NAG Library Routine Document
S19APF returns an array of values for the Kelvin function .
||N, IVALID(N), IFAIL
S19APF evaluates an approximation to the Kelvin function for an array of arguments , for .
Note: , so the approximation need only consider .
The routine is based on several Chebyshev expansions:
and , , , and are expansions in the variable .
When is sufficiently close to zero, the result is computed as . If this result would underflow, the result returned is .
For large , there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore 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 for an accurate result to be returned. contains zero. The threshold value is the same as for in S19ABF, 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, .
Since the function is oscillatory, the absolute error rather than the relative error is important. Let
be the absolute error in the function, and
be the relative error in the argument. If
is somewhat larger than the machine precision
, then we have:
is within machine bounds).
For small the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large , the error behaviour is oscillatory and its amplitude grows like . Therefore it is impossible to calculate the functions with any accuracy when . Note that this value of is much smaller than the minimum value of for which the function overflows.
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 (s19apfe.f90)
9.2 Program Data
Program Data (s19apfe.d)
9.3 Program Results
Program Results (s19apfe.r)