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NAG Toolbox: nag_specfun_bessel_k1_real_vector (s18ar)

Purpose

nag_specfun_bessel_k1_real_vector (s18ar) returns an array of values of the modified Bessel function K1(x)K1(x).

Syntax

[f, ivalid, ifail] = s18ar(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_k1_real_vector(x, 'n', n)

Description

nag_specfun_bessel_k1_real_vector (s18ar) evaluates an approximation to the modified Bessel function of the second kind K1(xi)K1(xi) for an array of arguments xixi, for i = 1,2,,ni=1,2,,n.
Note:  K1(x)K1(x) is undefined for x0x0 and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
For 0 < x10<x1,
K1(x) = 1/x + xlnx arTr(t)x brTr(t),   where ​t = 2x21.
r = 0 r = 0
K1(x)=1x+xlnxr=0arTr(t)-xr=0brTr(t),   where ​ t=2x2-1.
For 1 < x21<x2,
K1(x) = ex crTr(t),   where ​t = 2x3.
r = 0
K1(x)=e-xr=0crTr(t),   where ​t=2x-3.
For 2 < x42<x4,
K1(x) = ex drTr(t),   where ​t = x3.
r = 0
K1(x)=e-xr=0drTr(t),   where ​t=x-3.
For x > 4x>4,
K1(x) = (ex)/(sqrt(x)) erTr(t),   where ​t = (9x)/(1 + x).
r = 0
K1(x)=e-xx r=0erTr(t),   where ​t=9-x 1+x .
For xx near zero, K1(x)1/x K1(x) 1x . This approximation is used when xx is sufficiently small for the result to be correct to machine precision. For very small xx it is impossible to calculate 1/x 1x  without overflow and the function must fail.
For large xx, where there is a danger of underflow due to the smallness of K1K1, the result is set exactly to zero.

References

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

Parameters

Compulsory Input Parameters

1:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n0n0.
The argument xixi of the function, for i = 1,2,,ni=1,2,,n.
Constraint: x(i) > 0.0xi>0.0, for i = 1,2,,ni=1,2,,n.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
nn, the number of points.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     f(n) – double array
K1(xi)K1(xi), the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)ivalidi contains the error code for xixi, for i = 1,2,,ni=1,2,,n.
ivalid(i) = 0ivalidi=0
No error.
ivalid(i) = 1ivalidi=1
xi0.0xi0.0, K1(xi)K1(xi) is undefined. f(i)fi contains 0.00.0.
ivalid(i) = 2ivalidi=2
xixi is too small, there is a danger of overflow. f(i)fi contains zero. The threshold value is the same as for ifail = 2ifail=2 in nag_specfun_bessel_k1_real (s18ad), as defined in the Users' Note for your implementation.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
  ifail = 2ifail=2
Constraint: n0n0.

Accuracy

Let δδ and εε be the relative errors in the argument and result respectively.
If δδ is somewhat larger than the machine precision (i.e., if δδ is due to data errors etc.), then εε and δδ are approximately related by:
ε |( x K0(x) K1(x) )/(K1(x))|δ.
ε | x K0(x)- K1(x) K1(x) |δ.
Figure 1 shows the behaviour of the error amplification factor
|( xK0(x) K1 (x) )/(K1(x))|.
| xK0(x) - K1 (x) K1(x) |.
However if δδ is of the same order as the machine precision, then rounding errors could make εε slightly larger than the above relation predicts.
For small xx, εδεδ and there is no amplification of errors.
For large xx, εxδεxδ and we have strong amplification of the relative error. Eventually K1K1, which is asymptotically given by (ex)/(sqrt(x)) e-xx , becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large xx the errors will be dominated by those of the standard function exp.
Figure 1
Figure 1

Further Comments

None.

Example

function nag_specfun_bessel_k1_real_vector_example
x = [0.4; 0.6; 1.4; 1.6; 2.5; 3.5; 6; 8; 10; 1000];
[f, ivalid, ifail] = nag_specfun_bessel_k1_real_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
 

    X           Y
   4.000e-01   2.184e+00    0
   6.000e-01   1.303e+00    0
   1.400e+00   3.208e-01    0
   1.600e+00   2.406e-01    0
   2.500e+00   7.389e-02    0
   3.500e+00   2.224e-02    0
   6.000e+00   1.344e-03    0
   8.000e+00   1.554e-04    0
   1.000e+01   1.865e-05    0
   1.000e+03   0.000e+00    0

function s18ar_example
x = [0.4; 0.6; 1.4; 1.6; 2.5; 3.5; 6; 8; 10; 1000];
[f, ivalid, ifail] = s18ar(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
 

    X           Y
   4.000e-01   2.184e+00    0
   6.000e-01   1.303e+00    0
   1.400e+00   3.208e-01    0
   1.600e+00   2.406e-01    0
   2.500e+00   7.389e-02    0
   3.500e+00   2.224e-02    0
   6.000e+00   1.344e-03    0
   8.000e+00   1.554e-04    0
   1.000e+01   1.865e-05    0
   1.000e+03   0.000e+00    0


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Chapter Introduction
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