S18ATF (PDF version)
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S Chapter Introduction
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

S18ATF returns an array of values for the modified Bessel function I1x.

2  Specification

REAL (KIND=nag_wp)  X(N), F(N)

3  Description

S18ATF evaluates an approximation to the modified Bessel function of the first kind I1xi for an array of arguments xi, for i=1,2,,n.
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:     N – INTEGERInput
On entry: n, the number of points.
Constraint: N0.
2:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the argument xi of the function, for i=1,2,,N.
3:     F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: I1xi, the function values.
4:     IVALID(N) – INTEGER arrayOutput
On exit: IVALIDi contains the error code for xi, for i=1,2,,N.
No error.
xi is too large. Fi contains the approximate value of I1xi at the nearest valid argument. The threshold value is the same as for IFAIL=1 in S18AFF, as defined in the Users' Note for your implementation.
5:     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:
On entry, at least one value of X was invalid.
Check IVALID for more information.
On entry, N=value.
Constraint: N0.

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, for quite moderate values of x (x>x^, the threshold value), the routine must fail because I1x 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


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 (s18atfe.f90)

9.2  Program Data

Program Data (s18atfe.d)

9.3  Program Results

Program Results (s18atfe.r)

S18ATF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

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