logTaylor: Modified logarithm with derivatives

In smoothemplik: Smoothed Empirical Likelihood

Modified logarithm with derivatives

Description

Modified logarithm with derivatives

Usage

logTaylor(x, lower = NULL, upper = NULL, der = 0, order = 4)

Arguments

x	Numeric vector for which approximated logarithm is to be computed.
lower	Lower threshold below which approximation starts; can be a scalar of a vector of the same length as x.
upper	Upper threshold above which approximation starts; can be a scalar of a vector of the same length as x.
der	Non-negative integer: 0 yields the function, 1 and higher yields derivatives
order	Positive integer: Taylor approximation order. If NA, returns log(x) or its derivative.

Details

Provides a family of alternatives to -log() and derivative thereof in order to attain self-concordance and computes the modified negative logarithm and its first derivatives. For lower <= x <= upper, returns just the logarithm. For x < lower and x > upper, returns the Taylor approximation of the given order. 4th order is the lowest that gives self concordance.

Value

A numeric matrix with (order+1) columns containing the values of the modified log and its derivatives.

Examples

x <- seq(0.01^0.25, 2^0.25, length.out = 51)^4 - 0.11 # Denser where |f'| is higher
plot(x,  log(x)); abline(v = 0, lty = 2) # Observe the warning
lines(x, logTaylor(x, lower = 0.2), col = 2)
lines(x, logTaylor(x, lower = 0.5), col = 3)
lines(x, logTaylor(x, lower = 1, upper = 1.2, order = 6), col = 4)

# Substitute log with its Taylor approx. around 1
x <- seq(0.1, 2, 0.05)
ae <- abs(sapply(2:6, function(o) log(x) - logTaylor(x, lower=1, upper=1, order=o)))
matplot(x[x!=1], ae[x!=1,], type = "l", log = "y", lwd = 2,
  main = "Abs. trunc. err. of Taylor expansion at 1", ylab = "")

smoothemplik documentation built on Aug. 22, 2025, 1:11 a.m.