NAG C Library Function Document

, 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

NIST Digital Library of Mathematical Functions

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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

nag_bessel_y1 (s17adc) is not threaded in any implementation.

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 C 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

2

Specification

3

Description