NAG CL Interface
s17atc (bessel_j1_real_vector)
1
Purpose
s17atc returns an array of values of the Bessel function .
2
Specification
void |
s17atc (Integer n,
const double x[],
double f[],
Integer ivalid[],
NagError *fail) |
|
The function may be called by the names: s17atc, nag_specfun_bessel_j1_real_vector or nag_bessel_j1_vector.
3
Description
s17atc evaluates an approximation to the Bessel function of the first kind for an array of arguments , for .
Note: , so the approximation need only consider .
The function is based on three Chebyshev expansions:
For
,
For
,
where
,
and ,
with .
For near zero, . This approximation is used when is sufficiently small for the result to be correct to machine precision.
For very large
, it becomes impossible to provide results with any reasonable accuracy (see
Section 7), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of
; only the amplitude,
, can be determined and this is returned on failure. The range for which this occurs is roughly related to
machine precision; the function will fail if
(see the
Users' Note for your implementation for details).
4
References
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO
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.
- On entry, is too large. contains the amplitude of the oscillation, .
-
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
Let be the relative error in the argument and be the absolute error in the result. (Since oscillates about zero, absolute error and not relative error is significant.)
If
is somewhat larger than
machine precision (e.g., if
is due to data errors etc.), then
and
are approximately related by:
(provided
is also within machine bounds).
Figure 1 displays the behaviour of the amplification factor
.
However, if is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
For very large , the above relation ceases to apply. In this region, . The amplitude can be calculated with reasonable accuracy for all , but cannot. If is written as where is an integer and , then is determined by only. If , cannot be determined with any accuracy at all. Thus if is greater than, or of the order of, the reciprocal of machine precision, it is impossible to calculate the phase of and the function must fail.
8
Parallelism and Performance
s17atc 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