[MATH-364] Make Erf more precise in the tails by providing erfc

• Type: Improvement
• Status: Closed
• Priority: Minor
• Resolution: Fixed
• Affects Version/s: 1.1, 1.2, 2.0, 2.1
• Fix Version/s: 3.0
• Assignee: Unassigned
• Reporter: Christian Winter
• Created: 13/Apr/10 08:13
• Updated: 24/Mar/12 16:16
• Resolved: 01/Sep/11 01:26

Description

First I want to thank Phil Steitz for making Erf stable in the tails through adjusting the choices in calculating the regularized gamma functions, see Math-282. However, the precision of Erf in the tails is limitted to fixed point precision because of the closeness to +/-1.0, although the Gamma class could provide much more accuracy. Thus I propose to add the methods erfc(double) and erf(double, double) to the class Erf:

/**
 * Returns the complementary error function erfc(x).
 * @param x the value
 * @return the complementary error function erfc(x)
 * @throws MathException if the algorithm fails to converge
 */
public static double erfc(double x) throws MathException {
double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);
	if (x < 0) {
		ret = -ret;
	}
	return ret;
}

/**
 * Returns the difference of the error function values of x1 and x2.
 * @param x1 the first bound
 * @param x2 the second bound
 * @return erf(x2) - erf(x1)
 * @throws MathException
 */
public static double erf(double x1, double x2) throws MathException {
	if(x1>x2)
		return erf(x2, x1);
	if(x1==x2)
		return 0.0;
    	
	double f1 = erf(x1);
	double f2 = erf(x2);
	
	if(f2 > 0.5)
		if(f1 > 0.5)
			return erfc(x1) - erfc(x2);
		else
			return (0.5-erfc(x2)) + (0.5-f1);
	else
		if(f1 < -0.5)
			if(f2 < -0.5)
				return erfc(-x2) - erfc(-x1);
			else
				return (0.5-erfc(-x1)) + (0.5+f2);
		else
			return f2 - f1;
}

Further this can be used to improve the NormalDistributionImpl through

@Override
public double cumulativeProbability(double x0, double x1) throws MathException {
	return 0.5 * Erf.erf(
			(x0 - getMean()) / (getStandardDeviation() * sqrt2),
			(x1 - getMean()) / (getStandardDeviation() * sqrt2) );
}

Issue Links

• blocks: MATH-456 Erf.erf - handle infinities and large values