log1mexp {copula}

Compute f(a) = \mathrm{log}(1 +/- \mathrm{exp}(-a)) Numerically Optimally

Description

Compute f(a) = log(1 - exp(-a)), respectively g(x) = log(1 + exp(x)) quickly numerically accurately.

Usage

log1mexp(a, cutoff = log(2))
log1pexp(x, c0 = -37, c1 = 18, c2 = 33.3)

Arguments

a

numeric vector of positive values

x

numeric vector

cutoff

positive number; log(2) is “optimal”, but the exact value is unimportant, and anything in [0.5, 1] is fine.

c0, c1, c2

cutoffs for log1pexp; see below.

Value

f(a) == log(1 - exp(-a)) == log1p(-exp(-a)) == log(-expm1(-a))

or

g(x) == log(1 + exp(x)) == log1p(exp(x))

computed accurately and quickly

References

Martin Mächler (2012). Accurately Computing \log(1-\exp(-|a|)); https://CRAN.R-project.org/package=Rmpfr/vignettes/log1mexp-note.pdf.

Examples

a <- 2^seq(-58,10, length = 256)
fExpr <- expression(
          log(1 - exp(-a)),
          log(-expm1(-a)),
          log1p(-exp(-a)),
          log1mexp(a))
names(fExpr) <- c("DEF", "expm1", "log1p", "F")
str(fa <- do.call(cbind, as.list(fExpr)))
head(fa)# expm1() works here
tail(fa)# log1p() works here

## graphically:
lwd <- 1.5*(5:2); col <- adjustcolor(1:4, 0.4)
op <- par(mfcol=c(1,2), mgp = c(1.25, .6, 0), mar = .1+c(3,2,1,1))
  matplot(a, fa, type = "l", log = "x", col=col, lwd=lwd)
  legend("topleft", fExpr, col=col, lwd=lwd, lty=1:4, bty="n")
  # expm1() & log1mexp() work here

  matplot(a, -fa, type = "l", log = "xy", col=col, lwd=lwd)
  legend("left", paste("-",fExpr), col=col, lwd=lwd, lty=1:4, bty="n")
  # log1p() & log1mexp() work here
par(op)

curve(log1pexp, -10, 10, asp=1)
abline(0,1, h=0,v=0, lty=3, col="gray")

## Cutoff c1 for log1pexp() -- not often "needed":
curve(log1p(exp(x)) - log1pexp(x), 16, 20, n=2049)
## need for *some* cutoff:
x <- seq(700, 720, by=2)
cbind(x, log1p(exp(x)), log1pexp(x))

## Cutoff c2 for log1pexp():
curve((x+exp(-x)) - x, 20, 40, n=1025)
curve((x+exp(-x)) - x, 33.1, 33.5, n=1025)