The function may be called by the names: s17awc, nag_specfun_airy_ai_deriv_vector or nag_airy_ai_deriv_vector.
3Description
s17awc evaluates an approximation to 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 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 both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the function must fail. This occurs for , 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.
is too large and positive. contains zero. The threshold value is the same as for NE_REAL_ARG_GT in s17ajc
, 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 s17ajc
, 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).
6Error 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.
7Accuracy
For negative arguments the function is oscillatory and hence absolute error is the appropriate measure. In the positive region the function is essentially exponential in character and here 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 be roughly bounded by the machine precision.
For moderate to large negative , the error, like the function, is oscillatory; however, the amplitude of the 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 underflow. Thus in practice error amplification is limited.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s17awc is not threaded in any implementation.
9Further Comments
None.
10Example
This example reads values of x from a file, evaluates the function at each value of and prints the results.