How to Solve a System of Equations in R

R is a robust statistical software that provides various tools to analyze data. Beyond its wide applications in data analysis, it’s also an excellent tool for solving mathematical problems, such as a system of linear equations.

A system of linear equations is a collection of one or more linear equations involving the same set of variables. For instance, a system of three equations in three variables looks like this:

a1*x + b1*y + c1*z = d1
a2*x + b2*y + c2*z = d2
a3*x + b3*y + c3*z = d3

This article will guide you on how to solve a system of equations in R.

Basics of Solving a System of Equations

To solve a system of linear equations in R, we use the function solve(). This function works well for systems where the number of equations equals the number of unknowns.

Let’s consider a simple system of two equations:

3*x + 4*y = 24
5*x + 6*y = 36

The system of equations can be represented in matrix form as AX = B, where A is the matrix of coefficients, X is the vector of variables, and B is the constant vector.

A <- matrix(c(3, 4, 5, 6), nrow=2)
B <- c(24, 36)

To solve this system, we pass the matrix A and the constant vector B to the solve() function:

X <- solve(A, B)
print(X)

Handling Non-Square Systems

What if the system of equations is not square, meaning the number of equations doesn’t match the number of unknowns? For instance, an underdetermined system has fewer equations than unknowns, while an overdetermined system has more equations than unknowns.

In such scenarios, you can use the qr.solve() function, which is more versatile than the solve() function and can handle both underdetermined and overdetermined systems.

Consider an overdetermined system:

x + 2*y = 2
3*x + 4*y = 3
5*x + 6*y = 4

You can represent it in matrix form and solve as follows:

A <- matrix(c(1, 2, 3, 4, 5, 6), nrow=3)
B <- c(2, 3, 4)
X <- qr.solve(A, B)
print(X)

For underdetermined systems, qr.solve() will return a solution (not necessarily unique), and for overdetermined systems, it will return a least-squares solution.

Solving Nonlinear Systems

While R’s built-in functions are handy for linear systems of equations, what about nonlinear systems? You can use the nleqslv package in R, which provides a function for solving systems of nonlinear equations.

First, install and load the package:

install.packages("nleqslv")
library(nleqslv)

Consider a nonlinear system:

x^2 + y^2 = 1
x^2 - y = 0

Define the system of equations as a function in R:

system <- function(z) {
    x <- z[1]
    y <- z[2]
    f1 <- x^2 + y^2 - 1
    f2 <- x^2 - y
    c(f1, f2)
}

Use the nleqslv() function to solve this system. The function requires an initial guess for the solution, which you can often set as a vector of ones:

start <- c(1, 1)
solution <- nleqslv(start, system)
print(solution$root)

Conclusion

R offers a variety of tools and functions to solve both linear and nonlinear systems of equations. From using the solve() function for square systems, qr.solve() for non-square or singular systems, to leveraging the nleqslv package for nonlinear systems, you have a diverse toolset at your disposal.

While this tutorial provides the basics, remember that the complexity of your system could necessitate more advanced strategies. When it comes to real-world applications, systems of equations often represent complex models with numerous variables and constraints. In such cases, you may need to employ more sophisticated numerical methods or optimization techniques.