DPQ (version 0.3-3)

log1pmx: Accurate log(1+x) - x

Description

Compute $$\log(1+x) - x$$ accurately also for small \(x\), i.e., \(|x| \ll 1\).

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.

Value

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

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 \(t := \frac{x}{2 + x}\), and \(y := t^2\).

Note that the formulas based on \(t\) are based on the (fast converging) formula $$\log(1+x) = 2\left(r + \frac{r^3}{3}+ \frac{r^5}{5} + \ldots\right),$$ where \(r := x/(x+2)\), see the reference.

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

# NOT RUN {
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)
# }