Ratio of Bessel K Functions
Calculates the ratio of Bessel K functions of different orders, but the same value of the argument.
besselRatio(x, nu, orderDiff, useExpScaled = 700)
x |
Numeric, >= 0. Value at which the numerator and denominator Bessel functions are evaluated. |
nu |
Numeric. The order of the Bessel function in the denominator. |
orderDiff |
Numeric. The order of the numerator Bessel function minus the order of the denominator Bessel function. |
useExpScaled |
Numeric, >= 0. The smallest value of x for which the ratio is calculated using the exponentially-scaled Bessel function values. |
Uses the function besselK to calculate the ratio of two
modified Bessel function of the third kind whose orders are
different. The calculation of Bessel functions will underflow if the
value of x is greater than around 740. To avoid underflow the
exponentially-scaled Bessel functions can be returned by
besselK. The ratio is actually unaffected by exponential
scaling since the scaling cancels across numerator and denominator.
The Bessel function ratio is useful in calculating moments of the generalized inverse Gaussian distribution, and hence also for the moments of the hyperbolic and generalized hyperbolic distributions.
The ratio
K_(nu+k)(x)/K_nu(x)
of two modified Bessel functions of the third kind whose orders differ by k.
David Scott d.scott@auckland.ac.nz
nus <- c(0:5, 10, 20)
x <- seq(1, 4, length.out = 11)
k <- 3
raw <- matrix(nrow = length(nus), ncol = length(x))
scaled <- matrix(nrow = length(nus), ncol = length(x))
compare <- matrix(nrow = length(nus), ncol = length(x))
for (i in 1:length(nus)){
for (j in 1:length(x)) {
raw[i,j] <- besselRatio(x[j], nus[i],
orderDiff = k)
scaled[i,j] <- besselRatio(x[j], nus[i],
orderDiff = k, useExpScaled = 1)
compare[i,j] <- raw[i,j]/scaled[i,j]
}
}
raw
scaled
comparePlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.