Solving a system of two equations

Hi,

I am trying to find a simple way to numerically solve a system of two
equations equal to zero with two unknowns (x_loc and y_loc). Here is a mock
data set and below it, the equations I need to solve.

theta<-c(180,135,90)/(2*pi)
x<-c(0,0,15)
y<-c(20,0,0)

0 =
-sum((y_loc-y)*(sin(theta)*(x_loc-x)-cos(theta)*(y_loc-y))/(((x_loc-x)^2+(y_loc-y)^2)^0.5)^3)
0 =
-sum((x_loc-x)*(sin(theta)*(x_loc-x)-cos(theta)*(y_loc-y))/(((x_loc-x)^2+(y_loc-y)^2)^0.5)^3)

You can use package nleqslv for solving systems of nonlinear equations.
Example for your system:

library(nleqslv)

theta<-c(180,135,90)/(2*pi)
x<-c(0,0,15)
y<-c(20,0,0)

f <- function(par) {
    fval  <- numeric(length(par))
    x_loc <- par[1]
    y_loc <- par[2]

    fval[1] <- -sum((y_loc-y)*(sin(theta)*(x_loc-x)-cos(theta)*(y_loc-y))/(((x_loc-x)^2+(y_loc-y)^2)^0.5)^3)
    fval[2] <- -sum((x_loc-x)*(sin(theta)*(x_loc-x)-cos(theta)*(y_loc-y))/(((x_loc-x)^2+(y_loc-y)^2)^0.5)^3)

    fval
}

nleqslv(c(17,17), f)

I haven't got a clue about good starting values.
I get the impression is that your system is quite sensitive.
nleqslv finds a solution but it is very different from the starting value. It's up to you to verify the correctness of your system.

You can simplify the computation in your function by computing the common subexpressions only once

f1 <- function(par) {
    fval  <- numeric(length(par))
    x_loc <- par[1]
    y_loc <- par[2]
    
    A <- (sin(theta)*(x_loc-x)-cos(theta)*(y_loc-y))
    B <- (((x_loc-x)^2+(y_loc-y)^2)^0.5)^3
    C <- A/B
    fval[1] <- -sum((y_loc-y)*C)
    fval[2] <- -sum((x_loc-x)*C)

    fval
}    


Berend