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Last data update: 2014.03.03

R: Ratio of Bessel K Functions

Bessel K Ratio

R Documentation

Ratio of Bessel K Functions

Description

Calculates the ratio of Bessel K functions of different orders, but the same value of the argument.

Usage

besselRatio(x, nu, orderDiff, useExpScaled = 700)

Arguments

`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.

Details

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.

Value

The ratio

K_(nu+k)(x)/K_nu(x)

of two modified Bessel functions of the third kind whose orders differ by k.

Author(s)

David Scott d.scott@auckland.ac.nz

Examples

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
compare

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

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> library(DistributionUtils)
Loading required package: RUnit
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/DistributionUtils/besselRatio.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Bessel K Ratio
> ### Title: Ratio of Bessel K Functions
> ### Aliases: besselRatio
> ### Keywords: math
> 
> ### ** Examples
> 
> 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
            [,1]        [,2]         [,3]         [,4]        [,5]        [,6]
[1,]    16.86663    10.75385     7.781137     6.090468    5.024655    4.302128
[2,]    73.48710    39.35525    24.822473    17.380739   13.074196   10.355893
[3,]   222.15162   109.48831    64.098882    41.995961   29.773284   22.368514
[4,]   514.53360   244.54113   137.964988    87.165108   59.673423   43.368998
[5,]   999.42586   467.03943   258.597043   160.171259  107.450155   76.520698
[6,]  1724.70940   798.93325   437.859195   268.144833  177.715991  124.972350
[7,] 10633.28765  4862.90579  2623.886703  1578.095601 1024.973207  705.066531
[8,] 74052.62238 33747.89311 18129.755109 10846.870001 7002.415905 4783.910099
            [,7]        [,8]        [,9]       [,10]       [,11]
[1,]    3.785224    3.399798    3.102843    2.867909    2.677938
[2,]    8.525505    7.230002    6.276125    5.550945    4.984885
[3,]   17.564024   14.275849   11.927487   10.191042    8.869535
[4,]   33.006241   26.053230   21.180509   17.641581   14.993686
[5,]   57.081406   44.185346   35.249374   28.831590   24.082062
[6,]   92.029179   70.317805   55.375514   44.718344   36.886643
[7,]  507.122749  377.998297  290.107405  228.152598  183.181231
[8,] 3414.690278 2524.043094 1919.703872 1495.154973 1188.123170
> scaled
            [,1]        [,2]         [,3]         [,4]        [,5]        [,6]
[1,]    16.86663    10.75385     7.781137     6.090468    5.024655    4.302128
[2,]    73.48710    39.35525    24.822473    17.380739   13.074196   10.355893
[3,]   222.15162   109.48831    64.098882    41.995961   29.773284   22.368514
[4,]   514.53360   244.54113   137.964988    87.165108   59.673423   43.368998
[5,]   999.42586   467.03943   258.597043   160.171259  107.450155   76.520698
[6,]  1724.70940   798.93325   437.859195   268.144833  177.715991  124.972350
[7,] 10633.28765  4862.90579  2623.886703  1578.095601 1024.973207  705.066531
[8,] 74052.62238 33747.89311 18129.755109 10846.870001 7002.415905 4783.910099
            [,7]        [,8]        [,9]       [,10]       [,11]
[1,]    3.785224    3.399798    3.102843    2.867909    2.677938
[2,]    8.525505    7.230002    6.276125    5.550945    4.984885
[3,]   17.564024   14.275849   11.927487   10.191042    8.869535
[4,]   33.006241   26.053230   21.180509   17.641581   14.993686
[5,]   57.081406   44.185346   35.249374   28.831590   24.082062
[6,]   92.029179   70.317805   55.375514   44.718344   36.886643
[7,]  507.122749  377.998297  290.107405  228.152598  183.181231
[8,] 3414.690278 2524.043094 1919.703872 1495.154973 1188.123170
> compare
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
[1,]    1    1    1    1    1    1    1    1    1     1     1
[2,]    1    1    1    1    1    1    1    1    1     1     1
[3,]    1    1    1    1    1    1    1    1    1     1     1
[4,]    1    1    1    1    1    1    1    1    1     1     1
[5,]    1    1    1    1    1    1    1    1    1     1     1
[6,]    1    1    1    1    1    1    1    1    1     1     1
[7,]    1    1    1    1    1    1    1    1    1     1     1
[8,]    1    1    1    1    1    1    1    1    1     1     1
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>