NAG FL Interface
s17ajf (airy_​ai_​deriv)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

s17ajf returns a value of the derivative of the Airy function Ai(x), via the function name.

2 Specification

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 <nag.h>
double  s17ajf_ (const double *x, Integer *ifail)
The routine may be called by the names s17ajf or nagf_specfun_airy_ai_deriv.

3 Description

s17ajf evaluates an approximation to the derivative of the Airy function Ai(x). It is based on a number of Chebyshev expansions.
For x<-5,
Ai(x)=-x4 [a(t)cosz+b(t)ζsinz] ,  
where z= π4+ζ, ζ= 23-x3 and a(t) and b(t) 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 y(t) is an expansion in t=4 ( x9)-1.
For 4.5x<9,
where v(t) is an expansion in t=4 ( x9)-3.
For x9,
Ai(x) = x4 e-z u(t) ,  
where z= 23x3 and u(t) is an expansion in t=2 ( 18z)-1.
For |x|< the square of the machine precision, the result is set directly to Ai(0). 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.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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).

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:
On entry, x=value.
Constraint: xvalue.
x is too large and positive. The function returns zero.
On entry, x=value.
Constraint: xvalue.
x is too large and negative. The function returns zero.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 |x2Ai(x)|δε | x2 Ai(x) Ai(x) |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 |x|7/4> 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.

8 Parallelism and Performance

s17ajf is not threaded in any implementation.

9 Further Comments


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 (s17ajfe.f90)

10.2 Program Data

Program Data (s17ajfe.d)

10.3 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