NAG Library Routine Document

s17ajf (airy_ai_deriv)


    1  Purpose
    7  Accuracy


s17ajf returns a value of the derivative of the Airy function Aix, via the function name.


Fortran Interface
Function s17ajf ( x, ifail)
Real (Kind=nag_wp):: s17ajf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include nagmk26.h
double  s17ajf_ (const double *x, Integer *ifail)


s17ajf 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 routine 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 routine must fail.


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


1:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the function.
2:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
x is too large and positive. On soft failure, the routine returns zero. (see the Users' Note for your implementation for details)
x is too large and negative. On soft failure, the routine returns zero. See also the Users' Note for your implementation.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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.

Parallelism and Performance

s17ajf is not threaded in any implementation.

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.

Program Text

Program Text (s17ajfe.f90)

Program Data

Program Data (s17ajfe.d)

Program Results

Program Results (s17ajfe.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