f <- function(x){
  return ( (x[1] -3)^2 + (x[2]-1)^2 )
}

grad_f <- function(x) {
  return ( c(2*(x[1]-3), 2*(x[2]-1) ) )
}

descent_gradient <- function(f, gradf, x, sigma, precision) {
  while (abs(f(x)) > precision){
    x <- x + sigma * -gradf(x)
  }
  return (x)
}

g <- function(x){
  return (-x^3 + x^2 -2*x +5)
}

dg <- function(x){
  return (-3*x^2 +2*x -2)
}

newton <- function (g, dg,  x, precision){
  while(abs(g(x)) > precision){
    x <- x - g(x)/dg(x)
  }
  return (x)
}


h <- function(x){
  return (10*x^4 -20*x^3 -90*x^2 +20*x +80)
}

dh <- function(x){
  return (40*x^3 -60*x^2 -180*x +20)
}

ddh <- function(x){
  return (120*x^2 -120*x -180)
}

main <- function(){
  x <- c(4,5)
  #cat(f(x), "\n")
  #cat(grad_f(x), "\n")

  xm <- descent_gradient(f, grad_f, x, sigma = 0.1, precision = 10^-6)
  cat(xm, " ", f(xm), "\n")

  xn <- newton(g, dg, 5, precision = 10^-6)
  cat(xn, " " , g(xn), "\n")

  xm <- newton(dh, ddh, 2, precision = 10^-6)
  cat(xm, " " , dh(xm), " ", h(xm) ,"\n")

  x <- seq(2.5,3.5, by = 0.01)
  plot(x, h(x), type ="l")

}