# log1pmx

From DPQ v0.3-3
0th

Percentile

##### 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.

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

• log1pmx
##### 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)
# }

Documentation reproduced from package DPQ, version 0.3-3, License: GPL (>= 2)

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