, 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: $x$ – double scalar: The argument $x$ of the function.

Constraint: $x > 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – 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:

$ifail = 1$: x is too large. On soft failure the function returns the amplitude of the $Y_{1}$ oscillation, $\sqrt{\frac{2}{π x}}$ .

$ifail = 2$: $x \leq 0.0$ , $Y_{1}$ is undefined. On soft failure the function returns zero.

$ifail = 3$: x is too close to zero, there is a danger of overflow. On soft failure, 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.

$ifail = - 399$: 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 the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

Open in the MATLAB editor: s17ad_example

function s17ad_example


fprintf('s17ad example results\n\n');

x = [0.5     1    3    6   8    10   100];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s17ad(x(j));
end

disp('      x         Y_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s17ad_plot;



function s17ad_plot
x = [0.1:0.1:0.4,0.5:0.5:50];
  for j = 1:numel(x)
    [Y1(j), ifail] = s17ad(x(j));
  end

  fig1 = figure;
  plot(x,Y1,'-r');
  xlabel('x');
  ylabel('Y_1(x)');
  title('Bessel Function Y_1(x)');
  axis([0 50 -2 0.75]);

s17ad example results

      x         Y_1(x)
   5.000e-01  -1.471e+00
   1.000e+00  -7.812e-01
   3.000e+00   3.247e-01
   6.000e+00  -1.750e-01
   8.000e+00  -1.581e-01
   1.000e+01   2.490e-01
   1.000e+02  -2.037e-02

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

Chapter Contents

Chapter Introduction

NAG Toolbox