log1pmx {DPQ}R Documentation

Accurate log(1+x) - x

Description

Compute

log(1+x) - x

accurately also for small x, i.e., |x| << 1.

Usage

log1pmx(x, tol_logcf = 1e-14)

Arguments

x

numeric vector with values x > -1.

tol_logcf

a non-negative number indicating the tolerance (maximal relative error) for the auxiliary logcf() function.

Details

In order to provide full accuracy, the computations happens differently in three regions for x,

m_l = -0.79149064

is the first cutpoint,

x < ml or x > 1:

use log1pmx(x) := log1p(x) - x,

|x| < 0.01:

use t((((2/9 * y + 2/7)y + 2/5)y + 2/3)y - x),

x \in [ml,1], and |x| >= 0.01:

use t(2y logcf(y, 3, 2) - x),

where t := x/(2+x), and y := t^2.

Note that the formulas based on t are based on the (fast converging) formula

log(1+x) = 2(r + r^3/3 + r^5/5 + ...),

where r := x/(x+2), see the reference.

Value

a numeric vector (with the same attributes as x).

Author(s)

A translation of Morten Welinder's C code of Jan 2005, see R's bug issue PR#7307.

References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides links to the full text which is in public domain.
Formula (4.1.29), p.68.

See Also

logcf, the auxiliary function, lgamma1p which calls log1pmx, log1p

Examples

l1x <- curve(log1pmx, -.9999, 7, n=1001)
abline(h=0, v=-1:0, lty=3)
l1xz  <- curve(log1pmx, -.1, .1, n=1001); abline(h=0, v=0, lty=3)
l1xz2 <- curve(log1pmx, -.01, .01, n=1001); abline(h=0, v=0, lty=3)
l1xz3 <- curve(-log1pmx(x), -.002, .002, n=2001, log="y", yaxt="n")
sfsmisc::eaxis(2); abline(v=0, lty=3)

[Package DPQ version 0.4-2 Index]