NAG FL Interface
s19apf (kelvin_​bei_​vector)

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1 Purpose

s19apf returns an array of values for the Kelvin function beix.

2 Specification

Fortran Interface
Subroutine s19apf ( 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  s19apf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s19apf or nagf_specfun_kelvin_bei_vector.

3 Description

s19apf evaluates an approximation to the Kelvin function beixi for an array of arguments xi, for i=1,2,,n.
Note:  bei(-x)=beix, so the approximation need only consider x0.0.
The routine is based on several Chebyshev expansions:
For 0x5,
beix = x24 r=0 ar Tr (t) ,   with ​ t=2 (x5) 4 - 1 ;  
For x>5,
beix = e x/2 2πx [(1+1xa(t))sinα-1xb(t)cosα]  
+ e x/2 2π x [(1+1xc(t))cosβ-1xd(t)sinβ]  
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 computed as beix= x24 . If this result would underflow, the result returned is beix=0.0.
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: 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.
3: f(n) Real (Kind=nag_wp) array Output
On exit: beixi, 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
abs(xi) is too large for an accurate result to be returned. f(i) contains zero. The threshold value is the same as for ifail=1 in s19abf , as defined in the Users' Note for your implementation.
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

Since the function is oscillatory, the absolute error rather than the relative error is important. Let E be the absolute error in the function, 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 impossible to calculate the functions 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

s19apf is not threaded in any implementation.

9 Further Comments

None.

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 (s19apfe.f90)

10.2 Program Data

Program Data (s19apfe.d)

10.3 Program Results

Program Results (s19apfe.r)