log1pmx
Accurate log(1+x) - x
Compute $$\log(1+x) - x$$ accurately also for small \(x\), i.e., \(|x| \ll 1\).
- Keywords
- math
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 \(t := \frac{x}{2 + x}\), 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.
Value
a numeric vector (with the same attributes as x).
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)
# }