tlog: d-th derivative of the k-th-order Taylor expansion of log(x)

In smoothemplik: Smoothed Empirical Likelihood

d-th derivative of the k-th-order Taylor expansion of log(x)

Description

d-th derivative of the k-th-order Taylor expansion of log(x)

Usage

tlog(x, a = as.numeric(c(1)), k = 4L, d = 0L)

Arguments

x	Numeric: a vector of points for which the logarithm is to be evaluated
a	Scalar: the point at which the polynomial approximation is computed
k	Non-negative integer: maximum polynomial order in the Taylor expansion of the original function. k = 0 returns a constant.
d	Non-negative integer: derivative order Note that this function returns the d-th derivative of the k-th-order Taylor expansion, not the k-th-order approximation of the d-th derivative. Therefore, the degree of the resulting polynomial is d-k.

Value

The approximating Taylor polynomial around a of the order d-k evaluated at x.

Examples

cl <- rainbow(9, end = 0.8, v = 0.8, alpha = 0.8)
a <- 1.5
x <- seq(a*2, a/2, length.out = 101)
f <- function(x, d = 0)  if (d == 0) log(x) else ((d%%2 == 1)*2-1) * 1/x^d * gamma(d)
oldpar <- par(mfrow = c(2, 3), mar = c(2, 2, 2.5, 0.2))
for (d in 0:5) {
y <- f(x, d = d)
plot(x, y, type = "l", lwd = 7, bty = "n", ylim = range(0, y),
       main = paste0("d^", d, "/dx^", d, " Taylor(Log(x))"))
  for (k in 0:8) lines(x, tlog(x, a = a, k = k, d = d), col = cl[k+1], lwd = 1.5)
  points(a, f(a, d = d), pch = 16, cex = 1.5, col = "white")
}
legend("topright", as.character(0:8), title = "Order", col = cl, lwd = 1)
par(oldpar)

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