NAG Library Routine Document

S18AQF

+− 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

S18AQF returns an array of values of the modified Bessel function

K_{0} (x)

2 Specification

SUBROUTINE S18AQF (

N, X, F, IVALID, IFAIL)

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

3 Description

S18AQF evaluates an approximation to the modified Bessel function of the second kind

K_{0} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

K_{0} (x)

is undefined for

x \leq 0

and the routine will fail for such arguments.

The routine is based on five Chebyshev expansions:

For

0 < x \leq 1

K_{0} (x) = - \ln x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t) + \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   where ​ t = 2 x^{2} - 1 .

For

1 < x \leq 2

K_{0} (x) = e^{- x} \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t),   where ​ t = 2 x - 3 .

For

2 < x \leq 4

K_{0} (x) = e^{- x} \underset{r = 0}{\sum^{'}} d_{r} T_{r} (t),   where ​ t = x - 3 .

For

x > 4

K_{0} (x) = \frac{e^{- x}}{\sqrt{x}} \underset{r = 0}{\sum^{'}} e_{r} T_{r} (t), where ​ t = \frac{9 - x}{1 + x} .

For

x

near zero,

K_{0} (x) ≃ - γ - \ln (\frac{x}{2})

, where

γ

denotes Euler's constant. This approximation is used when

x

is sufficiently small for the result to be correct to machine precision.

For large

x

, where there is a danger of underflow due to the smallness of

K_{0}

, the result is set exactly to zero.

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:

K_{0} (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.
$IVALID (i) = 1$: $x_{i} \leq 0.0$ , $K_{0} (x_{i})$ is undefined. $F (i)$ contains $0.0$ .

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

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 K_{1} (x)}{K_{0} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x K_{1} (x)}{K_{0} (x)}| .

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{1}{\ln x}|

, 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 the relative error. Eventually

K_{0}

, which is asymptotically given by

\frac{e^{- x}}{\sqrt{x}}

, becomes so small that it cannot be calculated without underflow and hence the routine will return zero. Note that for large

x

the errors will be dominated by those of the standard function exp.

Figure 1

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 DocumentS18AQF

+− 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

S18AQF