gammaincinv

Inverse incomplete gamma function

Syntax

X = gammaincinv(Y,A)

X = gammaincinv(Y,A,type)

Description

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.

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

Plot Inverse of Lower Incomplete Gamma Function

Calculate the inverse of the lower incomplete gamma function for $$a$$ = 0.5, 1, 1.5, and 2 within the interval $$0\le y\le 1$$. 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')

Plot Inverse of Upper Incomplete Gamma Function

Calculate the inverse of the upper incomplete gamma function for $$a$$ = 0.5, 1, 1.5, and 2 within the interval $$0\le y\le 1$$. 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

Y — Input array

scalar | vector | matrix | multidimensional array

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

A — Input array

scalar | vector | matrix | multidimensional array

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 — Type of inverse incomplete gamma function

'lower' (default) | 'upper'

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

Inverse of Incomplete Gamma Function

The inverse of the lower incomplete gamma function is defined as $$x=P^{-1}(y,a)$$, such that

$$y=P(x,a)=\frac{1}{\Gamma \left(a\right)}{\displaystyle {\int }_0^x{t}^{a-1}{e}^{-t}dt}.$$

The inverse of the upper incomplete gamma function is defined as $$x=Q^{-1}(y,a)$$, such that

$$y=Q(x,a)=\frac{1}{\Gamma \left(a\right)}{\displaystyle {\int }_x^{\infty }{t}^{a-1}{e}^{-t}dt}.$$

The $$\Gamma \left(a\right)$$ term is the gamma function

$$\Gamma \left(a\right)={\displaystyle {\int }_0^{\infty }{t}^{a-1}{e}^{-t}dt}.$$

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:

• $$\underset{y\to 1}{\lim }{P}^{-1}(y,a)=\infty \text{for}a>0$$

• $$\underset{\begin{array}{l}y\to 1\\ a\to 0\end{array}}{\lim }{P}^{-1}(y,a)=0$$

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

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Output is always complex.

• Strict single-precision calculations are not supported. In the generated code, single-precision inputs produce single-precision outputs. However, variables inside the function might be double-precision.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).