NAG FL Interface
s17awf (airy_​ai_​deriv_​vector)

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1 Purpose

s17awf returns an array of values of the derivative of the Airy function Ai(x).

2 Specification

Fortran Interface
Subroutine s17awf ( n, x, f, ivalid, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(n)
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s17awf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s17awf or nagf_specfun_airy_ai_deriv_vector.

3 Description

s17awf evaluates an approximation to the derivative of the Airy function Ai(xi) for an array of arguments xi, for i=1,2,,n. 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,
Ai(x)=x2f(t)-g(t),  
where f and g are expansions in t=-2 ( x5) 3-1.
For 0<x<4.5,
Ai(x)=e-11x/8y(t),  
where y(t) is an expansion in t=4 ( x9)-1.
For 4.5x<9,
Ai(x)=e-5x/2v(t),  
where v(t) is an expansion in t=4 ( x9)-3.
For x9,
Ai(x) = x 4 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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x(n) Real (Kind=nag_wp) array Input
On entry: the argument xi of the function, for i=1,2,,n.
3: f(n) Real (Kind=nag_wp) array Output
On exit: Ai(xi), the function values.
4: ivalid(n) Integer array Output
On exit: ivalid(i) contains the error code for xi, for i=1,2,,n.
ivalid(i)=0
No error.
ivalid(i)=1
xi is too large and positive. f(i) contains zero. The threshold value is the same as for ifail=1 in s17ajf , as defined in the Users' Note for your implementation.
ivalid(i)=2
xi is too large and negative. f(i) contains zero. The threshold value is the same as for ifail=2 in s17ajf , as defined in the Users' Note for your implementation.
5: 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:
ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
ifail=2
On entry, n=value.
Constraint: n0.
ifail=-99
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.
ifail=-399
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.
ifail=-999
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
|x|7/4π.  
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:
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

s17awf is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

10.1 Program Text

Program Text (s17awfe.f90)

10.2 Program Data

Program Data (s17awfe.d)

10.3 Program Results

Program Results (s17awfe.r)