NAG C Library Function Document
nag_airy_bi_deriv (s17akc)
1
Purpose
nag_airy_bi_deriv (s17akc) returns a value for the derivative of the Airy function .
2
Specification
| #include <nag.h> |
| #include <nags.h> |
| double |
nag_airy_bi_deriv (double x,
NagError *fail) |
|
3
Description
nag_airy_bi_deriv (s17akc) calculates an approximate value for the derivative of 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 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
5
Arguments
- 1:
– doubleInput
-
On entry: the argument of the function.
- 2:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL_ARG_GT
-
On entry,
.
x is too large and positive. The function returns zero.
- NE_REAL_ARG_LT
-
On entry,
.
x is too large and negative. The function returns zero.
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
nag_airy_bi_deriv (s17akc) 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
Program Text (s17akce.c)
10.2
Program Data
Program Data (s17akce.d)
10.3
Program Results
Program Results (s17akce.r)