NAG Library Routine Document

S11ACF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S11ACF returns the value of the inverse hyperbolic cosine,

arccosh x

, via the function name. The result is in the principal positive branch.

2 Specification

FUNCTION S11ACF (

X, IFAIL)

REAL (KIND=nag_wp) S11ACF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X

3 Description

S11ACF calculates an approximate value for the inverse hyperbolic cosine,

arccosh x

. It is based on the relation

arccosh x = \ln (x + \sqrt{x^{2} - 1}) .

This form is used directly for

1 < x < 10^{k}

, where

k = n / 2 + 1

, and the machine uses approximately

n

decimal place arithmetic.

For

x \geq 10^{k}

\sqrt{x^{2} - 1}

is equal to

\sqrt{x}

to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate

arccosh x = \ln 2 + \ln x .

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.
Constraint: $X \geq 1.0$ .
2: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ 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 less than $1.0$ , for which $arccosh x$ is not defined. The result returned is zero.

7 Accuracy

δ

and

ε

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

|ε| ≃ |\frac{x}{\sqrt{x^{2} - 1} arccosh x} \times δ| .

That is the relative error in the argument is amplified by a factor at least

\frac{x}{\sqrt{x^{2} - 1} arccosh x}

in the result. The equality should apply if

δ

is greater than the machine precision (

δ

due to data errors etc.) but if

δ

is simply a result of round-off in the machine representation it is possible that an extra figure may be lost in internal calculation and round-off. The behaviour of the amplification factor is shown in the following graph:

Figure 1

It should be noted that for

x > 2

the factor is always less than

1.0

. For large

x

we have the absolute error

E

in the result, in principle, given by

E \sim δ .

This means that eventually accuracy is limited by machine precision. More significantly for

x

close to

1

x - 1 \sim δ

, the above analysis becomes inapplicable due to the fact that both function and argument are bounded,

x \geq 1

arccosh x \geq 0

. In this region we have

E \sim \sqrt{δ} .

That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.

8 Further Comments

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.

NAG Library Routine DocumentS11ACF