NAG Library Function Document

nag_bessel_y0 (s17acc)

▸▿ 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_y0 (s17acc) returns the value of the Bessel function

Y_{0} (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_bessel_y0 (double x, NagError *fail)

3 Description

nag_bessel_y0 (s17acc) evaluates an approximation to the Bessel function of the second kind

Y_{0} (x)

Note:

Y_{0} (x)

is undefined for

x \leq 0

and the function will fail for such arguments.

The function is based on four Chebyshev expansions:

For

0 < x \leq 8

Y_{0} (x) = \frac{2}{π} \ln x \sum_{r = 0}^{'} a_{r} T_{r} (t) + \sum_{r = 0}^{'} 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 function 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 failure. The range for which this occurs is roughly related to machine precision; the function 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 Arguments

1: $x$ – doubleInput: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $fail$ – 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_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.
NE_REAL_ARG_GT: On entry, $x = 〈value〉$ .
Constraint: $x \leq 〈value〉$ .
x is too large. On failure the function returns the amplitude of the $Y_{0}$ oscillation, $\sqrt{2 / (π x)}$ .

x is too large, the function returns the amplitude of the $Y_{0}$ oscillation, $\sqrt{\frac{2}{π x}}$ .
NE_REAL_ARG_LE: On entry, $x = 〈value〉$ .
Constraint: $x > 0.0$ .

$Y_{0}$ is undefined, the function returns zero.

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 function 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 function must fail.

Figure 1

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Function Documentnag_bessel_y0 (s17acc)