Solve equation using optim() - r

Question

I have 2 equation that need to be solved by using optim
{(4x^2-20x+1/4 y^2+8=0
1/2 xy^2+2x-5y+8=0)
I have already run the code,but I'm confused if there should be 1 answer or 2 because function will only return the results for the last line
Should I do like this
> myfunc=function(x){
+ 4*x[1]^2-20*x[1]+(x[2]^2/4)+8
+ }
> optim(c(0,0),myfunc,method="BFGS")
and
> myfunc=function(x){
+ (1/2)*(x[1]*x[2]^2)+2*x[1]-5*x[2]+8
+ }
> optim(c(0,0),myfunc,method="BFGS")
or should I do like this
> myfunc=function(x){
+ 4*x[1]^2-20*x[1]+(x[2]^2/4)+8
+ (1/2)*(x[1]*x[2]^2)+2*x[1]-5*x[2]+8
+ }
> optim(c(0,0),myfunc,method="BFGS")
For the second one it still give me only the answer for the second function so which method is correct.

Answer 1

Minimize the sum of the squares of the two expressions that should equal zero and ensure that the value at the optimum equals 0 (up to floating point approximation).
myfunc <- function(z) {
x <- z[1]
y <- z[2]
(4*x^2-20* x + 1/4*y^2 + 8)^2 + (1/2 * x*y^2 + 2*x- 5*y + 8)^2
}
optim(c(0, 0), myfunc)
giving:
$par
[1] 0.5000553 2.0002986
$value
[1] 1.291233e-06
$counts
function gradient
67 NA
$convergence
[1] 0
$message
NULL

Answer 2

You can also use a package for solving systems of non linear equations such as nleqslv.
Slightly redefine your function by making it return a vector containing the result for each equation
myfunc <- function(x){
y <- numeric(length(x))
y[1] <- 4*x[1]^2-20*x[1]+(x[2]^2/4)+8
y[2] <- (1/2)*(x[1]*x[2]^2)+2*x[1]-5*x[2]+8
y
}
Define a starting value for the solver
xstart <- c(0,0)
Then do this
library(nleqslv)
nleqslv(xstart,myfunc)
giving
$x
[1] 0.5 2.0
$fvec
[1] -1.472252e-09 -7.081979e-10
$termcd
[1] 1
$message
[1] "Function criterion near zero"
$scalex
[1] 1 1
$nfcnt
[1] 7
$njcnt
[1] 1
$iter
[1] 7
There are more packages that can solve equation systems such as BB and pracma.