Alex Herbert created NUMBERS-177:
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Summary: Scaled complementary error function
Key: NUMBERS-177
URL: https://issues.apache.org/jira/browse/NUMBERS-177
Project: Commons Numbers
Issue Type: New Feature
Components: gamma
Affects Versions: 1.1
Reporter: Alex Herbert
Fix For: 1.1
Add a scaled complementary error function:
{noformat}
erfcx(x) = exp(x^2) * erfc(x)
{noformat}
For small x this can use the current rational function approximations for erf
or erfc and remove the scaling down by exp(-x^2). For large x this can use the
[continued fraction expansion of
erfc|https://en.wikipedia.org/wiki/Error_function#Continued_fraction_expansion]
and remove the scaling by e^-z:
{noformat}
z 1
erfc z = -------- e^-z^2 -----------------------------
sqrt(pi) z^2 + 1/2
-----------------------
1 + 1
-------------------
z^2 + 3/2
-------------
1 + 2
---------
z^2 + ...
{noformat}
At very large x the rational function cannot be evaluated as {{z^2 + 0.5 ~
z^2}} and the following approximation can be used:
{noformat}
1 1
erfcx z = -------- * -
sqrt(pi) z
{noformat}
This occurs at 2^26 or approximately 6.71e7.
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