NAG Library Function Document

nag_cos_integral (s13acc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_cos_integral (s13acc) returns the value of the cosine integral
Cix=γ+lnx+0xcosu-1udu,  x>0  
where γ denotes Euler's constant.

2
Specification

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

3
Description

nag_cos_integral (s13acc) calculates an approximate value for Cix.
For 0<x16 it is based on the Chebyshev expansion
Cix=lnx+r=0arTrt,t=2 x16 2-1.  
For 16<x<xhi where the value of xhi is given in the Users' Note for your implementation,
Cix=fxsinxx-gxcosxx2  
where fx=r=0frTrt and gx=r=0grTrt, t=2 16x 2-1.
For xxhi, Cix=0 to within the accuracy possible (see Section 7).

4
References

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

5
Arguments

1:     x doubleInput
On entry: the argument x of the function.
Constraint: x>0.0.
2:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR
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.
NE_NO_LICENCE
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.
NE_REAL_ARG_LE
On entry, x=value.
Constraint: x>0.0.
The function has been called with an argument less than or equal to zero for which Cix is not defined.

7
Accuracy

If E and ε are the absolute and relative errors in the result and δ is the relative error in the argument then in principle these are related by
E δ cosx and ​ ε δ cosx Cix .  
That is accuracy will be limited by machine precision near the origin and near the zeros of cosx, but near the zeros of Cix only absolute accuracy can be maintained.
The behaviour of this amplification is shown in Figure 1.
Figure 1
Figure 1
For large values of x, Cix sinxx  therefore εδxcotx and since δ is limited by the finite precision of the machine it becomes impossible to return results which have any relative accuracy. That is, when x1/δ we have that Cix1/xE and hence is not significantly different from zero.
Hence xhi is chosen such that for values of xxhi, Cix in principle would have values less than the machine precision and so is essentially zero.

8
Parallelism and Performance

nag_cos_integral (s13acc) is not threaded in any implementation.

9
Further Comments

None.

10
Example

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

10.1
Program Text

Program Text (s13acce.c)

10.2
Program Data

Program Data (s13acce.d)

10.3
Program Results

Program Results (s13acce.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −1.5 −1 −0.5 0 0.5 1 0 5 10 15 20 25 Ci(x) x Example Program Returned Values for the Cosine Integral Ci(x) gnuplot_plot_1
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017