NAG CL Interface
s17auc (airy_ai_real_vector)
1
Purpose
s17auc returns an array of values for the Airy function, .
2
Specification
void |
s17auc (Integer n,
const double x[],
double f[],
Integer ivalid[],
NagError *fail) |
|
The function may be called by the names: s17auc, nag_specfun_airy_ai_real_vector or nag_airy_ai_vector.
3
Description
s17auc 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 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 function 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 function must fail.
4
References
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of points.
Constraint:
.
-
2:
– const double
Input
-
On entry: the argument of the function, for .
-
3:
– double
Output
-
On exit: , the function values.
-
4:
– Integer
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 NE_REAL_ARG_GT in s17agc, as defined in the Users' Note for your implementation.
- is too large and negative. contains zero. The threshold value is the same as for NE_REAL_ARG_LT in s17agc, as defined in the Users' Note for your implementation.
-
5:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NW_IVALID
-
On entry, at least one value of
x was invalid.
Check
ivalid for more 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
s17auc 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