S18AFF (PDF version)
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NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

S18AFF returns a value for the modified Bessel function I1x, via the function name.

2  Specification

REAL (KIND=nag_wp) S18AFF
REAL (KIND=nag_wp)  X

3  Description

S18AFF evaluates an approximation to the modified Bessel function of the first kind I1x.
Note:  I1-x=-I1x, so the approximation need only consider x0.
The routine is based on three Chebyshev expansions:
For 0<x4,
I1x=xr=0arTrt,   where ​t=2 x4 2-1;
For 4<x12,
I1x=exr=0brTrt,   where ​t=x-84;
For x>12,
I1x=exx r=0crTrt,   where ​t=2 12x -1.
For small x, I1xx. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, the routine must fail because I1x cannot be represented without overflow.

4  References

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

5  Parameters

1:     X – REAL (KIND=nag_wp)Input
On entry: the argument x of the function.
2:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
X is too large. On soft failure the routine returns the approximate value of I1x at the nearest valid argument. See also the Users' Note for your implementation.

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 xI0x- I1x I1 x δ.
Figure 1 shows the behaviour of the error amplification factor
xI0x - I1x I1x .
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, εδ and there is no amplification of errors.
For large x, εxδ and we have strong amplification of errors. However the routine must fail for quite moderate values of x because I1x 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  Further Comments


9  Example

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

9.1  Program Text

Program Text (s18affe.f90)

9.2  Program Data

Program Data (s18affe.d)

9.3  Program Results

Program Results (s18affe.r)

S18AFF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

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