NAG Library Routine Document

S18ATF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

S18ATF returns an array of values for the modified Bessel function

I_{1} (x)

2 Specification

SUBROUTINE S18ATF (

N, X, F, IVALID, IFAIL)

INTEGER	N, IVALID(N), IFAIL
REAL (KIND=nag_wp)	X(N), F(N)

3 Description

S18ATF evaluates an approximation to the modified Bessel function of the first kind

I_{1} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

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

, so the approximation need only consider

x \geq 0

The routine 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 routine 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 Parameters

1: $N$ – INTEGERInput

On entry:

n

, the number of points.

Constraint:

N \geq 0

2: $X (N)$ – REAL (KIND=nag_wp) arrayInput

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, N

3: $F (N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

I_{1} (x_{i})

, the function values.

4: $IVALID (N)$ – INTEGER arrayOutput

On exit:

IVALID (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, N

$IVALID (i) = 0$: No error.
$IVALID (i) = 1$: $x_{i}$ is too large. $F (i)$ contains the approximate value of $I_{1} (x_{i})$ 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

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, at least one value of X was invalid.
Check IVALID for more information.

$IFAIL = 2$: On entry, $N = 〈value〉$ .
Constraint: $N \geq 0$ .

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further 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_{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, for quite moderate values of

x

(

x > \hat{x}

, the threshold value), the routine must fail because

I_{1} (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 Routine DocumentS18ATF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

S18ATF