, it becomes impossible to provide results with any reasonable accuracy (see Accuracy), 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 soft failure. The range for which this occurs is roughly related to machine precision; the function will fail if

x ≳ 1 / machine precision

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

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array: The argument $x_{i}$ of the function, for $i = 1, 2, \dots, n$ .

Constraint: $x (i) > 0.0$ , for $i = 1, 2, \dots, n$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
$n$ , the number of points.

Constraint: $n \geq 0$ .

Output Parameters

1: $f (n)$ – double array

Y_{1} (x_{i})

, the function values.

2: $ivalid (n)$ – int64int32nag_int array

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_{1}

oscillation,

\sqrt{\frac{2}{π x_{i}}}

$ivalid (i) = 2$

On entry,

x_{i} \leq 0.0

Y_{1}

is undefined.

f (i)

contains

0.0

$ivalid (i) = 3$

x_{i}

is too close to zero, there is a danger of overflow. On soft failure,

f (i)

contains the value of

Y_{1} (x)

at the smallest valid argument.

3: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: On entry, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: Constraint: $n \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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

Further Comments

None.

Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

Open in the MATLAB editor: s17ar_example

function s17ar_example


fprintf('s17ar example results\n\n');

x = [0.5; 1; 3; 6; 8; 10; 1000];

[f, ivalid, ifail] = s17ar(x);

fprintf('     x          Y_1(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

s17ar example results

     x          Y_1(x)   ivalid
   5.000e-01  -1.471e+00    0
   1.000e+00  -7.812e-01    0
   3.000e+00   3.247e-01    0
   6.000e+00  -1.750e-01    0
   8.000e+00  -1.581e-01    0
   1.000e+01   2.490e-01    0
   1.000e+03  -2.478e-02    0

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox