NAG FL Interfaces10acf (cosh)

1Purpose

s10acf returns the value of the hyperbolic cosine, $\mathrm{cosh}x$, via the function name.

2Specification

Fortran Interface
 Function s10acf ( x,
 Real (Kind=nag_wp) :: s10acf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s10acf_ (const double *x, Integer *ifail)
The routine may be called by the names s10acf or nagf_specfun_cosh.

3Description

s10acf calculates an approximate value for the hyperbolic cosine, $\mathrm{cosh}x$.
For $\left|x\right|\le {E}_{1}\text{, }\mathrm{cosh}x=\frac{1}{2}\left({e}^{x}+{e}^{-x}\right)$.
For $\left|x\right|>{E}_{1}$, the routine fails owing to danger of setting overflow in calculating ${e}^{x}$. The result returned for such calls is $\mathrm{cosh}{E}_{1}$, i.e., it returns the result for the nearest valid argument. The value of machine-dependent constant ${E}_{1}$ may be given in the Users' Note for your implementation.

4References

NIST Digital Library of Mathematical Functions

5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
2: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value 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 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: $\left|{\mathbf{x}}\right|\le {E}_{1}$.
The routine has been called with an argument too large in absolute magnitude. There is a danger of overflow. The result returned is the value of $\mathrm{cosh}x$ at the nearest valid argument.
${\mathbf{ifail}}=-99$
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 $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε≃xtanh⁡x×δ.$
That is, the relative error in the argument, $x$, is amplified by a factor, at least $x\mathrm{tanh}x$. The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is due to data errors etc.) but if $\delta$ is simply a result of round-off in the machine representation of $x$ then it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the error amplification factor is shown by the following graph:
Figure 1
It should be noted that near $x=0$ where this amplification factor tends to zero the accuracy will be limited eventually by the machine precision. Also, for $\left|x\right|\ge 2$
 $ε∼xδ=Δ$
where $\Delta$ is the absolute error in the argument $x$.

8Parallelism and Performance

s10acf 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 (s10acfe.f90)

10.2Program Data

Program Data (s10acfe.d)

10.3Program Results

Program Results (s10acfe.r)