NAG Library Routine Document
S19AQF returns an array of values for the Kelvin function .
||N, IVALID(N), IFAIL
S19AQF evaluates an approximation to the Kelvin function for an array of arguments , for .
Note: for the function is undefined and at it is infinite so we need only consider .
The routine is based on several Chebyshev expansions:
are expansions in the variable
is an expansion in the variable
are expansions in the variable
is sufficiently close to zero, the result is computed as
is even closer to zero, simply as
For large , is asymptotically given by and this becomes so small that it cannot be computed without underflow and the routine fails.
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 .
, 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, the result underflows. contains zero. The threshold value is the same as for in S19ACF, as defined in the Users' Note for your implementation.
- , the function is undefined. contains .
- 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, .
be the absolute error in the result,
be the relative error in the result and
be the relative error in the argument. If
is somewhat larger than the machine precision
, then we have:
For very small
, the relative error amplification factor is approximately given by
, which implies a strong attenuation of relative error. However,
in general cannot be less than the machine precision
For small , errors are damped by the function and hence are limited by the machine precision.
For medium and large , the error behaviour, like the function itself, is oscillatory, and hence only the absolute accuracy for the function can be maintained. For this range of , the amplitude of the absolute error decays like which implies a strong attenuation of error. Eventually, , which asymptotically behaves like , becomes so small that it cannot be calculated without causing underflow, and the routine returns zero. Note that for large the errors are dominated by those of the standard function exp.
Underflow may occur for a few values of close to the zeros of , below the limit which causes a failure with .
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 (s19aqfe.f90)
9.2 Program Data
Program Data (s19aqfe.d)
9.3 Program Results
Program Results (s19aqfe.r)