NAG Library Routine Document

S17AQF

+− 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

S17AQF returns an array of values of the Bessel function

Y_{0} (x)

2 Specification

SUBROUTINE S17AQF (

N, X, F, IVALID, IFAIL)

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

3 Description

S17AQF evaluates an approximation to the Bessel function of the second kind

Y_{0} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

Y_{0} (x)

is undefined for

x \leq 0

and the routine will fail for such arguments.

The routine is based on four Chebyshev expansions:

For

0 < x \leq 8

Y_{0} (x) = \frac{2}{π} \ln x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t) + \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   with ​ t = 2 {(\frac{x}{8})}^{2} - 1 .

For

x > 8

Y_{0} (x) = \sqrt{\frac{2}{π x}} \{P_{0} (x) \sin (x - \frac{π}{4}) + Q_{0} (x) \cos (x - \frac{π}{4})\}

where

P_{0} (x) = \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t)

and

Q_{0} (x) = \frac{8}{x} \underset{r = 0}{\sum^{'}} d_{r} T_{r} (t), with ​ t = 2 {(\frac{8}{x})}^{2} - 1 .

For

x

near zero,

Y_{0} (x) ≃ \frac{2}{π} (\ln (\frac{x}{2}) + γ)

, 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 very large

x

, it becomes impossible to provide results with any reasonable accuracy (see Section 7), hence the routine fails. Such arguments contain insufficient information to determine the phase of oscillation of

Y_{0} (x)

; only the amplitude,

\sqrt{\frac{2}{π n}}

, can be determined and this is returned on soft failure. The range for which this occurs is roughly related to machine precision; the routine will fail if

x ≳ 1 / machine precision

(see the Users' Note for your implementation for details).

4 References

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

Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

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:

Y_{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.

$IVALID (i) = 1$

On entry,

x_{i}

is too large.

F (i)

contains the amplitude of the

Y_{0}

oscillation,

\sqrt{\frac{2}{π x_{i}}}

$IVALID (i) = 2$

On entry,

x_{i} \leq 0.0

Y_{0}

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

be the relative error in the argument and

E

be the absolute error in the result. (Since

Y_{0} (x)

oscillates about zero, absolute error and not relative error is significant, except for very small

x

δ

is somewhat larger than the machine representation error (e.g., if

δ

is due to data errors etc.), then

E

and

δ

are approximately related by

E ≃ |x Y_{1} (x)| δ

(provided

E

is also within machine bounds). Figure 1 displays the behaviour of the amplification factor

|x Y_{1} (x)|

However, if

δ

is of the same order as the machine representation errors, then rounding errors could make

E

slightly larger than the above relation predicts.

For very small

x

, the errors are essentially independent of

δ

and the routine should provide relative accuracy bounded by the machine precision.

For very large

x

, the above relation ceases to apply. In this region,

Y_{0} (x) ≃ \sqrt{\frac{2}{π x}} \sin (x - \frac{π}{4})

. The amplitude

\sqrt{\frac{2}{π x}}

can be calculated with reasonable accuracy for all

x

, but

\sin (x - \frac{π}{4})

cannot. If

x - \frac{π}{4}

is written as

2 N π + θ

where

N

is an integer and

0 \leq θ < 2 π

, then

\sin (x - \frac{π}{4})

is determined by

θ

only. If

x ≳ δ^{- 1}

θ

cannot be determined with any accuracy at all. Thus if

x

is greater than, or of the order of the inverse of machine precision, it is impossible to calculate the phase of

Y_{0} (x)

and the routine must fail.

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 DocumentS17AQF

+− 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

S17AQF