, 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_{1} (x)

; only the amplitude,

\sqrt{\frac{2}{π x}}

, 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

NIST Digital Library of Mathematical Functions

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

5 Arguments

1: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $x = 〈value〉$ .
Constraint: $x \leq 〈value〉$ .
$x$ is too large, the function returns the amplitude of the $Y_{1}$ oscillation, $\sqrt{2 / (π x)}$ .

$ifail = 2$: On entry, $x = 〈value〉$ .
Constraint: $x > 0.0$ .
$Y_{1}$ is undefined, the function returns zero.

$ifail = 3$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 routine must fail.

Figure 1

8 Parallelism and Performance

s17adf 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.

s17ad: FL CL CPP AD

NAG FL Interfaces17adf (bessel_​y1_​real)

▸▿ 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

NAG FL Interface
s17adf (bessel_y1_real)