nag_bessel_i0_vector (s18asc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_bessel_i0_vector (s18asc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_bessel_i0_vector (s18asc) returns an array of values of the modified Bessel function I0x.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_bessel_i0_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_bessel_i0_vector (s18asc) evaluates an approximation to the modified Bessel function of the first kind I0xi for an array of arguments xi, for i=1,2,,n.
Note:  I0-x=I0x, so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x4,
I0x=exr=0arTrt,   where ​ t=2 x4 -1.
For 4<x12,
I0x=exr=0brTrt,   where ​ t=x-84.
For x>12,
I0x=exx r=0crTrt,   where ​ t=2 12x -1.
For small x, I0x1. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, the function must fail because of the danger of overflow in calculating ex.

4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5  Arguments

1:     nIntegerInput
On entry: n, the number of points.
Constraint: n0.
2:     x[n]const doubleInput
On entry: the argument xi of the function, for i=1,2,,n.
3:     f[n]doubleOutput
On exit: I0xi, the function values.
4:     ivalid[n]IntegerOutput
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
xi is too large. f[i-1] contains the approximate value of I0xi at the nearest valid argument. The threshold value is the same as for fail.code= NE_REAL_ARG_GT in nag_bessel_i0 (s18aec), as defined in the Users' Note for your implementation.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NW_IVALID
On entry, at least one value of x was invalid.
Check ivalid for more information.

7  Accuracy

Let δ and ε be the relative errors in the argument and result respectively.
If δ is somewhat larger than the machine precision (i.e., if δ is due to data errors etc.), then ε and δ are approximately related by:
ε x I1x I0 x δ.
Figure 1 shows the behaviour of the error amplification factor
xI1x I0x .
Figure 1
Figure 1
However if δ is of the same order as machine precision, then rounding errors could make ε slightly larger than the above relation predicts.
For small x the amplification factor is approximately x22 , which implies strong attenuation of the error, but in general ε can never be less than the machine precision.
For large x, εxδ and we have strong amplification of errors. However, for quite moderate values of x (x>x^, the threshold value), the function must fail because I0x would overflow; hence in practice the loss of accuracy for x close to x^ is not excessive and the errors will be dominated by those of the standard function exp.

8  Further Comments

None.

9  Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.

9.1  Program Text

Program Text (s18asce.c)

9.2  Program Data

Program Data (s18asce.d)

9.3  Program Results

Program Results (s18asce.r)


nag_bessel_i0_vector (s18asc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012