NAG Library Routine Document

S18CRF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S18CRF returns an array of values of the scaled modified Bessel function

e^{x} K_{1} (x)

2 Specification

SUBROUTINE S18CRF (

N, X, F, IVALID, IFAIL)

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

3 Description

S18CRF evaluates an approximation to

e^{x_{i}} K_{1} (x_{i})

, where

K_{1}

is a modified Bessel function of the second kind for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

. The scaling factor

e^{x}

removes most of the variation in

K_{1} (x)

The routine uses the same Chebyshev expansions as S18ARF, which returns an array of the unscaled values of

K_{1} (x)

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

Constraint:

X (i) > 0.0

, for

i = 1, 2, \dots, N

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

On exit:

e^{x_{i}} K_{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.

$IVALID (i) = 1$

On entry,

x_{i} \leq 0.0

K_{1} (x_{i})

is undefined.

F (i)

contains

0.0

$IVALID (i) = 2$

x_{i}

is too close to zero, as determined by the value of the safe-range parameter X02AMF: there is a danger of causing overflow.

F (i)

contains the reciprocal of the safe-range parameter.

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: $N \geq 0$ .

7 Accuracy

Relative errors in the argument are attenuated when propagated into the function value. When the accuracy of the argument is essentially limited by the machine precision, the accuracy of the function value will be similarly limited by at most a small multiple of the machine precision.

8 Further Comments

None.

9 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 DocumentS18CRF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S18CRF