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From "Christian Winter (JIRA)" <j...@apache.org>
Subject [jira] [Commented] (MATH-364) Make Erf more precise in the tails by providing erfc
Date Tue, 16 Aug 2011 22:09:32 GMT

    [ https://issues.apache.org/jira/browse/MATH-364?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13086014#comment-13086014
] 

Christian Winter commented on MATH-364:
---------------------------------------

Here's the new suggestion:
{code}
    public static double erf(double x1, double x2) {
        if(x1 > x2) {
            return -erf(x2, x1);
        }
        /**
         * The number {@code x_crit} solves {@code erf(x)=0.5} within 1ulp.
         * More precisely, the current implementations of
         * {@link #erf(double)} and {@link #erfc(double)} satisfy:<br/>
         * {@code erf(x_crit) < 0.5},<br/>
         * {@code erf(Math.nextUp(x_crit) > 0.5},<br/>
         * {@code erfc(x_crit) = 0.5}, and<br/>
         * {@code erfc(Math.nextUp(x_crit) < 0.5}
         */
        double x_crit = 0.4769362762044697;
        
        return
        x1 < -x_crit && x2 < 0.0 ?
            erfc(-x2) - erfc(-x1)
        : x2 > x_crit && x1 > 0.0 ?
            erfc(x1) - erfc(x2)
        :
            erf(x2) - erf(x1);
    }
{code}
Following the idea to keep numbers small during calculation, the stragegies for {{x1,x2 <
-x_crit}},&emsp;{{x1,x2 > x_crit}}, and {{|x1|,|x2| &le; x_crit}} are straightforward
and mandatory. In the other cases, numbers &ge; 0.5 cannot be avoided and there is some
freedom of choice. The suggested code avoids number &ge; 1 where the final result is &lt;
1.

> Make Erf more precise in the tails by providing erfc
> ----------------------------------------------------
>
>                 Key: MATH-364
>                 URL: https://issues.apache.org/jira/browse/MATH-364
>             Project: Commons Math
>          Issue Type: Improvement
>    Affects Versions: 1.1, 1.2, 2.0, 2.1
>            Reporter: Christian Winter
>            Priority: Minor
>             Fix For: 3.0
>
>
> 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|https://issues.apache.org/jira/browse/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:
> {code:borderStyle=solid}
> /**
>  * 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;
> }
> {code} 
> Further this can be used to improve the NormalDistributionImpl through
> {code:borderStyle=solid}
> @Override
> public double cumulativeProbability(double x0, double x1) throws MathException {
> 	return 0.5 * Erf.erf(
> 			(x0 - getMean()) / (getStandardDeviation() * sqrt2),
> 			(x1 - getMean()) / (getStandardDeviation() * sqrt2) );
> }
> {code}

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