Automatically "recognizing" (calculus) derivatives - r

Question

I am using the R programming language. Using the "optim" library and the "BFGS" optimization algorithm, I am interested in optimizing the following function (also called the "Rosenbrock Function"):
If you define this function and the derivative of this function, it is pretty straightforward to optimize with the "optim" library and the BFGS algorithm (note: the BFGS algorithm requires knowledge of the function's derivative):
fr <- function(x) { ## Rosenbrock Banana function
x1 <- x[1]
x2 <- x[2]
100 * (x2 - x1 * x1)^2 + (1 - x1)^2
}
grr <- function(x) { ## Gradient of 'fr'
x1 <- x[1]
x2 <- x[2]
c(-400 * x1 * (x2 - x1 * x1) - 2 * (1 - x1),
200 * (x2 - x1 * x1))
}
res <- optim(c(-1.2,1), fr, grr, method = "BFGS")
> res
$par
[1] 1 1
$value
[1] 9.594956e-18
$counts
function gradient
110 43
$convergence
[1] 0
$message
NULL
Suppose you are working with a high dimensional complicated function - the derivative of this function will be difficult to manually evaluate and then write a function for this derivative (i.e. an additional place where you can make a mistake). Are there any "automatic" ways in R, such that if you write a mathematical function - R can automatically "infer" the derivative of this function?
For instance, in a new R session - would there have been some way to run the BFGS algorithm without explicitly defining the derivative?
fr <- function(x) { ## Rosenbrock Banana function
x1 <- x[1]
x2 <- x[2]
100 * (x2 - x1 * x1)^2 + (1 - x1)^2
}
#pseudo code
res <- optim(c(-1.2,1), fr, ??? , method = "BFGS")
Does anyone know if something like this is possible? Can R automatically infer the derivative?
I thought of an approach where you could use a preexisting "numerical differentiation" function in R to approximate the derivative at each iteration, and then feed this approximation into the BFGS algorithm, but that sounds very complicated and unnecessary.
It would have been nice if R could somehow automatically infer the derivative of a function.
References:
https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/optim

Answer 1

This is already built into the optim() function. If you don't specify the derivative it will be calculated numerically e.g.
fr <- function(x) { ## Rosenbrock Banana function
x1 <- x[1]
x2 <- x[2]
100 * (x2 - x1 * x1)^2 + (1 - x1)^2
}
optim(c(-1.2,1), fr, method = "BFGS")
# $par
# [1] 0.9998044 0.9996084
#
# $value
# [1] 3.827383e-08
#
# $counts
# function gradient
# 118 38
#
# $convergence
# [1] 0
#
# $message
# NULL
Note that (at least for this case) the solution is very close to that found using the analytical derivatives.