besselJs {Bessel} R Documentation

## Bessel J() function Simple Series Representation

### Description

Computes the modified Bessel J function, using one of its basic definitions as an infinite series, e.g. A. & S., p.360, (9.1.10). The implementation is pure R, working for numeric, complex, but also e.g., for objects of class "mpfr" from package Rmpfr.

### Usage

besselJs(x, nu, nterm = 800, log = FALSE,
Ceps = if (isNum) 8e-16 else 2^(-x@.Data[[1]]@prec))


### Arguments

 x numeric or complex vector, or of another class for which arithmetic methods are defined, notably objects of class mpfr. nu non-negative numeric (scalar). nterm integer indicating the number of terms to be used. Should be in the order of abs(x), but can be smaller for large x. A warning is given, when nterm was possibly too small. (Currently, many of these warnings are wrong, as log logical indicating if the logarithm log J.() is required. Ceps a relative error tolerance for checking if nterm has been sufficient. The default is “correct” for double precision and also for multiprecision objects.

### Value

a “numeric” (or complex or "mpfr") vector of the same class and length as x.

Martin Maechler

### References

Abramowitz, M., and Stegun, I. A. (1955, etc). Handbook of mathematical functions (NBS AMS series 55, U.S. Dept. of Commerce). http://people.math.sfu.ca/~cbm/aands/page_360.htm

This package BesselJ, base besselJ, etc

### Examples

stopifnot(all.equal(besselJs(1:10, 1), # our R code
besselJ (1:10, 1)))# internal C code w/ different algorithm

## Large 'nu' ...
x <- (0:20)/4
(bx <- besselJ(x, nu=200))# base R's -- gives (mostly wrong) warnings
if(require("Rmpfr")) { ## Use high precision, notably large exponent range, numbers:
Bx <- besselJs(mpfr(x, 64), nu=200)
all.equal(Bx, bx, tol = 1e-15)# TRUE -- warnings were mostly wrong; specifically:
cbind(bx, Bx)
signif(asNumeric(1 - (bx/Bx)[19:21]), 4) # only [19] had lost accuracy

## With*out* mpfr numbers -- using log -- is accurate (here)
lbx <- besselJs(     x,      nu=200, log=TRUE)
lBx <- besselJs(mpfr(x, 64), nu=200, log=TRUE)
cbind(x, lbx, lBx)
stopifnot(all.equal(asNumeric(log(Bx)), lbx, tol=1e-15),
all.equal(lBx, lbx, tol=4e-16))
} # Rmpfr


[Package Bessel version 0.6-0 Index]