NAG Library Function Document

nag_bessel_i1 (s18afc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_bessel_i1 (s18afc) returns a value for the modified Bessel function

I_{1} (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_bessel_i1 (double x, NagError *fail)

3 Description

nag_bessel_i1 (s18afc) evaluates an approximation to the modified Bessel function of the first kind

I_{1} (x)

Note:

I_{1} (- x) = - I_{1} (x)

, so the approximation need only consider

x \geq 0

The function is based on three Chebyshev expansions:

For

0 < x \leq 4

I_{1} (x) = x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t),   where ​ t = 2 {(\frac{x}{4})}^{2} - 1;

For

4 < x \leq 12

I_{1} (x) = e^{x} \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   where ​ t = \frac{x - 8}{4};

For

x > 12

I_{1} (x) = \frac{e^{x}}{\sqrt{x}} \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t),   where ​ t = 2 (\frac{12}{x}) - 1 .

For small

x

I_{1} (x) ≃ x

. 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

I_{1} (x)

cannot be represented without overflow.

4 References

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

5 Arguments

1: $x$ – doubleInput: On entry: the argument $x$ of the function.
2: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL_ARG_GT: On entry, $x = 〈value〉$ .
Constraint: $|x| \leq 〈value〉$ .
$|x|$ is too large and the function returns the approximate value of $I_{1} (x)$ at the nearest valid argument.

7 Accuracy

Let

δ

and

ε

be the relative errors in the argument and result respectively.

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 ≃ |\frac{x I_{0} (x) - I_{1} (x)}{I_{1} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x I_{0} (x) - I_{1} (x)}{I_{1} (x)}| .

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

ε ≃ δ

and there is no amplification of errors.

For large

x

ε ≃ x δ

and we have strong amplification of errors. However the function must fail for quite moderate values of

x

because

I_{1} (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 math library function exp.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

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.

NAG Library Function Documentnag_bessel_i1 (s18afc)