NAG FL Interface
s13acf (integral_​cos)

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1 Purpose

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

2 Specification

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 <nag.h>
double  s13acf_ (const double *x, Integer *ifail)
The routine may be called by the names s13acf or nagf_specfun_integral_cos.

3 Description

s13acf 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 Real (Kind=nag_wp) Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: 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, x=value.
Constraint: x>0.0.
The routine has been called with an argument less than or equal to zero for which Ci(x) is not defined.
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

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.
s13acf 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 (s13acfe.f90)

10.2 Program Data

Program Data (s13acfe.d)

10.3 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