bI {Bessel} | R Documentation |

## Bessel I() function Simple Series Representation

### Description

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.

### Usage

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

### Arguments

`x` |
numeric or complex vector, or of another |

`nu` |
non-negative numeric (scalar). |

`nterm` |
integer indicating the number of terms to be used.
Should be in the order of |

`expon.scaled` |
logical indicating if the result should be scaled
by |

`log` |
logical indicating if the logarithm |

`Ceps` |
a relative error tolerance for checking if |

### Value

a “numeric” (or complex or `"mpfr"`

)
vector of the same class and length as `x`

.

### Author(s)

Martin Maechler

### References

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

### Examples

```
(nus <- c(outer((0:3)/4, 1:5, `+`)))
stopifnot(
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
```

*Bessel*version 0.6-0 Index]