LAL  6.21.0.1-1696c80
XLALMarcumQ.c File Reference

Prototypes

static double Q (double mu, double x)
 
static double c_mu (double mu, double xi)
 
static double lnA (int n, double mu)
 
static double half_zeta_2 (double x, double y)
 
static double zeta (double x, double y)
 
static double theta_over_sin_theta (double theta)
 
static double theta_over_sin_theta_primed_sin_theta (double theta)
 
static double rho (double theta_over_sin_theta_, double xi)
 
static double r (double theta, double y, double xi)
 
static double r_primed_sin_theta (double theta, double y, double xi)
 
static double f (double theta, double y, double xi)
 
static double psi (double theta, double xi)
 
static void f1_f2 (double x, double M, double *f1, double *f2)
 
static double MarcumQ_small_x (double M, double x, double y)
 
static double MarcumQ_large_xy (double M, double x, double y, double xi)
 
static double MarcumQ_recurrence (double M, double x, double y, double xi)
 
static double MarcumQ_large_M (double M, double x, double y)
 
static double integrand (double theta, void *integrand_params)
 
static double MarcumQ_quadrature (double M, double x, double y, double xi)
 
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.

Data Structures

struct  integrand_params
 

Function Documentation

◆ Q()

static double Q ( double  mu,
double  x 
)
static

Definition at line 99 of file XLALMarcumQ.c.

◆ c_mu()

static double c_mu ( double  mu,
double  xi 
)
static

Definition at line 117 of file XLALMarcumQ.c.

◆ lnA()

static double lnA ( int  n,
double  mu 
)
static

Definition at line 128 of file XLALMarcumQ.c.

◆ half_zeta_2()

static double half_zeta_2 ( double  x,
double  y 
)
static

Definition at line 141 of file XLALMarcumQ.c.

◆ zeta()

static double zeta ( double  x,
double  y 
)
static

Definition at line 184 of file XLALMarcumQ.c.

◆ theta_over_sin_theta()

static double theta_over_sin_theta ( double  theta)
static

Definition at line 196 of file XLALMarcumQ.c.

◆ theta_over_sin_theta_primed_sin_theta()

static double theta_over_sin_theta_primed_sin_theta ( double  theta)
static

Definition at line 204 of file XLALMarcumQ.c.

◆ rho()

static double rho ( double  theta_over_sin_theta_,
double  xi 
)
static

Definition at line 222 of file XLALMarcumQ.c.

◆ r()

static double r ( double  theta,
double  y,
double  xi 
)
static

Definition at line 228 of file XLALMarcumQ.c.

◆ r_primed_sin_theta()

static double r_primed_sin_theta ( double  theta,
double  y,
double  xi 
)
static

Definition at line 237 of file XLALMarcumQ.c.

◆ f()

static double f ( double  theta,
double  y,
double  xi 
)
static

Definition at line 258 of file XLALMarcumQ.c.

◆ psi()

static double psi ( double  theta,
double  xi 
)
static

Definition at line 273 of file XLALMarcumQ.c.

◆ f1_f2()

static void f1_f2 ( double  x,
double  M,
double *  f1,
double *  f2 
)
static

Definition at line 288 of file XLALMarcumQ.c.

◆ MarcumQ_small_x()

static double MarcumQ_small_x ( double  M,
double  x,
double  y 
)
static

Definition at line 311 of file XLALMarcumQ.c.

◆ MarcumQ_large_xy()

static double MarcumQ_large_xy ( double  M,
double  x,
double  y,
double  xi 
)
static

Definition at line 336 of file XLALMarcumQ.c.

◆ MarcumQ_recurrence()

static double MarcumQ_recurrence ( double  M,
double  x,
double  y,
double  xi 
)
static

Definition at line 455 of file XLALMarcumQ.c.

◆ MarcumQ_large_M()

static double MarcumQ_large_M ( double  M,
double  x,
double  y 
)
static

Definition at line 490 of file XLALMarcumQ.c.

◆ integrand()

static double integrand ( double  theta,
void *  integrand_params 
)
static

Definition at line 557 of file XLALMarcumQ.c.

◆ MarcumQ_quadrature()

static double MarcumQ_quadrature ( double  M,
double  x,
double  y,
double  xi 
)
static

Definition at line 569 of file XLALMarcumQ.c.

◆ 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.