NAG Library Routine Document
S18ASF returns an array of values of the modified Bessel function .
||N, IVALID(N), IFAIL
S18ASF evaluates an approximation to the modified Bessel function of the first kind for an array of arguments , for .
Note: , so the approximation need only consider .
The routine is based on three Chebyshev expansions:
. This approximation is used when
is sufficiently small for the result to be correct to machine precision
For large , the routine must fail because of the danger of overflow in calculating .
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. contains the approximate value of at the nearest valid argument. The threshold value is the same as for in S18AEF, 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, .
Let and be the relative errors in the argument and result respectively.
is somewhat larger than the machine precision
is due to data errors etc.), then
are approximately related by:
shows the behaviour of the error amplification factor
However if is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small the amplification factor is approximately , which implies strong attenuation of the error, but in general can never be less than the machine precision.
For large , and we have strong amplification of errors. However, for quite moderate values of (, the threshold value), the routine must fail because would overflow; hence in practice the loss of accuracy for close to is not excessive and the errors will be dominated by those of the standard function exp.
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 (s18asfe.f90)
9.2 Program Data
Program Data (s18asfe.d)
9.3 Program Results
Program Results (s18asfe.r)