NAG C Library Function Document

nag_shifted_log (s01bac)

1
Purpose

nag_shifted_log (s01bac) returns a value of the shifted logarithmic function, ln1+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 ln1+x, retaining full relative precision even when x is small. The function is based on the Chebyshev expansion
ln1+p2+2px- 1+p2-2px- =4k=0p2k+1 2k+1 T2k+1x-.  
Setting x-= x1+p2 2px+2 , and choosing p= q-1 q+1 , q=24 the expansion is valid in the domain x 12-1,2-1 .
Outside this domain, ln1+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 3.7 in How to Use the NAG Library and its Documentation).

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 ln1+x, in nag_shifted_log (s01bac).

8
Parallelism and Performance

nag_shifted_log (s01bac) is not threaded in any implementation.

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.

10.1
Program Text

Program Text (s01bace.c)

10.2
Program Data

Program Data (s01bace.d)

10.3
Program Results

Program Results (s01bace.r)