NAG Library Function Document

nag_bessel_i1 (s18afc)


    1  Purpose
    7  Accuracy


nag_bessel_i1 (s18afc) returns a value for the modified Bessel function I1x.


#include <nag.h>
#include <nags.h>
double  nag_bessel_i1 (double x, NagError *fail)


nag_bessel_i1 (s18afc) evaluates an approximation to the modified Bessel function of the first kind I1x.
Note:  I1-x=-I1x, so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x4,
I1x=xr=0arTrt,   where ​t=2 x4 2-1;  
For 4<x12,
I1x=exr=0brTrt,   where ​t=x-84;  
For x>12,
I1x=exx r=0crTrt,   where ​t=2 12x -1.  
For small x, I1xx. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, the function must fail because I1x cannot be represented without overflow.


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications


1:     x doubleInput
On entry: the argument x of the function.
2:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, x=value.
Constraint: xvalue.
x is too large and the function returns the approximate value of I 1 x  at the nearest valid argument.


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 xI0x- I1x I1 x δ.  
Figure 1 shows the behaviour of the error amplification factor
xI0x - I1x I1x .  
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, εδ and there is no amplification of errors.
For large x, εxδ and we have strong amplification of errors. However the function must fail for quite moderate values of x because I1x would overflow; hence in practice the loss of accuracy for large x is not excessive. Note that for large x, the errors will be dominated by those of the standard math library function exp.

Parallelism and Performance

nag_bessel_i1 (s18afc) is not threaded in any implementation.

Further Comments



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

Program Text

Program Text (s18afce.c)

Program Data

Program Data (s18afce.d)

Program Results

Program Results (s18afce.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 I1(x) x Example Program Returned Values for the Bessel Function I1(x) gnuplot_plot_1
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017