Generalized Marcum Q function

`Q = marcumq(a,b)`

Q = marcumq(a,b,m)

`Q = marcumq(a,b)`

computes
the Marcum Q function of `a`

and `b`

,
defined by

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

where `a`

and `b`

are nonnegative
real numbers. In this expression, *I*_{0} is
the modified Bessel function of the first kind of zero order.

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

computes
the generalized Marcum Q, defined by

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

where `a`

and `b`

are nonnegative
real numbers, and `m`

is a positive integer. In this
expression, *I*_{m–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.

`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.

[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.