Directly Plotting Mathematical Functions in R

Question

I am working with the R programming language.

In a previous question that I asked (Understanding 3D traces in Plotly and R: Evaluating and Plotting Functions Over a Grid), I learned how to plot mathematical functions by first evaluating the mathematical function at different points, then by plotting these points on a 3D grid, and finally "interpolating" a 3D surface over these points:

# set seed for reproducibility 
#load libraries
set.seed(123)
library(dplyr)
library(plotly)

#create more data
n <- 50
my_grid <- expand.grid(i1 = 1:n, i2 = 1:n)
my_grid$final_value = with(my_grid, sin(i1) + cos(i2) )

#make plot
plot_ly(data = my_grid,  x=~i1, y=~i2, z=~final_value, type='mesh3d', intensity = ~final_value, colors = colorRamp(c("blue", "grey", "red")))

I am trying to use this same approach to plot the following function (https://en.wikipedia.org/w/index.php?title=Test_functions_for_optimization&oldid=1030693803, https://en.wikipedia.org/w/index.php?title=Test_functions_for_optimization&oldid=1030693803#/media/File:ConstrTestFunc03.png) :

I first defined the function:

  my_function <- function(x,y) {
    
    final_value = (1 - x)^2 + 100*((y - x^2)^2)
     
    }

Then, I defined the "grid":

input_1 <- seq(-1.5, 1.5,0.1)
input_2 <- seq(-1.5, 1.5,0.1)


my_grid <- data.frame(input_1, input_2)
my_grid$final_value =   (1 - input_1)^2 + 100*((input_2 - input_1^2)^2)

Then, I tried to plot this function:

 x <- my_grid$input_1
y <- my_grid$input_2

z <- matrix(my_grid$final_value, nrow = length(x), ncol = length(y)) # proper matrix & dimensions

plot_ly(x = x, y = y, z = z) %>% add_surface()

My Problem: The final result does not look similar to the result from the Wikipedia page:

Can someone please show me what I am doing wrong? Is there an easier way to do this?

Thanks!

Answer 1

Your problem is that you are not actually creating a grid, you are creating a single vector of equal x, y points and running your formula on that, so your matrix is wrong (every column will be the same due to it being repeated). The easiest fix is to run outer on your function to evaluate it at every pair of input 1 and input 2:

z <- outer(input_1, input_2, my_function)

plot_ly(x = input_1, y = input_2, z = z) %>% add_surface()