LAL  7.1.7.1-37cf38b
LALMarcumQ.h File Reference

## Prototypes

double XLALMarcumQmodified (double M, double x, double y)
The modified form of the Marcum Q function. More...

double XLALMarcumQ (double M, double a, double b)
The function defined by J. Marcum,

$Q_{M}(a, b) = \int_{b}^{\infty} x \left( \frac{x}{a} \right)^{M - 1} \exp \left( -\frac{x^{2} + a^{2}}{2} \right) I_{M - 1}(a x) \,\mathrm{d}x,$

where $$I_{M - 1}$$ is the modified Bessel function of order $$M - 1$$. More...

Go to the source code of this file.

## ◆ XLALMarcumQmodified()

 double XLALMarcumQmodified ( double M, double x, double y )

The modified form of the Marcum Q function.

Used by Gil et al. in

A. Gil, J. Segura, and N. M. Temme. Algorithm 939: Computation of the Marcum Q-Function. ACM Transactions on Mathematical Software (TOMS), Volume 40 Issue 3, April 2014, Article No. 20. arXiv:1311.0681

The relationship between this function and the standard Marcum Q function is

 XLALMarcumQmodified(M, x, y) = XLALMarcumQ(M, sqrt(2. * x), sqrt(2. * y)).


The function is defined for $$1 \leq M$$, $$0 \leq x$$, $$0 \leq y$$. Additionally, the implementation here becomes inaccurate when $$M$$, $$x$$, or $$y$$ is $$\geq 10000$$.

Definition at line 626 of file XLALMarcumQ.c.

## ◆ XLALMarcumQ()

 double XLALMarcumQ ( double M, double a, double b )

The function defined by J. Marcum,

$Q_{M}(a, b) = \int_{b}^{\infty} x \left( \frac{x}{a} \right)^{M - 1} \exp \left( -\frac{x^{2} + a^{2}}{2} \right) I_{M - 1}(a x) \,\mathrm{d}x,$

where $$I_{M - 1}$$ is the modified Bessel function of order $$M - 1$$.

The CCDF for the random variable $$x$$ distributed according to the noncentral $$\chi^{2}$$ distribution with $$k$$ degrees-of-freedom and noncentrality parameter $$\lambda$$ is $$Q_{k/2}(\sqrt{\lambda}, \sqrt{x})$$.

The CCDF for the random variable $$x$$ distributed according to the Rice distribution with noncentrality parameter $$\nu$$ and width $$\sigma$$ is $$Q_{1}(\nu/\sigma, x/\sigma)$$.

The probability that a signal that would be seen in a two-phase matched filter with |SNR| $$\rho_{0}$$ is seen to have matched filter |SNR| $$\geq \rho$$ in stationary Gaussian noise is $$Q_{1}(\rho_{0}, \rho)$$.

This function is implemented by computing the modified form used by Gil et al.,

 XLALMarcumQ(M, a, b) = XLALMarcumQmodified(M, a * a / 2., b * b / 2.).

Definition at line 742 of file XLALMarcumQ.c.