NAG FL Interface
s17avf (airy_bi_real_vector)
1
Purpose
s17avf returns an array of values of the Airy function, .
2
Specification
Fortran Interface
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 |
s17avf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
s17avf_ (const Integer &n, const double x[], double f[], Integer ivalid[], Integer &ifail) |
}
|
The routine may be called by the names s17avf or nagf_specfun_airy_bi_real_vector.
3
Description
s17avf evaluates an approximation to the Airy function for an array of arguments , for . 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 avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate the phase of the oscillating function with any accuracy so the routine must fail. This occurs if , where is the machine precision.
For large positive arguments, there is a danger of causing overflow since Bi grows in an essentially exponential manner, so the routine must fail.
4
References
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of points.
Constraint:
.
-
2:
– Real (Kind=nag_wp) array
Input
-
On entry: the argument of the function, for .
-
3:
– Real (Kind=nag_wp) array
Output
-
On exit: , the function values.
-
4:
– Integer array
Output
-
On exit:
contains the error code for
, for
.
- No error.
- is too large and positive. contains zero. The threshold value is the same as for in s17ahf, as defined in the Users' Note for your implementation.
- is too large and negative. contains zero. The threshold value is the same as for in s17ahf, as defined in the Users' Note for your implementation.
-
5:
– 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, at least one value of
x was invalid.
Check
ivalid for more information.
-
On entry, .
Constraint: .
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 and hence will be bounded essentially by the machine precision.
For moderate to large negative , the error behaviour is clearly oscillatory but the amplitude of the error grows like amplitude .
However, the phase error will be growing roughly as and hence all accuracy will be lost for large negative arguments. This is 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 causing overflow and errors are therefore limited in practice.
8
Parallelism and Performance
s17avf is not threaded in any implementation.
None.
10
Example
This example reads values of
x 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