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NAG Toolbox: nag_specfun_airy_bi_deriv (s17ak)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_airy_bi_deriv (s17ak) returns a value for the derivative of the Airy function Bix, via the function name.


[result, ifail] = s17ak(x)
[result, ifail] = nag_specfun_airy_bi_deriv(x)


nag_specfun_airy_bi_deriv (s17ak) calculates an approximate value for the derivative of the Airy function Bix. It is based on a number of Chebyshev expansions.
For x<-5,
Bix=-x4 -atsinz+btζcosz ,  
where z= π4+ζ, ζ= 23-x3 and at and bt are expansions in the variable t=-2 5x 3-1.
For -5x0,
where f and g are expansions in t=-2 x5 3-1.
For 0<x<4.5,
where yt is an expansion in t=4x/9-1.
For 4.5x<9,
where ut is an expansion in t=4x/9-3.
For x9,
where z= 23x3 and vt is an expansion in t=2 18z-1.
For x< the square of the machine precision, the result is set directly to Bi0. This saves time and avoids possible underflows in calculation.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy so the function must fail. This occurs for x<- πε 4/7 , where ε is the machine precision.
For large positive arguments, where Bi grows in an essentially exponential manner, there is a danger of overflow so the function must fail.


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications


Compulsory Input Parameters

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

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 and positive. On soft failure, the function returns zero.
x is too large and negative. 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.


For negative arguments the function is oscillatory and hence absolute error is appropriate. In the positive region the function has essentially exponential behaviour and hence relative error is needed. The absolute error, E, and the relative error ε, are related in principle to the relative error in the argument δ, by
E x2 Bix δ ε x2 Bix Bix δ.  
In practice, approximate equality is the best that can be expected. When δ, ε or E is of the order of the machine precision, the errors in the result will be somewhat larger.
For small x, positive or negative, errors are strongly attenuated by the function and hence will effectively be bounded by the machine precision.
For moderate to large negative x, the error is, like the function, oscillatory. However, the amplitude of the absolute error grows like x7/4π . Therefore it becomes impossible to calculate the function with any accuracy if x7/4> πδ .
For large positive x, the relative error amplification is considerable: εδx3. However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.

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 s17ak_example

fprintf('s17ak example results\n\n');

x = [-10    -1    0    1    5    10   20];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s17ak(x(j));

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


function s17ak_plot
  x = [-10:0.1:3];
  for j = 1:numel(x)
    [Bid(j), ifail] = s17ak(x(j));

  fig1 = figure;
  title('Derivative of Airy Function Bi(x)');
  axis([-10 4 -2 10]);

s17ak example results

      x         Bi'(x)
  -1.000e+01   1.194e-01
  -1.000e+00   5.924e-01
   0.000e+00   4.483e-01
   1.000e+00   9.324e-01
   5.000e+00   1.436e+03
   1.000e+01   1.429e+09
   2.000e+01   9.382e+25

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

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