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NAG Toolbox: nag_specfun_fresnel_c_vector (s20ar)

Purpose

nag_specfun_fresnel_c_vector (s20ar) returns an array of values for the Fresnel integral C(x)C(x).

Syntax

[f, ifail] = s20ar(x, 'n', n)
[f, ifail] = nag_specfun_fresnel_c_vector(x, 'n', n)

Description

nag_specfun_fresnel_c_vector (s20ar) evaluates an approximation to the Fresnel integral
xi
C(xi) = cos(π/2t2)dt
0
C(xi)=0xicos(π2t2)dt
for an array of arguments xixi, for i = 1,2,,ni=1,2,,n.
Note:  C(x) = C(x)C(x)=-C(-x), so the approximation need only consider x0.0x0.0.
The function is based on three Chebyshev expansions:
For 0 < x30<x3,
C(x) = x arTr(t),   with ​t = 2(x/3)41.
r = 0
C(x)=xr=0arTr(t),   with ​ t=2 (x3) 4-1.
For x > 3x>3,
C(x) = (1/2) + (f(x))/xsin(π/2x2)(g(x))/(x3)cos(π/2x2) ,
C(x)=12+f(x)xsin(π2x2)-g(x)x3cos(π2x2) ,
where f(x) = r = 0 brTr(t)f(x)=r=0brTr(t),
and g(x) = r = 0 crTr(t)g(x)=r=0crTr(t),
with t = 2 (3/x)41t=2 ( 3x) 4-1.
For small xx, C(x)xC(x)x. This approximation is used when xx is sufficiently small for the result to be correct to machine precision.
For large xx, f(x)1/π f(x) 1π  and g(x)1/(π2) g(x) 1π2 . Therefore for moderately large xx, when 1/(π2x3) 1π2x3  is negligible compared with (1/2) 12 , the second term in the approximation for x > 3x>3 may be dropped. For very large xx, when 1/(πx) 1πx  becomes negligible, C(x)(1/2) C(x) 12 . However there will be considerable difficulties in calculating sin(π/2x2)sin( π2x2) accurately before this final limiting value can be used. Since sin(π/2x2)sin( π2x2) is periodic, its value is essentially determined by the fractional part of x2x2. If x2 = N + θx2=N+θ, where NN is an integer and 0θ < 10θ<1, then sin(π/2x2)sin( π2x2) depends on θθ and on NN modulo 44. By exploiting this fact, it is possible to retain some significance in the calculation of sin(π/2x2)sin( π2x2) either all the way to the very large xx limit, or at least until the integer part of x/2 x2  is equal to the maximum integer allowed on the machine.

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.

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
C(xi)C(xi), the function values.
2:     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
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 cos(π/2x2) )/(C(x))|δ.
ε | x cos( π2 x2 ) C(x) |δ.
Figure 1 shows the behaviour of the error amplification factor |( x cos(π/2x2) )/(C(x))| | x cos( π2 x2 ) C(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 relative error.
For moderately large values of xx,
ε |2xcos(π/2x2)| δ
ε | 2x cos( π2 x2 ) | δ
and the result will be subject to increasingly large amplification of errors. However the above relation breaks down for large values of xx (i.e., when 1/(x2) 1x2  is of the order of the machine precision); in this region the relative error in the result is essentially bounded by 2/(πx) 2πx .
Hence the effects of error amplification are limited and at worst the relative error loss should not exceed half the possible number of significant figures.
Figure 1
Figure 1

Further Comments

None.

Example

function nag_specfun_fresnel_c_vector_example
x = [0; 0.5; 1; 2; 4; 5; 6; 8; 10; -1; 1000];
[f, ifail] = nag_specfun_fresnel_c_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e\n', x(i), f(i));
end
 

    X           Y
   0.000e+00   0.000e+00
   5.000e-01   4.923e-01
   1.000e+00   7.799e-01
   2.000e+00   4.883e-01
   4.000e+00   4.984e-01
   5.000e+00   5.636e-01
   6.000e+00   4.995e-01
   8.000e+00   4.998e-01
   1.000e+01   4.999e-01
  -1.000e+00  -7.799e-01
   1.000e+03   5.000e-01

function s20ar_example
x = [0; 0.5; 1; 2; 4; 5; 6; 8; 10; -1; 1000];
[f, ifail] = s20ar(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e\n', x(i), f(i));
end
 

    X           Y
   0.000e+00   0.000e+00
   5.000e-01   4.923e-01
   1.000e+00   7.799e-01
   2.000e+00   4.883e-01
   4.000e+00   4.984e-01
   5.000e+00   5.636e-01
   6.000e+00   4.995e-01
   8.000e+00   4.998e-01
   1.000e+01   4.999e-01
  -1.000e+00  -7.799e-01
   1.000e+03   5.000e-01


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