Gamma inverse cumulative distribution function

`X = gaminv(P,A,B)`

[X,XLO,XUP] = gaminv(P,A,B,pcov,alpha)

`X = gaminv(P,A,B)`

computes the inverse of the gamma
cdf with shape parameters in `A`

and scale parameters in `B`

for the corresponding probabilities in `P`

. `P`

,
`A`

, and `B`

can be vectors, matrices, or multidimensional
arrays that all have the same size. A scalar input is expanded to a constant array with the
same dimensions as the other inputs. The parameters in `A`

and
`B`

must all be positive, and the values in `P`

must lie
on the interval `[0 1]`

.

The gamma inverse function in terms of the gamma cdf is

$$x={F}^{-1}(p|a,b)=\{x:F(x|a,b)=p\}$$

where

$$p=F(x|a,b)=\frac{1}{{b}^{a}\Gamma (a)}{\displaystyle \underset{0}{\overset{x}{\int}}{t}^{a-1}{e}^{\frac{-t}{b}}dt}$$

`[X,XLO,XUP] = gaminv(P,A,B,pcov,alpha)`

produces confidence bounds for
`X`

when the input parameters `A`

and `B`

are estimates. `pcov`

is a 2-by-2 matrix containing the covariance matrix of
the estimated parameters. `alpha`

has a default value of 0.05, and specifies
`100(1-alpha)`

% confidence bounds. `XLO`

and
`XUP`

are arrays of the same size as `X`

containing the
lower and upper confidence bounds.

This example shows the relationship between the gamma cdf and its inverse function.

a = 1:5; b = 6:10; x = gaminv(gamcdf(1:5,a,b),a,b) x = 1.0000 2.0000 3.0000 4.0000 5.0000

There is no known analytical solution to the integral equation
above. `gaminv`

uses an iterative approach (Newton's
method) to converge on the solution.