedia.org/wiki/Logit

public static Logit ( double p ) : double
p	double	The parameter for which to compute the logit function. This number should be /// between 0 and 1.
return	double

Multinomial() public static method

Computes the multinomial coefficient: n choose n1, n2, n3, ...

if is . If or any of the are negative. If the sum of all is not equal to .

public static Multinomial ( int n, int ni ) : double
n	int	A nonnegative value n.
ni	int	An array of nonnegative values that sum to .
return	double