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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_bessel_y1_real (s17ad)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_bessel_y1_real (s17ad) returns the value of the Bessel function Y1x, via the function name.


[result, ifail] = s17ad(x)
[result, ifail] = nag_specfun_bessel_y1_real(x)


nag_specfun_bessel_y1_real (s17ad) evaluates an approximation to the Bessel function of the second kind Y1x.
Note:  Y1x is undefined for x0 and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
For 0<x8,
Y1x=2π lnxx8r=0arTrt-2πx +x8r=0brTrt,   with ​t=2 x8 2-1.  
For x>8,
Y1x=2πx P1xsinx-3π4+Q1xcosx-3π4  
where P1x=r=0crTrt,
and Q1x= 8xr=0drTrt, with t=2 8x 2-1.
For x near zero, Y1x- 2πx . This approximation is used when x is sufficiently small for the result to be correct to machine precision. For extremely small x, there is a danger of overflow in calculating - 2πx  and for such arguments the function will fail.
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 Y1x; only the amplitude, 2πx , 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 Y1 oscillation, 2πx .
x0.0, Y1 is undefined. On soft failure the function returns zero.
x is too close to zero, there is a danger of overflow. On soft failure, the function returns the value of Y1x at the smallest valid argument.
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 Y1x oscillates about zero, absolute error and not relative error is significant, except for very small x.)
If 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 Y0 x - Y1 x δ  
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xY0x-Y1x.
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, Y1 x 2 πx sinx- 3π4 . The amplitude 2 πx  can be calculated with reasonable accuracy for all x, but sinx- 3π4 cannot. If x- 3π4  is written as 2Nπ+θ where N is an integer and 0θ<2π, then sinx- 3π4 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 Y1x 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 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));

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


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));

  fig1 = figure;
  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

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Chapter Contents
Chapter Introduction
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