workspace/sqrt.R

In JonasMoss/univariateML: Maximum Likelihood Estimation for Univariate Densities

y <- 2

# g <- function(x) {
#   (x^3 - x*y) / (2*y)
# }

# g <- function(x) {
#   (x^3 - x*y) / (3*x^2 - y)
# }

g <- function(x) {
  (x * (x^2 - y)) * (x * y + 1) / (2 * x^3 * y + x^2 + y)
}

h <- function(x) {
  (x^2 - y) / (2 * x)
}

f <- function(x) {
  x * (x^2 - y) / (x^2 + y)
}

f <- function(x) {
  p <- 1 / (2 * x)
  f <- (x^2 - y)
  df <- 2 * x
  f / (df - p * f)
  2 * (x^3 - x * y) / (3 * x^2 + y)
}

x0 <- 7
newton_raphson_1d(g, x0)
newton_raphson_1d(h, x0)
newton_raphson_1d(f, x0)



x <- seq(.1, 2, by = 0.000001)
y <- sqrt(x)
mod <- lm(y ~ x + I(x^2))
mod <- lm(y ~ x)
plot(x, y)
lines(x, predict(mod), col = "red")