NAG FL Interface
s18asf (bessel_​i0_​real_​vector)

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1 Purpose

s18asf returns an array of values of the modified Bessel function I0(x).

2 Specification

Fortran Interface
Subroutine s18asf ( 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  s18asf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s18asf or nagf_specfun_bessel_i0_real_vector.

3 Description

s18asf evaluates an approximation to the modified Bessel function of the first kind I0(xi) for an array of arguments xi, for i=1,2,,n.
Note:  I0(-x)=I0(x), so the approximation need only consider x0.
The routine is based on three Chebyshev expansions:
For 0<x4,
I0(x)=exr=0arTr(t),   where ​ t = 2 (x4) -1.  
For 4<x12,
I0(x)=exr=0brTr(t),   where ​ t=x-84.  
For x>12,
I0(x)=exx r=0crTr(t),   where ​ t=2(12x) -1.  
For small x, I0(x)1. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, the routine must fail because of the danger of overflow in calculating ex.

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: I0(xi), 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. f(i) contains the approximate value of I0(xi) at the nearest valid argument. The threshold value is the same as for ifail=1 in s18aef , as defined in the 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

Let δ and ε be the relative errors in the argument and result respectively.
If δ is somewhat larger than the machine precision (i.e., if δ is due to data errors etc.), then ε and δ are approximately related by:
ε | x I1(x) I0 (x) |δ.  
Figure 1 shows the behaviour of the error amplification factor
| xI1(x) I0(x) |.  
Figure 1
Figure 1
However, if δ is of the same order as machine precision, then rounding errors could make ε slightly larger than the above relation predicts.
For small x the amplification factor is approximately x22 , which implies strong attenuation of the error, but in general ε can never be less than the machine precision.
For large x, εxδ and we have strong amplification of errors. However, for quite moderate values of x (x>x^, the threshold value), the routine must fail because I0(x) would overflow; hence in practice the loss of accuracy for x close to x^ is not excessive and the errors will be dominated by those of the standard function exp.

8 Parallelism and Performance

s18asf 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 (s18asfe.f90)

10.2 Program Data

Program Data (s18asfe.d)

10.3 Program Results

Program Results (s18asfe.r)