NAG CL Interface
s13acc (integral_​cos)

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1 Purpose

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

2 Specification

#include <nag.h>
double  s13acc (double x, NagError *fail)
The function may be called by the names: s13acc, nag_specfun_integral_cos or nag_cos_integral.

3 Description

s13acc calculates an approximate value for Ci(x).
For 0<x16 it is based on the Chebyshev expansion
Ci(x)=lnx+r=0arTr(t),t=2 (x16) 2-1.  
For 16<x<xhi where the value of xhi is given in the Users' Note for your implementation,
where f(x)=r=0frTr(t) and g(x)=r=0grTr(t), t=2 ( 16x) 2-1.
For xxhi, Ci(x)=0 to within the accuracy possible (see Section 7).

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x double Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL 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 CL Interface for further information.
On entry, x=value.
Constraint: x>0.0.
The function has been called with an argument less than or equal to zero for which Ci(x) 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| |δcosx|and ​ |ε| | δ cosx Ci(x) | .  
That is accuracy will be limited by machine precision near the origin and near the zeros of cosx, but near the zeros of Ci(x) 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, Ci(x) 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 |Ci(x)|1/xE and hence is not significantly different from zero.
Hence xhi is chosen such that for values of xxhi, Ci(x) in principle would have values less than the machine precision and so is essentially zero.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s13acc is not threaded in any implementation.

9 Further Comments


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 5.4 patchlevel 6 −1.5 −1 −0.5 0 0.5 1 0 5 10 15 20 25 Ci(x) x "s13acfe.r" Example Program Returned Values for the Cosine Integral Ci(x)