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NAG Library Manual

# NAG Library Routine DocumentS10ABF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S10ABF returns the value of the hyperbolic sine, $\mathrm{sinh}x$, via the function name.

## 2  Specification

 FUNCTION S10ABF ( X, IFAIL)
 REAL (KIND=nag_wp) S10ABF
 INTEGER IFAIL REAL (KIND=nag_wp) X

## 3  Description

S10ABF calculates an approximate value for the hyperbolic sine of its argument, $\mathrm{sinh}x$.
For $\left|x\right|\le 1$ it uses the Chebyshev expansion
 $sinh⁡x=x×yt=x∑′r=0arTrt$
where $t=2{x}^{2}-1$.
For $1<\left|x\right|\le {E}_{1}\text{, }\mathrm{sinh}x=\frac{1}{2}\left({e}^{x}-{e}^{-x}\right)$
where ${E}_{1}$ is a machine-dependent constant, details of which are given in the Users' Note for your implementation.
For $\left|x\right|>{E}_{1}$, the routine fails owing to the danger of setting overflow in calculating ${e}^{x}$. The result returned for such calls is $\mathrm{sinh}\left(\mathrm{sign}x{E}_{1}\right)$, i.e., it returns the result for the nearest valid argument.

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5  Parameters

1:     X – REAL (KIND=nag_wp)Input
On entry: the argument $x$ of the function.
2:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
The routine has been called with an argument too large in absolute magnitude. There is a danger of setting overflow. The result is the value of $\mathrm{sinh}x$ at the closest argument for which a valid call could be made. (See Section 3 and the Users' Note for your implementation.)

## 7  Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε≃ xcoth⁡x×δ.$
That is the relative error in the argument, $x$, is amplified by a factor, approximately $x\mathrm{coth}x$. The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is a result of 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 can be seen in the following graph:
Figure 1
It should be noted that for $\left|x\right|\ge 2$
 $ε∼xδ=Δ$
where $\Delta$ is the absolute error in the argument.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s10abfe.f90)

### 9.2  Program Data

Program Data (s10abfe.d)

### 9.3  Program Results

Program Results (s10abfe.r)