NAG Library Function Document
nag_bessel_k1_vector (s18arc) returns an array of values of the modified Bessel function .
||nag_bessel_k1_vector (Integer n,
const double x,
nag_bessel_k1_vector (s18arc) evaluates an approximation to the modified Bessel function of the second kind for an array of arguments , for .
Note: is undefined for and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
. This approximation is used when
is sufficiently small for the result to be correct to machine precision
. For very small
it is impossible to calculate
without overflow and the function must fail.
For large , where there is a danger of underflow due to the smallness of , the result is set exactly to zero.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
n – IntegerInput
On entry: , the number of points.
x[n] – const doubleInput
On entry: the argument of the function, for .
, for .
f[n] – doubleOutput
On exit: , the function values.
ivalid[n] – IntegerOutput
contains the error code for
- No error.
- , is undefined. contains .
- is too small, there is a danger of overflow. contains zero. The threshold value is the same as for NE_REAL_ARG_TOO_SMALL in nag_bessel_k1 (s18adc), as defined in the Users' Note for your implementation.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, argument had an illegal value.
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
On entry, at least one value of x
for more information.
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 the machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small , and there is no amplification of errors.
For large , and we have strong amplification of the relative error. Eventually , which is asymptotically given by , becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large the errors will be dominated by those of the standard function exp.
8 Parallelism and Performance
This example reads values of x
from a file, evaluates the function at each value of
and prints the results.
10.1 Program Text
Program Text (s18arce.c)
10.2 Program Data
Program Data (s18arce.d)
10.3 Program Results
Program Results (s18arce.r)