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NAG Toolbox: nag_specfun_airy_ai_deriv_vector (s17aw)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_airy_ai_deriv_vector (s17aw) returns an array of values of the derivative of the Airy function Aix.

Syntax

[f, ivalid, ifail] = s17aw(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_airy_ai_deriv_vector(x, 'n', n)

Description

nag_specfun_airy_ai_deriv_vector (s17aw) evaluates an approximation to the derivative of the Airy function Aixi for an array of arguments xi, for i=1,2,,n. 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,
Aix=x2ft-gt,  
where f and g are expansions in t=-2 x5 3-1.
For 0<x<4.5,
Aix=e-11x/8yt,  
where yt is an expansion in t=4 x9-1.
For 4.5x<9,
Aix=e-5x/2vt,  
where vt is an expansion in t=4 x9-3.
For x9,
Aix = x 4 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.

References

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

Parameters

Compulsory Input Parameters

1:     xn – double array
The argument xi of the function, for i=1,2,,n.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of points.
Constraint: n0.

Output Parameters

1:     fn – double array
Aixi, the function values.
2:     ivalidn int64int32nag_int array
ivalidi contains the error code for xi, for i=1,2,,n.
ivalidi=0
No error.
ivalidi=1
xi is too large and positive. fi contains zero. The threshold value is the same as for ifail=1 in nag_specfun_airy_ai_deriv (s17aj), as defined in the Users' Note for your implementation.
ivalidi=2
xi is too large and negative. fi contains zero. The threshold value is the same as for ifail=2 in nag_specfun_airy_ai_deriv (s17aj), as defined in the Users' Note for your implementation.
3:     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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
   ifail=2
Constraint: n0.
   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

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
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 underflow. Thus in practice error amplification is limited.

Further Comments

None.

Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.
function s17aw_example


fprintf('s17aw example results\n\n');

x = [-10; -1; 0; 1; 5; 10; 20];

[f, ivalid, ifail] = s17aw(x);

fprintf('     x           Ai''(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end


s17aw example results

     x           Ai'(x)   ivalid
  -1.000e+01   9.963e-01    0
  -1.000e+00  -1.016e-02    0
   0.000e+00  -2.588e-01    0
   1.000e+00  -1.591e-01    0
   5.000e+00  -2.474e-04    0
   1.000e+01  -3.521e-10    0
   2.000e+01  -7.586e-27    0

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