gammaincinv

Inverse incomplete gamma function

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Syntax

X = gammaincinv(Y,A)
X = gammaincinv(Y,A,type)

Description

example

X = gammaincinv(Y,A) returns the inverse of the lower incomplete gamma function evaluated at the elements of Y and A, such that Y = gammainc(X,A). Both Y and A must be real. The elements of Y must be in the closed interval [0,1] and A must be nonnegative.

example

X = gammaincinv(Y,A,type) returns the inverse of the lower or upper incomplete gamma function. The choices for type are 'lower' (the default) and 'upper'.

Examples

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Open Live Script

Calculate the inverse of the lower incomplete gamma function for a = 0.5, 1, 1.5, and 2 within the interval 0≤y≤1. Loop over values of a, evaluate the inverse function at each one, and assign each result to a column of X.

A = [0.5 1 1.5 2];
Y = 0:0.005:1;
X = zeros(201,4);
for i = 1:4
    X(:,i) = gammaincinv(Y,A(i));
end

Plot all of the inverse functions in the same figure.

plot(Y,X)
grid on
legend('$a = 0.5$','$a = 1$','$a = 1.5$','$a = 2$','interpreter','latex')
title('Lower inverse incomplete gamma function for $a = 0.5, 1, 1.5,$ and $2$','interpreter','latex')
xlabel('$y$','interpreter','latex')
ylabel('$P^{-1}(y,a)$','interpreter','latex')

Open Live Script

Calculate the inverse of the upper incomplete gamma function for a = 0.5, 1, 1.5, and 2 within the interval 0≤y≤1. Loop over values of a, evaluate the inverse function at each one, and assign each result to a column of X.

A = [0.5 1 1.5 2];
Y = 0:0.005:1;
X = zeros(201,4);
for i = 1:4
    X(:,i) = gammaincinv(Y,A(i),'upper');
end

Plot all of the inverse functions in the same figure.

plot(Y,X)
grid on
legend('$a = 0.5$','$a = 1$','$a = 1.5$','$a = 2$','interpreter','latex')
title('Upper inverse incomplete gamma function for $a = 0.5, 1, 1.5,$ and $2$','interpreter','latex')
xlabel('$y$','interpreter','latex')
ylabel('$Q^{-1}(y,a)$','Interpreter','latex')

Input Arguments

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Input array, specified as a scalar, vector, matrix, or multidimensional array. The elements of Y must be real and within the closed interval [0,1]. Y and A must be the same size, or else one of them must be a scalar.

Data Types: single | double

Input array, specified as a scalar, vector, matrix, or multidimensional array. The elements of A must be real and nonnegative. Y and A must be the same size, or else one of them must be a scalar.

Data Types: single | double

Type of inverse incomplete gamma function, specified as 'lower' or 'upper'. If type is 'lower', then gammainc returns the inverse of the lower incomplete gamma function. If type is 'upper', then gammainc returns the inverse of the upper incomplete gamma function.

More About

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Inverse of Incomplete Gamma Function

The inverse of the lower incomplete gamma function is defined as x=P−1(y,a), such that

y=P(x,a)=1Γ(a)∫0xta−1e−tdt.

The inverse of the upper incomplete gamma function is defined as x=Q−1(y,a), such that

y=Q(x,a)=1Γ(a)∫x∞ta−1e−tdt.

The Γ(a) term is the gamma function

Γ(a)=∫0∞ta−1e−tdt.

MATLAB® uses the normalized definition of the incomplete gamma function, where P(x,a)+Q(x,a)=1.

Some properties of the inverse of the lower incomplete gamma function are:

  • limy→1P−1(y,a)=∞ for a>0

  • limy→1a→0P−1(y,a)=0

Tips

  • When the upper incomplete gamma function is close to 0, specifying the 'upper' option to calculate the upper inverse function is more accurate than subtracting the lower incomplete gamma function from 1 and then taking the lower inverse function.

References

[1] Olver, F. W. J., A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., Chapter 8. Incomplete Gamma and Related Functions, NIST Digital Library of Mathematical Functions, Release 1.0.22, Mar. 15, 2018.

Extended Capabilities

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