NAG CL Interface
s19aac (kelvin_​ber)

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1 Purpose

s19aac returns a value for the Kelvin function berx.

2 Specification

#include <nag.h>
double  s19aac (double x, NagError *fail)
The function may be called by the names: s19aac, nag_specfun_kelvin_ber or nag_kelvin_ber.

3 Description

s19aac evaluates an approximation to the Kelvin function berx.
Note:  ber(-x)=berx, so the approximation need only consider x0.0.
The function 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 function must fail.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x double Input
On entry: the argument x of the function.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 for assistance.
See Section 7.5 in the Introduction to the NAG Library CL 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 CL Interface for further information.
On entry, x=value.
Constraint: |x|value.
|x| is too large for an accurate result to be returned and the function returns zero.

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.
s19aac 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 (s19aace.c)

10.2 Program Data

Program Data (s19aace.d)

10.3 Program Results

Program Results (s19aace.r)