Zoom Talk Draw Fractals from Root Finding Iterations

Samples codes:

Use NewtonRaphson root finding methods to draw some fractals for some Complex functions.

Create function newtonraphson

# ftn is the name of a function that has two output including f(x) and f'(x) 
# x0 is the starting point for the algorithm  
# tol is a good stop condition when |f(x)| <= tol for the algorithm, the default here is 1e-9 
# max.iter is a stop condition for the algorithm when n = max.itr 

newtonraphson <- function(ftn, x0, tol = 1e-9, max.iter) {
  # initialize 
  x <- x0
  fx <- ftn(x)
  iter <- 0
  
  # continue iterating until stopping conditions are met  
  while((abs(fx[1]) > tol) && (iter < max.iter)) {
    x <- x - fx[1]/fx[2]
    fx <- ftn(x)
    iter <- iter + 1
    # cat("At iteration", iter, "value of x is:", x, "\n")
  }
  
  # output depends on the success of the algorithm  
  if (abs(fx[1]) > tol){
    # cat("Algorithm failed to converge\n")
    return(data.frame(x0, root = NA, iter = NA))
  } else {
    # cat("Algorithm converged\n")
    return(data.frame(x0, root = x, iter))
  }
  
}

Draw graph for x^3-1

F1 <- function(x){
  return(c(x^3-1, 3*(x^2))) 
}

# create complex numbers 
x <- seq(-1, 1, length.out = 500)
y <- seq(-1, 1, length.out = 500)
z <- outer(x, 1i*y,'+')

# parallel processing using furrr
plan(multiprocess)

df <- z %>% future_map_dfr(~newtonraphson(F1,.,1e-9, 40), .progress = TRUE)

df$x <- Re(df$x0)
df$y <- Im(df$x0)

# color by iteration  
df %>% ggplot(aes(x = x, y = y)) + geom_raster(aes(fill = iter), interpolate = TRUE)+ scale_fill_gradientn(colors=brewer.pal(12, "Paired"))+ theme_void() + theme(legend.position = "none")

df %>% ggplot(aes(x = x, y = y)) + geom_raster(aes(fill = iter), interpolate = TRUE) + scale_fill_gradientn(colors = carto.pal("multi.pal"))+ theme_void() + theme(legend.position = "none")

df %>% ggplot(aes(x = x, y = y)) + geom_raster(aes(fill = iter), interpolate = TRUE) + scale_fill_gradientn(colors = carto.pal("turquoise.pal"))+ theme_void() + theme(legend.position = "none")

df %>% ggplot(aes(x = x, y = y)) + geom_raster(aes(fill = iter), interpolate = TRUE) + scale_fill_gradientn(colors = wes_palette("BottleRocket2")) + theme_void() + theme(legend.position = "none")

df %>% ggplot(aes(x = x, y = y)) + geom_raster(aes(fill = iter), interpolate = TRUE) + scale_fill_gradientn(colors = wes_palette("Rushmore1")) + theme_void() + theme(legend.position = "none")

Sample codes for Secant method

secant <- function(ftn, x0, x1, tol = 1e-9, max.iter) {
  # initialize 
  x_n0 <- x0
  x_n1 <- x1
  ftn_n0 <- ftn(x_n0)
  ftn_n1 <- ftn(x_n1)
  iter <- 0
  
  # continue iterating until stopping conditions are met  
  while((abs(ftn_n1) > tol) && (iter < max.iter)) {
    x_n2 <- x_n1 - ftn_n1*(x_n1 - x_n0)/(ftn_n1 - ftn_n0)
    x_n0 <- x_n1
    ftn_n0 <- ftn(x_n0)
    x_n1 <- x_n2
    ftn_n1 <- ftn(x_n1)
    iter <- iter + 1
    # cat("At iteration", iter, "value of x is:", x_n1, "\n")
  }
  
  return(c(x_n1, iter))
}   

Art Gallary

plots <- list.files("/img/fractals/")

for(i in plots){
  filename <- file.path("/img/fractals", i)
  cat("![text](",filename,")")
}