Bessel_mpfr: Bessel functions of Integer Order in multiple precisions
In Rmpfr: Interface R to MPFR - Multiple Precision Floating-Point Reliable

Bessel_mpfr

R Documentation

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"))

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`.

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.

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

Rmpfr documentation built on Aug. 8, 2025, 7:49 p.m.