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# 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

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

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

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

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.

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### Algorithms

marcumq uses the algorithm developed in [3]. The paper describes two error criteria: a relative error criterion and an absolute error criterion. marcumq utilizes the absolute error criterion.

## References

[1] 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.

[2] 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.

[3] 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.