s17ahf 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 routine 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 routine must fail.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
1: – Real (Kind=nag_wp)Input
On entry: the argument of the function.
2: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
x is too large and positive. On soft failure, the routine returns zero. (see the Users' Note for your implementation for details)
x is too large and negative. On soft failure, the routine returns zero. See also the Users' Note for your implementation.
An unexpected error has been triggered by this routine. Please
See Section 3.9 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 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
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.
Parallelism and Performance
s17ahf is not threaded in any implementation.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.