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gammaincinv

Inverse incomplete gamma function

Syntax

x = gammaincinv(y,a)
x = gammaincinv(y,a,tail)

Description

x = gammaincinv(y,a)evaluates the inverse incomplete gamma function for corresponding elements ofyanda, such thaty = gammainc(x,a). The elements ofymust be in the closed interval[0,1], and those ofamust be nonnegative.yandamust be real and the same size (or either can be a scalar).

x = gammaincinv(y,a,tail)specifies the tail of the incomplete gamma function. Choices are'lower'(the default) to use the integral from 0 tox, or'upper'to use the integral fromxto infinity.

These two choices are related as:

gammaincinv(y,a,'upper') = gammaincinv(1-y,a,'lower').

Whenyis close to 0, the'upper'option provides a way to computexmore accurately than by subtractingyfrom 1.

More About

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Inverse Incomplete Gamma Function

The lower incomplete gamma function is defined as:

gammainc ( x,a ) = 1 Γ ( a ) ∫ 0 x t a − 1 e − t d t

where Γ ( a ) is the gamma function,gamma(a). The upper incomplete gamma function is defined as:

gammainc ( x,a,'upper' ) = 1 Γ ( a ) ∫ x ∞ t a − 1 e − t d t

gammaincinvcomputes the inverse of the incomplete gamma function with respect to the integration limitxusing Newton's method.

For anya>0, asyapproaches 1,gammaincinv(y,a)approaches infinity. For smallxanda,gammainc(x,a) ≅ x a , sogammaincinv (1,0) = 0.

References

[1] Cody, J., An Overview of Software Development for Special Functions, Lecture Notes in Mathematics, 506, Numerical Analysis Dundee, G. A. Watson (ed.), Springer Verlag, Berlin, 1976.

[2] Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sec. 6.5.

Extended Capabilities

See Also

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