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NAG Library Routine Document

s10acf (cosh)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

s10acf returns the value of the hyperbolic cosine,

\cosh x

, via the function name.

2

Specification

Fortran Interface

Function s10acf (

x, ifail)

Real (Kind=nag_wp)	::	s10acf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include nagmk26.h

double

s10acf_ ( const double *x, Integer *ifail)

3

Description

s10acf calculates an approximate value for the hyperbolic cosine,

\cosh x

For

|x| \leq E_{1}, \cosh x = \frac{1}{2} (e^{x} + e^{- x})

For

|x| > E_{1}

, the routine fails owing to danger of setting overflow in calculating

e^{x}

. The result returned for such calls is

\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.

4

References

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

5

Arguments

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 $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . 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

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: 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 $\cosh x$ at the nearest valid argument.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

δ

and

ε

are the relative errors in the argument and result, respectively, then in principle

ε ≃ x \tanh x \times δ .

That is, the relative error in the argument,

x

, is amplified by a factor, at least

x \tanh x

. The equality should hold if

δ

is greater than the machine precision (

δ

is due to data errors etc.) but if

δ

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| \geq 2

ε \sim x δ = Δ

where

Δ

is the absolute error in the argument

x

8

Parallelism and Performance

s10acf is not threaded in any implementation.

9

Further Comments

None.

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.

NAG Library Routine Document

s10acf (cosh)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results