INTERFACE SpecialFunction;
* The interface is Public Domain. The supporting implementations are copyrighted, but may be used free of charge so long as appropriate credit is given.

WARNING: USE AT YOUR OWN RISK. The authors accept no responsibility for the accuracy, appropriateness or fitness for use of any of this material.

Abstract: This is a Modula-3 rendition of a collection of numerical analysis routines.

12/13/95 Harry George Initial version 1/22/96 Harry George Change to m3na project 2/17/96 Harry George Convert to separate Real* modules

FROM Arithmetic IMPORT Error;

TYPE T = LONGREAL;               (* IEEE 64-bit real *)
---- Really special functions ----

PROCEDURE Factorial (n: CARDINAL; ): T;
n! as a real

PROCEDURE LnFactorial (n: CARDINAL; ): T;
ln(n!) as a real

PROCEDURE Gamma (x: T; ): T;
Euler's Gamma function

PROCEDURE LnGamma (x: T; ): T;

Binomial coefficient for n over k

PROCEDURE GammaP (a, x: T; ): T RAISES {Error};
incomplete Gamma P(a,x)=Gamma(a,x)/Gamma(a)

PROCEDURE GammaQ (a, x: T; ): T RAISES {Error};
incomplete Gamma Q(a,x)=Gamma(a,x)/Gamma(a)
also, Q(a,x)=1-P(a,x) 

Notes for in-lines:

      1. Cumulative Poisson Probability:
        Px(<k)=probability that the number of events will be
        between 0 and k-1 inclusive, given mean=x.
      2. Chi-Square Probability:
        P(X2|df)=probability that observed chi-square should be
        less than X2, given df degrees of freedom.
          P(X2|df)=GammaP(df/2.0,X2/2.0); P(0|df)=0, P(inf|df)=1
        Complementary form:
          Q(X2|df)=GammaQ(df/2.0,X2/2.0); Q(0|df)=1, Q(inf|df)=0

PROCEDURE Erf (x: T; ): T RAISES {Error};
error function of x

PROCEDURE ErfC (x: T; ): T RAISES {Error};

PROCEDURE Beta (x, y: T; ): T;

PROCEDURE BetaI (a, b, x: T; ): T RAISES {Error};
incomplete Beta Ix(a,b)
Notes for in-lines:
      1. Student's t-test distribution for df degrees of freedom is
          A(t|df) = 1.0-BetaI(df/2,1/2,df/(df+t^2))
        In other words, big A means t should probably be smaller
      2. F-test distribution for df1 and df2 degrees of freedom is
          A(F|df1,df2) = BetaI(df1/2,df2/2,df2/(df2+df1*F))
      3. Cumulative binomial probability for event which has
        probability p of occurring in each trial,
        having the event occur k or moe times in n trials is
          P(= BetaI(k,n-k+1,p)

END SpecialFunction.