NAG Library Routine Document
S17AUF returns an array of values for the Airy function, .
||N, IVALID(N), IFAIL
S17AUF evaluates an approximation to the Airy function, for an array of arguments , for . It is based on a number of Chebyshev expansions:
are expansions in the variable
are expansions in
is an expansion in
is an expansion in
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 routine 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 routine must fail.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
- 1: N – INTEGERInput
On entry: , the number of points.
- 2: X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the argument of the function, for .
- 3: F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: , the function values.
- 4: IVALID(N) – INTEGER arrayOutput
contains the error code for
- No error.
- is too large and positive. contains zero. The threshold value is the same as for in S17AGF, as defined in the Users' Note for your implementation.
- is too large and negative. contains zero. The threshold value is the same as for in S17AGF, as defined in the Users' Note for your implementation.
- 5: IFAIL – INTEGERInput/Output
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.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
On entry, at least one value of X
for more information.
On entry, .
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,
In practice, approximate equality is the best that can be expected. When
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.
This example reads values of X
from a file, evaluates the function at each value of
and prints the results.
9.1 Program Text
Program Text (s17aufe.f90)
9.2 Program Data
Program Data (s17aufe.d)
9.3 Program Results
Program Results (s17aufe.r)