Alex Herbert created NUMBERS-169:
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Summary: Function to evaluate the complement of the RegularizedBeta
Key: NUMBERS-169
URL: https://issues.apache.org/jira/browse/NUMBERS-169
Project: Commons Numbers
Issue Type: New Feature
Components: gamma
Affects Versions: 1.0
Reporter: Alex Herbert
The RegularizedBeta class computes the regularized beta function I(x, a, b).
There is a functional identity for this:
{code:java}
1 - I(x, a, b) = I(1 - x, b, a)
{code}
See [Wolfram Regularized Incomplete Beta
Identities|https://functions.wolfram.com/06.21.17.0008.01]
The RegularizedBeta class currently exploits this identity to compute the value
by switching to call '1.0 - value(1 - x, b, a)' depending on the parameters.
I suggest a complement for this function to compute 1 - I(x, a, b):
{code:java}
public class RegularizedBetac {
public static double value(double x,
final double a,
final double b,
double epsilon,
int maxIterations);
}
{code}
This can call RegularizedBeta.value using the same inversion logic and avoid a
potential computation of the complement using multiple inversions with loss of
precision:
{noformat}
1 - I(x, a, b) == 1 - [ 1 - I(1 - x, b, a) ] = I(1 - x, b, a)
or
I(1 - x, b, a) == 1 - I(x, a, b) = 1 - [ 1 - I(1 - x, b, a) ]
{noformat}
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