NAG FL Interface
s20aqf (fresnel_​s_​vector)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

s20aqf returns an array of values for the Fresnel integral S(x).

2 Specification

Fortran Interface
Subroutine s20aqf ( n, x, f, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s20aqf_ (const Integer *n, const double x[], double f[], Integer *ifail)
The routine may be called by the names s20aqf or nagf_specfun_fresnel_s_vector.

3 Description

s20aqf evaluates an approximation to the Fresnel integral
for an array of arguments xi, for i=1,2,,n.
Note:  S(x)=-S(-x), so the approximation need only consider x0.0.
The routine is based on three Chebyshev expansions:
For 0<x3,
S(x)=x3r=0arTr(t),   with ​ t=2 (x3) 4-1.  
For x>3,
S(x)=12-f(x)xcos(π2x2)-g(x)x3sin(π2x2) ,  
where f(x)=r=0brTr(t),
and g(x)=r=0crTr(t),
with t=2 ( 3x) 4-1.
For small x, S(x) π6x3. This approximation is used when x is sufficiently small for the result to be correct to machine precision. For very small x, this approximation would underflow; the result is then set exactly to zero.
For large x, f(x) 1π and g(x) 1π2 . Therefore, for moderately large x, when 1π2x3 is negligible compared with 12 , the second term in the approximation for x>3 may be dropped. For very large x, when 1πx becomes negligible, S(x)12 . However, there will be considerable difficulties in calculating cos( π2x2) accurately before this final limiting value can be used. Since cos( π2x2) is periodic, its value is essentially determined by the fractional part of x2. If x2=N+θ where N is an integer and 0θ<1, then cos( π2x2) depends on θ and on N modulo 4. By exploiting this fact, it is possible to retain significance in the calculation of cos( π2x2) either all the way to the very large x limit, or at least until the integer part of x2 is equal to the maximum integer allowed on the machine.

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: S(xi), the function values.
4: 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:
On entry, n=value.
Constraint: n0.
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.
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.
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 sin( π2x2) S(x) |δ.  
Figure 1 shows the behaviour of the error amplification factor | x sin( π2x2) S(x) | .
However, if δ is of the same order as the machine precision, then rounding errors could make ε slightly larger than the above relation predicts.
For small x, ε3δ and hence there is only moderate amplification of relative error. Of course for very small x where the correct result would underflow and exact zero is returned, relative error-control is lost.
For moderately large values of x,
ε |2xsin(π2x2)| δ  
and the result will be subject to increasingly large amplification of errors. However, the above relation breaks down for large values of x (i.e., when 1x2 is of the order of the machine precision); in this region the relative error in the result is essentially bounded by 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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s20aqf is not threaded in any implementation.

9 Further Comments


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 (s20aqfe.f90)

10.2 Program Data

Program Data (s20aqfe.d)

10.3 Program Results

Program Results (s20aqfe.r)