NAG Library Function Document

nag_shifted_log (s01bac)

+− 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

nag_shifted_log (s01bac) returns a value of the shifted logarithmic function,

\ln (1 + x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_shifted_log (double x, NagError *fail)

3 Description

nag_shifted_log (s01bac) computes values of

\ln (1 + x)

, retaining full relative precision even when

|x|

is small. The function is based on the Chebyshev expansion

\ln \frac{1 + p^{2} + 2 p \bar{x}}{1 + p^{2} - 2 p \bar{x}} = 4 \sum_{k = 0}^{\infty} \frac{p^{2 k + 1}}{2 k + 1} T_{2 k + 1} (\bar{x}) .

Setting

\bar{x} = \frac{x (1 + p^{2})}{2 p (x + 2)}

, and choosing

p = \frac{q - 1}{q + 1}

q = \sqrt[4]{2}

the expansion is valid in the domain

x \in [\frac{1}{\sqrt{2}} - 1, \sqrt{2} - 1]

Outside this domain,

\ln (1 + x)

is computed by the standard logarithmic function.

4 References

Lyusternik L A, Chervonenkis O A and Yanpolskii A R (1965) Handbook for Computing Elementary Functions p. 57 Pergamon Press

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: $x > - 1.0$ .
2: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_REAL_ARG_LE: On entry, $x = ⟨value⟩$ .
Constraint: $x > - 1.0$ .

7 Accuracy

The returned result should be accurate almost to machine precision, with a limit of about

20

significant figures due to the precision of internal constants. Note however that if

x

lies very close to

- 1.0

and is not exact (for example if

x

is the result of some previous computation and has been rounded), then precision will be lost in the computation of

1 + x

, and hence

\ln (1 + x)

, in nag_shifted_log (s01bac).

8 Parallelism and Performance

Not applicable.

9 Further Comments

Empirical tests show that the time taken for a call of nag_shifted_log (s01bac) usually lies between about

1.25

and

2.5

times the time for a call to the standard logarithm function.

10 Example

The example program reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Function Documentnag_shifted_log (s01bac)