NAG FL Interface
s19aqf (kelvin_​ker_​vector)

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1 Purpose

s19aqf returns an array of values for the Kelvin function kerx.

2 Specification

Fortran Interface
Subroutine s19aqf ( n, x, f, ivalid, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(n)
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s19aqf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s19aqf or nagf_specfun_kelvin_ker_vector.

3 Description

s19aqf evaluates an approximation to the Kelvin function kerxi for an array of arguments xi, for i=1,2,,n.
Note:  for x<0 the function is undefined and at x=0 it is infinite so we need only consider x>0.
The routine is based on several Chebyshev expansions:
For 0<x1,
kerx=-f(t)log(x)+π16x2g(t)+y(t)  
where f(t), g(t) and y(t) are expansions in the variable t=2x4-1.
For 1<x3,
kerx=exp(-1116x) q(t)  
where q(t) is an expansion in the variable t=x-2.
For x>3,
kerx=π 2x e-x/2 [(1+1xc(t))cosβ-1xd(t)sinβ]  
where β= x2+ π8 , and c(t) and d(t) are expansions in the variable t= 6x-1.
When x is sufficiently close to zero, the result is computed as
kerx=-γ-log(x2)+(π-38x2) x216  
and when x is even closer to zero, simply as kerx=-γ-log( x2) .
For large x, kerx is asymptotically given by π 2x e-x/2 and this becomes so small that it cannot be computed without underflow and the routine fails.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x(n) Real (Kind=nag_wp) array Input
On entry: the argument xi of the function, for i=1,2,,n.
Constraint: x(i)>0.0, for i=1,2,,n.
3: f(n) Real (Kind=nag_wp) array Output
On exit: kerxi, the function values.
4: ivalid(n) Integer array Output
On exit: ivalid(i) contains the error code for xi, for i=1,2,,n.
ivalid(i)=0
No error.
ivalid(i)=1
xi is too large, the result underflows. f(i) contains zero. The threshold value is the same as for ifail=1 in s19acf , as defined in the Users' Note for your implementation.
ivalid(i)=2
xi0.0, the function is undefined. f(i) contains 0.0.
5: 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:
ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
ifail=2
On entry, n=value.
Constraint: n0.
ifail=-99
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.
ifail=-399
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Let E be the absolute error in the result, ε be the relative 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(ker1x+kei1x)|δ,  
ε |x2 ker1x + kei1x kerx | δ.  
For very small x, the relative error amplification factor is approximately given by 1|log(x)| , which implies a strong attenuation of relative error. However, ε in general cannot be less than the machine precision.
For small x, errors are damped by the function and hence are limited by the machine precision.
For medium and large x, the error behaviour, like the function itself, is oscillatory, and hence only the absolute accuracy for the function can be maintained. For this range of x, the amplitude of the absolute error decays like πx2e-x/2 which implies a strong attenuation of error. Eventually, kerx, which asymptotically behaves like π2x e-x/2, becomes so small that it cannot be calculated without causing underflow, and the routine returns zero. Note that for large x the errors are dominated by those of the standard function exp.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s19aqf is not threaded in any implementation.

9 Further Comments

Underflow may occur for a few values of x close to the zeros of kerx, below the limit which causes a failure with ifail=1.

10 Example

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

10.1 Program Text

Program Text (s19aqfe.f90)

10.2 Program Data

Program Data (s19aqfe.d)

10.3 Program Results

Program Results (s19aqfe.r)