NAG CL Interface
s18aec (bessel_​i0_​real)

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1 Purpose

s18aec returns the value of the modified Bessel function I0(x).

2 Specification

#include <nag.h>
double  s18aec (double x, NagError *fail)
The function may be called by the names: s18aec, nag_specfun_bessel_i0_real or nag_bessel_i0.

3 Description

s18aec evaluates an approximation to the modified Bessel function of the first kind I0(x).
Note:  I0(-x)=I0(x), so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x4,
I0(x)=exr=0arTr(t),   where ​ t=2 (x4) -1.  
For 4<x12,
I0(x)=exr=0brTr(t),   where ​ t=x-84.  
For x>12,
I0(x)=exx r=0crTr(t),   where ​ t=2 (12x) -1.  
For small x, I0(x)1. 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

NIST Digital Library of Mathematical Functions

5 Arguments

1: x double Input
On entry: the argument x of the function.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, x=value.
Constraint: |x|value.
|x| is too large and the function returns the approximate value of I0(x) at the nearest valid argument.

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 I1(x) I0 (x) |δ.  
Figure 1 shows the behaviour of the error amplification factor
| xI1(x) I0(x) |.  
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, the function must fail for quite moderate values of x, because I0(x) would overflow; hence in practice the loss of accuracy for large x is not excessive. Note that for large x the errors will be dominated by those of the standard function exp.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s18aec is not threaded in any implementation.

9 Further Comments


10 Example

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

10.1 Program Text

Program Text (s18aece.c)

10.2 Program Data

Program Data (s18aece.d)

10.3 Program Results

Program Results (s18aece.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 3.5 4 I0(x) x "s18aefe.r" Example Program Returned Values for the Bessel Function I0(x)