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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_bessel_y0_real (s17ac)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_bessel_y0_real (s17ac) returns the value of the Bessel function Y0x, via the function name.


[result, ifail] = s17ac(x)
[result, ifail] = nag_specfun_bessel_y0_real(x)


nag_specfun_bessel_y0_real (s17ac) evaluates an approximation to the Bessel function of the second kind Y0x.
Note:  Y0x is undefined for x0 and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
For 0<x8,
Y0x=2π lnxr=0arTrt+r=0brTrt,   with ​t=2 x8 2-1.  
For x>8,
Y0x=2πx P0xsinx-π4+Q0xcosx-π4  
where P0x=r=0crTrt,
and Q0x= 8xr=0drTrt,with ​ t=2 8x 2-1.
For x near zero, Y0x2π lnx2+γ , where γ denotes Euler's constant. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For very large x, 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 Y0x; only the amplitude, 2πn , 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 x1/machine precision.


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


Compulsory Input Parameters

1:     x – double scalar
The argument x of the function.
Constraint: x>0.0.

Optional Input Parameters


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:
x is too large. On soft failure the function returns the amplitude of the Y0 oscillation, 2/πx.
x0.0, Y0 is undefined. On soft failure the function returns zero.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Let δ be the relative error in the argument and E be the absolute error in the result. (Since Y0x oscillates about zero, absolute error and not relative error is significant, except for very small x.)
If δ is somewhat larger than the machine representation error (e.g., if δ is due to data errors etc.), then E and δ are approximately related by
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xY1x.
However, if δ is of the same order as the machine representation errors, then rounding errors could make E slightly larger than the above relation predicts.
For very small x, the errors are essentially independent of δ and the function should provide relative accuracy bounded by the machine precision.
For very large x, the above relation ceases to apply. In this region, Y0x 2πx sinx- π4. The amplitude 2πx  can be calculated with reasonable accuracy for all x, but sinx-π4 cannot. If x- π4  is written as 2Nπ+θ where N is an integer and 0θ<2π, then sinx- π4 is determined by θ only. If xδ-1, θ cannot be determined with any accuracy at all. Thus if x is greater than, or of the order of the inverse of machine precision, it is impossible to calculate the phase of Y0x and the function must fail.
Figure 1
Figure 1

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s17ac_example

fprintf('s17ac 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] = s17ac(x(j));

disp('      x         Y_0(x)');
fprintf('%12.3e%12.3e\n',[x; result]);


function s17ac_plot
x = [0.1:0.1:0.4,0.5:0.5:50];
  for j = 1:numel(x)
    [Y0(j), ifail] = s17ac(x(j));

  fig1 = figure;
  title('Bessel Function Y_0(x)');
  axis([0 50 -0.75 0.75]);

s17ac example results

      x         Y_0(x)
   5.000e-01  -4.445e-01
   1.000e+00   8.826e-02
   3.000e+00   3.769e-01
   6.000e+00  -2.882e-01
   8.000e+00   2.235e-01
   1.000e+01   5.567e-02
   1.000e+02  -7.724e-02

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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