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.
where , and and are expansions in the variable .
where and are expansions in .
where is an expansion in .
where is an expansion in .
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.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
x – doubleInput
On entry: the argument of the function.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
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.
On entry, . Constraint: . x is too large and positive. The function returns zero.
On entry, . Constraint: . x is too large and negative. The function returns zero.
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
9 Further Comments
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.