NAG FL Interfaces13acf (integral_​cos)

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1Purpose

s13acf returns the value of the cosine integral
 $Ci(x)=γ+ln⁡x+∫0xcos⁡u-1udu, x>0$
via the routine name where $\gamma$ denotes Euler's constant.

2Specification

Fortran Interface
 Function s13acf ( x,
 Real (Kind=nag_wp) :: s13acf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s13acf_ (const double *x, Integer *ifail)
The routine may be called by the names s13acf or nagf_specfun_integral_cos.

3Description

s13acf calculates an approximate value for $\mathrm{Ci}\left(x\right)$.
For $0 it is based on the Chebyshev expansion
 $Ci(x)=ln⁡x+∑r=0′arTr(t),t=2 (x16) 2-1.$
For $16 where the value of ${x}_{\mathrm{hi}}$ is given in the Users' Note for your implementation,
 $Ci(x)=f(x)sin⁡xx-g(x)cos⁡xx2$
where $f\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{f}_{r}{T}_{r}\left(t\right)$ and $g\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{g}_{r}{T}_{r}\left(t\right)$, $t=2{\left(\frac{16}{x}\right)}^{2}-1$.
For $x\ge {x}_{\mathrm{hi}}$, $\mathrm{Ci}\left(x\right)=0$ to within the accuracy possible (see Section 7).

4References

NIST Digital Library of Mathematical Functions

5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2: $\mathbf{ifail}$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 $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}>0.0$.
The routine has been called with an argument less than or equal to zero for which $\mathrm{Ci}\left(x\right)$ is not defined.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7Accuracy

If $E$ and $\epsilon$ are the absolute and relative errors in the result and $\delta$ is the relative error in the argument then in principle these are related by
 $|E|≃ |δcos⁡x|and ​ |ε|≃ | δ cos⁡x Ci(x) | .$
That is accuracy will be limited by machine precision near the origin and near the zeros of $\mathrm{cos}x$, but near the zeros of $\mathrm{Ci}\left(x\right)$ only absolute accuracy can be maintained.
The behaviour of this amplification is shown in Figure 1.
For large values of $x$, $\mathrm{Ci}\left(x\right)\sim \frac{\mathrm{sin}x}{x}$, therefore, $\epsilon \sim \delta x\mathrm{cot}x$ and since $\delta$ is limited by the finite precision of the machine it becomes impossible to return results which have any relative accuracy. That is, when $x\ge 1/\delta$ we have that $|\mathrm{Ci}\left(x\right)|\le 1/x\sim E$ and hence is not significantly different from zero.
Hence ${x}_{\mathrm{hi}}$ is chosen such that for values of $x\ge {x}_{\mathrm{hi}}$, $\mathrm{Ci}\left(x\right)$ in principle would have values less than the machine precision and so is essentially zero.

8Parallelism and Performance

s13acf is not threaded in any implementation.

None.

10Example

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

10.1Program Text

Program Text (s13acfe.f90)

10.2Program Data

Program Data (s13acfe.d)

10.3Program Results

Program Results (s13acfe.r)