nag_airy_ai (s17agc) evaluates an approximation to 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 result is set directly to . This both saves time and guards against underflow in intermediate calculations.
For large negative arguments, it becomes impossible to calculate the phase of the oscillatory function with any precision and so the function must fail. This occurs if , 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.
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 the appropriate measure. In the positive region the function is essentially exponential-like and here relative error is appropriate. 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 , errors are strongly damped by the function and hence will be bounded by the machine precision.
For moderate negative , the error behaviour is oscillatory but the amplitude of the error grows like
However the phase error will be growing roughly like and hence all accuracy will be lost for large negative arguments due to the impossibility of calculating sin and cos to any accuracy if .
For large positive arguments, the relative error amplification is considerable:
This means a loss of roughly two decimal places accuracy for arguments in the region of . However very large arguments are not possible due to the danger of setting underflow and so the errors are limited in practice.
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.