Method digamma

JSAT/src/jsat/math/FastMath.java:205–224

Computes the digamma function of the input @param x the input value @return ψ(x)

(double x)

     * @return ψ(x)
     */
    public static double digamma(double x)
    {
        if(x == 0)
            return Double.NaN;//complex infinity
        else if(x < 0)//digamma(1-x) == digamma(x)+pi/tan(pi*x), to make x positive
        {
            if(Math.rint(x) == x)
                return Double.NaN;//the zeros are complex infinity
            return digamma(1-x)-PI/tan(PI*x); 
        }
        
        /*
         * shift over 2 values to the left and use truncated approximation 
         * log(x+2)-1/(2 (x+2))-1/(12 (x+2)^2) -1/x-1/(x+1), 
         * the x+2 and x and x+1 are grouped sepratly 
         */
        double xp2 = x+2;
        
        return log(xp2)-(6*x+13)/(12*xp2*xp2)-(2*x+1)/(x*x+x);
    }
}