NAG Library Function Document

nag_bessel_i0_vector (s18asc)

▸▿ 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_i0_vector (s18asc) returns an array of values of the modified Bessel function

I_{0} (x)

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

I_{0} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

I_{0} (- x) = I_{0} (x)

, so the approximation need only consider

x \geq 0

The function is based on three Chebyshev expansions:

For

0 < x \leq 4

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

For

4 < x \leq 12

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

For

x > 12

I_{0} (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_{0} (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

e^{x}

4 References

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

5 Arguments

1: $n$ – IntegerInput

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $x [n]$ – const doubleInput

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

3: $f [n]$ – doubleOutput

On exit:

I_{0} (x_{i})

, the function values.

4: $ivalid [n]$ – IntegerOutput

On exit:

ivalid [i - 1]

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $x_{i}$ is too large. $f [i - 1]$ contains the approximate value of $I_{0} (x_{i})$ 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: $fail$ – 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_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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.
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.

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_{1} (x)}{I_{0} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

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

the amplification factor is approximately

\frac{x^{2}}{2}

, which implies strong attenuation of the error, but in general

ε

can never be less than the machine precision.

For large

x

ε ≃ x δ

and we have strong amplification of errors. However, for quite moderate values of

x

(

x > \hat{x}

, the threshold value), the function must fail because

I_{0} (x)

would overflow; hence in practice the loss of accuracy for

x

close to

\hat{x}

is not excessive and the errors will be dominated by those of the standard function exp.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

NAG Library Function Documentnag_bessel_i0_vector (s18asc)