NAG Library Function Document
nag_airy_bi_deriv_vector (s17axc)
1 Purpose
nag_airy_bi_deriv_vector (s17axc) returns an array of values for the derivative of the Airy function .
2 Specification
#include <nag.h> |
#include <nags.h> |
void |
nag_airy_bi_deriv_vector (Integer n,
const double x[],
double f[],
Integer ivalid[],
NagError *fail) |
|
3 Description
nag_airy_bi_deriv_vector (s17axc) calculates an approximate value for the derivative of 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 square of the machine precision, the result is set directly to . This saves time and avoids possible underflows in calculation.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy so the function must fail. This occurs for , where is the machine precision.
For large positive arguments, where grows in an essentially exponential manner, there is a danger of overflow so the function must fail.
4 References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
5 Arguments
- 1:
n – IntegerInput
On entry: , the number of points.
Constraint:
.
- 2:
x[n] – const doubleInput
On entry: the argument of the function, for .
- 3:
f[n] – doubleOutput
On exit: , the function values.
- 4:
ivalid[n] – IntegerOutput
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 nag_airy_bi_deriv (s17akc), 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 nag_airy_bi_deriv (s17akc), as defined in the Users' Note for your implementation.
- 5:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- 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.
- 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 appropriate. In the positive region the function has essentially exponential behaviour and hence relative error is needed. 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 , positive or negative, errors are strongly attenuated by the function and hence will effectively be bounded by the machine precision.
For moderate to large negative , the error is, like the function, oscillatory. However, the amplitude of the absolute error grows like . Therefore it becomes impossible to calculate the function with any accuracy if .
For large positive , the relative error amplification is considerable: . However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.
8 Parallelism and Performance
Not applicable.
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
Program Text (s17axce.c)
10.2 Program Data
Program Data (s17axce.d)
10.3 Program Results
Program Results (s17axce.r)