NAG FL Interface
s17agf (airy_ai_real)
1
Purpose
s17agf returns a value for the Airy function, , via the function name.
2
Specification
Fortran Interface
| Real (Kind=nag_wp) |
:: |
s17agf |
| Integer, Intent (Inout) |
:: |
ifail |
| Real (Kind=nag_wp), Intent (In) |
:: |
x |
|
C Header Interface
|
#include <nag.h>
| double |
s17agf_ (const double *x, Integer *ifail) |
|
C++ Header Interface
|
#include <nag.h> extern "C" {
| double |
s17agf_ (const double &x, Integer &ifail) |
}
|
The routine may be called by the names s17agf or nagf_specfun_airy_ai_real.
3
Description
s17agf evaluates an approximation to the Airy function, . It is based on a number of Chebyshev expansions:
For
,
where
, and
and
are expansions in the variable
.
For
,
where
and
are expansions in
For
,
where
is an expansion in
.
For
,
where
is an expansion in
.
For
,
where
and
is an expansion in
.
For , the result is set directly to . This both saves time and guards against underflow in intermediate calculations.
For large negative arguments, it becomes impossible to calculate the phase of the oscillatory function with any precision and so the routine must fail. This occurs if , where is the machine precision.
For large positive arguments, where decays in an essentially exponential manner, there is a danger of underflow so the routine must fail.
4
References
5
Arguments
-
1:
– Real (Kind=nag_wp)
Input
-
On entry: the argument of the function.
-
2:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
.
When the value is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
or
, 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 is too large and positive. The function returns zero.
-
On entry,
.
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-like and here relative error is appropriate. The absolute error,
, and the relative error,
, are related in principle to the relative error in the argument,
, by
In practice, approximate equality is the best that can be expected. When
,
or
is of the order of the
machine precision, the errors in the result will be somewhat larger.
For small , errors are strongly damped by the function and hence will be bounded by the machine precision.
For moderate negative
, the error behaviour is oscillatory but the amplitude of the error grows like
However, the phase error will be growing roughly like
and hence all accuracy will be lost for large negative arguments due to the impossibility of calculating sin and cos to any accuracy if
.
For large positive arguments, the relative error amplification is considerable:
This means a loss of roughly two decimal places accuracy for arguments in the region of
. However, very large arguments are not possible due to the danger of setting underflow and so the errors are limited in practice.
8
Parallelism and Performance
s17agf is not threaded in any implementation.
None.
10
Example
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
10.1
Program Text
10.2
Program Data
10.3
Program Results