library(numDeriv)

k <- 3

# obliczenie wartości x
f <- function(k,x){
  return( x^(k-1) * (x + 3) * (x - 4) * (x - 7)^k )
}

# funkcja wzór
function_formula = expression(x^(k-1) * (x + 3) * (x - 4) * (x - 7)^k)

# pochodna wzó
derivative_formula <- D(function_formula, 'x') 

x <- seq(-3, 8.5, by=0.1)
y <- f(k,x)
g <- eval(derivative_formula)
startPoint <- -2


grad.descent <- function(x0 = startPoint, 
                         epsilon = 0.0001, 
                         alpha = 0.00001, 
                         i.max = 1e6,
                         k = 3){
  # gradient
  x <- x0
  gradient <- eval(function_formula)
  x.path <- x0 
  loss <- c()
  for (i in 1:i.max){
    x.new <- x0 + alpha * gradient # Update
    x <- x.new
    gradient <- eval(function_formula) # Gradient in new point
    points(x = x.new, y = f(k,x.new), pch = 20, col = 'green', cex = 0.5)
    currentLoss <- (f(k, x0) - f(k,x.new))^2
    print(currentLoss)
    loss <- append(loss, currentLoss )
    if (currentLoss < epsilon){ # STOP
      break 
    }
    x0 <- x.new
    x.path <- rbind(x.path, x.new)
  }
  return(list(x.new, x.path, i, loss))
}


plot(x, y, type="l", ylim = c(-15000, 30000))
lines(x, g, col="yellow")
abline(h = 0, col="gray")

result <- grad.descent(k = k)

# Wartość funkcji w znalezionym punkcie
round(f(k,result[[1]][1]), 3)
# Znaleziony punkt
round(result[[1]], 2) 

# Staring point
points(x = startPoint, y = f(k,startPoint), pch = 20, col = 'red', cex = 2)
points(x = result[[1]], y = f(k,result[[1]]), pch = 20, col = 'blue', cex = 2)

# loss
plot(result[[4]], type="l")