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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_airy_ai_deriv (s17aj)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_airy_ai_deriv (s17aj) returns a value of the derivative of the Airy function Aix, via the function name.


[result, ifail] = s17aj(x)
[result, ifail] = nag_specfun_airy_ai_deriv(x)


nag_specfun_airy_ai_deriv (s17aj) evaluates an approximation to the derivative of the Airy function Aix. It is based on a number of Chebyshev expansions.
For x<-5,
Aix=-x4 atcosz+btζsinz ,  
where z= π4+ζ, ζ= 23-x3 and at and bt are expansions in 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=4 x9-1.
For 4.5x<9,
where vt is an expansion in t=4 x9-3.
For x9,
Aix = x4 e-z ut ,  
where z= 23x3 and ut is an expansion in t=2 18z-1.
For x< the square of the machine precision, the result is set directly to Ai0. This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the function must fail. This occurs for x<- πε 4/7 , where ε is the machine precision.
For large positive arguments, where Ai decays in an essentially exponential manner, there is a danger of underflow 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 the appropriate measure. In the positive region the function is essentially exponential in character and here 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 Aix δε x2 Aix Aix 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 be roughly bounded by the machine precision.
For moderate to large negative x, the error, like the function, is oscillatory; however the amplitude of the error grows like
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:
However, very large arguments are not possible due to the danger of underflow. Thus in practice error amplification 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 s17aj_example

fprintf('s17aj 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] = s17aj(x(j));

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


function s17aj_plot
  x = [-15:0.1:5];
  for j = 1:numel(x)
    [Aid(j), ifail] = s17aj(x(j));

  fig1 = figure;
  title('Derivative of Airy Function Ai(x)');
  axis([-15 5 -1.5 1.5]);

s17aj example results

      x         Ai'(x)
  -1.000e+01   9.963e-01
  -1.000e+00  -1.016e-02
   0.000e+00  -2.588e-01
   1.000e+00  -1.591e-01
   5.000e+00  -2.474e-04
   1.000e+01  -3.521e-10
   2.000e+01  -7.586e-27

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

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