# Zoom Talk Draw Fractals from Root Finding Iterations

## 2020/04/23

I did a zoom talk for LA R Users April Meetup on an interesting class project: draw fractals in R.

I attached my slides here.

Also welcome to check out the youtube recording:

## 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,")")
}