NAG FL Interfaces10acf (cosh)

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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 $|x|\le {E}_{1}\text{, }\mathrm{cosh}x=\frac{1}{2}\left({e}^{x}+{e}^{-x}\right)$.
For $|x|>{E}_{1}$, 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 $0$, $-1$ 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: $|{\mathbf{x}}|\le {E}_{1}$.
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 $|x|\ge 2$
 $ε∼xδ=Δ$
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)