NAG FL Interface
s19aaf (kelvin_​ber)

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1 Purpose

s19aaf returns a value for the Kelvin function berx via the function name.

2 Specification

Fortran Interface
Function s19aaf ( x, ifail)
Real (Kind=nag_wp) :: s19aaf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s19aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s19aaf or nagf_specfun_kelvin_ber.

3 Description

s19aaf evaluates an approximation to the Kelvin function berx.
Note:  ber(-x)=berx, so the approximation need only consider x0.0.
The routine is based on several Chebyshev expansions:
For 0x5,
berx=r=0arTr(t),   with ​ t=2 (x5) 4-1.  
For x>5,
berx= ex/22πx [(1+ 1xa(t))cosα+ 1xb(t)sinα] + e-x/22πx [(1+ 1xc(t))sinβ+ 1xd(t)cosβ] ,  
where α= x2- π8 , β= x2+ π8 ,
and a(t), b(t), c(t), and d(t) are expansions in the variable t= 10x-1.
When x is sufficiently close to zero, the result is set directly to ber0=1.0.
For large x, 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.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, x=value.
Constraint: |x|value.
|x| is too large for an accurate result to be returned and the function returns zero.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Since the function is oscillatory, the absolute error rather than the relative error is important. Let E be the absolute error in the result and δ be the relative error in the argument. If δ is somewhat larger than the machine precision, then we have:
E |x2(ber1x+bei1x)|δ  
(provided E is within machine bounds).
For small x the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large x, the error behaviour is oscillatory and its amplitude grows like x2π ex/2. Therefore, it is not possible to calculate the function with any accuracy when xex/2> 2πδ . Note that this value of x is much smaller than the minimum value of x for which the function overflows.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s19aaf is not threaded in any implementation.

9 Further Comments


10 Example

This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

10.1 Program Text

Program Text (s19aafe.f90)

10.2 Program Data

Program Data (s19aafe.d)

10.3 Program Results

Program Results (s19aafe.r)