nag_airy_bi (s17ahc) 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 avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate the phase of the oscillating function with any accuracy so the function must fail. This occurs if , where is the machine precision.
For large positive arguments, there is a danger of causing overflow since Bi grows in an essentially exponential manner, so the function must fail.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
On entry: the argument of the function.
– NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 184.108.40.206 in How to Use the NAG Library and its Documentation for further information.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.
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 and hence will be bounded essentially by the machine precision.
For moderate to large negative , the error behaviour is clearly oscillatory but the amplitude of the error grows like amplitude .
However the phase error will be growing roughly as and hence all accuracy will be lost for large negative arguments. This is 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 causing overflow and errors are therefore limited in practice.
8 Parallelism and Performance
nag_airy_bi (s17ahc) is not threaded in any implementation.
9 Further Comments
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.