Rmpfr (version 0.9-5)

Bessel_mpfr: Bessel functions of Integer Order in multiple precisions

Description

Bessel functions of integer orders, provided via arbitrary precision algorithms from the MPFR library.

Note that the computation can be very slow when n and x are large (and of similar magnitude).

Usage

Ai(x)
j0(x)
j1(x)
jn(n, x, rnd.mode = c("N","D","U","Z","A"))
y0(x)
y1(x)
yn(n, x, rnd.mode = c("N","D","U","Z","A"))

Value

Computes multiple precision versions of the Bessel functions of

integer order, \(J_n(x)\) and \(Y_n(x)\), and---when using MPFR library 3.0.0 or newer---also of the Airy function

\(Ai(x)\). Note that currently Ai(x) is very slow to compute for large x.

Arguments

x

a numeric or mpfr vector.

n

non-negative integer (vector).

rnd.mode

a 1-letter string specifying how rounding should happen at C-level conversion to MPFR, see mpfr.

See Also

besselJ, and besselY compute the same bessel functions but for arbitrary real order and only precision of a bit more than ten digits.

Examples

x <- (0:100)/8 # (have exact binary representation)
stopifnot(exprs = {
   all.equal(besselY(x, 0), bY0 <- y0(x))
   all.equal(besselJ(x, 1), bJ1 <- j1(x))
   all.equal(yn(0,x), bY0)
   all.equal(jn(1,x), bJ1)
})

mpfrVersion() # now typically 4.1.0
if(mpfrVersion() >= "3.0.0") { ## Ai() not available previously
  print( aix <- Ai(x) )
  plot(x, aix, log="y", type="l", col=2)
  stopifnot(
    all.equal(Ai (0) , 1/(3^(2/3) * gamma(2/3)))
    , # see https://dlmf.nist.gov/9.2.ii
    all.equal(Ai(100), mpfr("2.6344821520881844895505525695264981561e-291"), tol=1e-37)
  )
  two3rd <- 2/mpfr(3, 144)
  print( all.equal(Ai(0), 1/(3^two3rd * gamma(two3rd)), tol=0) ) # 1.7....e-40
  if(Rmpfr:::doExtras()) withAutoprint({ # slowish:
     system.time(ai1k <- Ai(1000)) # 1.4 sec (on 2017 lynne)
     stopifnot(all.equal(print(log10(ai1k)),
                         -9157.031193409585185582, tol=2e-16)) # seen 8.8..e-17 | 1.1..e-16
  })
} # ver >= 3.0