NAG Library Routine Document

s13acf (integral_cos)


s13acf returns the value of the cosine integral
Cix=γ+lnx+0xcosu-1udu,  x>0  
via the routine name where γ denotes Euler's constant.


Fortran Interface
Function s13acf ( x, ifail)
Real (Kind=nag_wp):: s13acf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include <nagmk26.h>
double  s13acf_ (const double *x, Integer *ifail)


s13acf 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,
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).


NIST Digital Library of Mathematical Functions


1:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the function.
Constraint: x>0.0.
2:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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).

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, x=value.
Constraint: x>0.0.
The routine has been called with an argument less than or equal to zero for which Cix is not defined.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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.

Parallelism and Performance

s13acf is not threaded in any implementation.

Further Comments



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

Program Text

Program Text (s13acfe.f90)

Program Data

Program Data (s13acfe.d)

Program Results

Program Results (s13acfe.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