NAG Library Function Document

nag_bessel_y1 (s17adc)

+− 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_y1 (s17adc) returns the value of the Bessel function

Y_{1} (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_bessel_y1 (double x, NagError *fail)

3 Description

nag_bessel_y1 (s17adc) evaluates an approximation to the Bessel function of the second kind

Y_{1} (x)

Note:

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

For

x > 8

Y_{1} (x) = \sqrt{\frac{2}{π x}} \{P_{1} (x) \sin (x - 3 \frac{π}{4}) + Q_{1} (x) \cos (x - 3 \frac{π}{4})\}

where

P_{1} (x) = \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t)

and

Q_{1} (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_{1} (x) ≃ - \frac{2}{π x}

. This approximation is used when

x

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

x

, there is a danger of overflow in calculating

- \frac{2}{π x}

and for such arguments the function will fail.

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_{1} (x)

; only the amplitude,

\sqrt{\frac{2}{π x}}

, 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_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.
NE_REAL_ARG_GT: On entry, $x = ⟨value⟩$ .
Constraint: $x \leq ⟨value⟩$ .
$x$ is too large, the function returns the amplitude of the $Y_{1}$ oscillation, $\sqrt{2 / (π x)}$ .
NE_REAL_ARG_LE: On entry, $x = ⟨value⟩$ .
Constraint: $x > 0.0$ .
$Y_{1}$ is undefined, the function returns zero.
NE_REAL_ARG_TOO_SMALL: x is too close to zero and there is danger of overflow, $x = ⟨value⟩$ .
Constraint: $x > ⟨value⟩$ .
The function returns the value of $Y_{1} (x)$ at the smallest valid argument.

7 Accuracy

Let

δ

be the relative error in the argument and

E

be the absolute error in the result. (Since

Y_{1} (x)

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

x

δ

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

δ

is due to data errors etc.), then

E

and

δ

are approximately related by:

E ≃ |x Y_{0} (x) - Y_{1} (x)| δ

(provided

E

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

|x Y_{0} (x) - Y_{1} (x)|

However, if

δ

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

E

slightly larger than the above relation predicts.

For very small

x

, absolute error becomes large, but the relative error in the result is of the same order as

δ

For very large

x

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

Y_{1} (x) ≃ \sqrt{\frac{2}{π x}} \sin (x - \frac{3 π}{4})

. The amplitude

\sqrt{\frac{2}{π x}}

can be calculated with reasonable accuracy for all

x

, but

\sin (x - \frac{3 π}{4})

cannot. If

x - \frac{3 π}{4}

is written as

2 N π + θ

where

N

is an integer and

0 \leq θ < 2 π

, then

\sin (x - \frac{3 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 the machine precision, it is impossible to calculate the phase of

Y_{1} (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_y1 (s17adc)