NAG Library Function Document

nag_airy_ai_deriv (s17ajc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_airy_ai_deriv (s17ajc) returns a value of the derivative of the Airy function Aix.

2
Specification

#include <nag.h>
#include <nags.h>
double  nag_airy_ai_deriv (double x, NagError *fail)

3
Description

nag_airy_ai_deriv (s17ajc) 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,
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 = 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.

4
References

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

5
Arguments

1:     x doubleInput
On entry: the argument x of the function.
2:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_GT
On entry, x=value.
Constraint: xvalue.
x is too large and positive. The function returns zero.
NE_REAL_ARG_LT
On entry, x=value.
Constraint: xvalue.
x is too large and negative. The function returns zero.

7
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.

8
Parallelism and Performance

nag_airy_ai_deriv (s17ajc) is not threaded in any implementation.

9
Further Comments

None.

10
Example

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

10.1
Program Text

Program Text (s17ajce.c)

10.2
Program Data

Program Data (s17ajce.d)

10.3
Program Results

Program Results (s17ajce.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −1.5 −1 −0.5 0 0.5 1 1.5 −15 −10 −5 0 5 Ai(x) x Example Program Returns a Value for the Derivative of the Airy Function Ai(x) gnuplot_plot_1
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017