bI {Bessel}R Documentation

Bessel I() function Simple Series Representation


Computes the modified Bessel I function, using one of its basic definitions as an infinite series. The implementation is pure R, working for numeric, complex, but also e.g., for objects of class "mpfr" from package Rmpfr.


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



numeric or complex vector, or of another class for which arithmetic methods are defined, notably objects of class mpfr (package Rmpfr).


non-negative numeric (scalar).


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 chosen too small.


logical indicating if the result should be scaled by exp(-abs(x)).


logical indicating if the logarithm log I.() is required. This allows even more precision than expon.scaled=TRUE in some cases.


a relative error tolerance for checking if nterm has been sufficient. The default is “correct” for double precision and also for multiprecision objects.


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


Martin Maechler


Abramowitz, M., and Stegun, I. A. (1955, etc). Handbook of mathematical functions (NBS AMS series 55, U.S. Dept. of Commerce).

See Also

This package BesselI, base besselI, etc


(nus <- c(outer((0:3)/4, 1:5, `+`)))
  all.equal(besselIs(1:10, 1), # our R code
            besselI (1:10, 1)) # internal C code w/ different algorithm
  sapply(nus, function(nu)
   all.equal(besselIs(1:10, nu, expon.scale=TRUE), # our R code
             BesselI (1:10, nu, expon.scale=TRUE)) # TOMS644 code
  ## complex argument [gives warnings  'nterm=800' may be too small]
  sapply(nus, function(nu)
   all.equal(besselIs((1:10)*(1+1i), nu, expon.scale=TRUE), # our R code
             BesselI ((1:10)*(1+1i), nu, expon.scale=TRUE)) # TOMS644 code

## Large 'nu' ...
x <- (0:20)/4
(bx <- besselI(x, nu=200))# base R's -- gives (mostly wrong) warnings
if(require("Rmpfr")) { ## Use high precision (notably large exponent range) numbers:
  Bx <- besselIs(mpfr(x, 64), nu=200)
  all.equal(Bx, bx, tol = 1e-15)# TRUE -- warning 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 <- besselIs(     x,      nu=200, log=TRUE))
  lBx <-  besselIs(mpfr(x, 64), nu=200, log=TRUE)
  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]