NAG Library Routine Document
S17AXF returns an array of values for the derivative of the Airy function .
||N, IVALID(N), IFAIL
S17AXF calculates an approximate value for the derivative of 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 square of the machine precision, the result is set directly to . This saves time and avoids possible underflows in calculation.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy so the routine must fail. This occurs for , where is the machine precision.
For large positive arguments, where grows in an essentially exponential manner, there is a danger of overflow 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 S17AKF, 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 S17AKF, 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 appropriate. In the positive region the function has essentially exponential behaviour and hence relative error is needed. 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 , positive or negative, errors are strongly attenuated by the function and hence will effectively be bounded by the machine precision.
For moderate to large negative , the error is, like the function, oscillatory. However, the amplitude of the absolute 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 overflow. Thus in practice the actual amplification that occurs is limited.
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 (s17axfe.f90)
9.2 Program Data
Program Data (s17axfe.d)
9.3 Program Results
Program Results (s17axfe.r)