# marcumq

Generalized Marcum Q function

## Syntax

```Q = marcumq(a,b) Q = marcumq(a,b,m) ```

## Description

`Q = marcumq(a,b)` computes the Marcum Q function of `a` and `b`, defined by

`$Q\left(a,b\right)=\underset{b}{\overset{\infty }{\int }}x\mathrm{exp}\left(-\frac{\left({x}^{2}+{a}^{2}\right)}{2}\right)\text{\hspace{0.17em}}{I}_{0}\left(ax\right)\text{\hspace{0.17em}}dx$`

where `a` and `b` are nonnegative real numbers. In this expression, I0 is the modified Bessel function of the first kind of zero order.

`Q = marcumq(a,b,m)` computes the generalized Marcum Q, defined by

`$Q\left(a,b\right)=\frac{1}{{a}^{m-1}}\underset{b}{\overset{\infty }{\int }}{x}^{m}\mathrm{exp}\left(-\frac{\left({x}^{2}+{a}^{2}\right)}{2}\right){I}_{m-1}\left(ax\right)\text{\hspace{0.17em}}dx$`

where `a` and `b` are nonnegative real numbers, and `m` is a positive integer. In this expression, Im–1 is the modified Bessel function of the first kind of order m–1.

If any of the inputs is a scalar, it is expanded to the size of the other inputs.

## Algorithms

`marcumq` uses the algorithm developed in . The paper describes two error criteria: a relative error criterion and an absolute error criterion. `marcumq` utilizes the absolute error criterion.

## References

 Cantrell, P. E., and A. K. Ojha, “Comparison of Generalized Q-Function Algorithms,” IEEE® Transactions on Information Theory, Vol. IT-33, July, 1987, pp. 591–596.

 Marcum, J. I., “A Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix,” RAND Corporation, Santa Monica, CA, Research Memorandum RM-753, July 1, 1948. Reprinted in IRE Transactions on Information Theory, Vol. IT-6, April, 1960, pp. 59–267.

 Shnidman, D. A., “The Calculation of the Probability of Detection and the Generalized Marcum Q-Function,” IEEE Transactions on Information Theory, Vol. IT-35, March, 1989, pp. 389–400.