S17AJF evaluates an approximation to the derivative of the Airy function . It is based on a number of Chebyshev expansions.
where , and and are expansions in 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 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 routine 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 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 parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).
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 3.8 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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 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.
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.