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: 2.1
Reporter: Christian Winter
Priority: Minor
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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