1. Newton's Method for Functions of Several Variables
By Nick Bulinski and Justin Gilmore

2. Systems of Equations Multivariate Newton’s Method
Solving for a system of equations is not all that complicated for a system of linear equations, but not all equations are linear. The combination of nonlinear and more then one equation raises the difficulty significantly.
Multivariate Newton’s Method
One way to solve systems of equations with multiple variables is using multivariate Newton’s method. This method comes from
Newton’s original method which is
Newton's Method for Functions of Several Variables finds the roots for a system of nonlinear equations.

3. Multivariate Newton’s Method (cont)
Newton's one-variable method provides an outline for how the multi-variable case will work. Both are derived from the linear approximation given by the Taylor expansion. Before we get to that however we need define a few terms.
We will also need to take the derivative of For that we will need to compute what is known as the Jacobian matrix.

4. Jacobian Matrix
The Jacobian matrix is an analog to the derivative of f in the one variable case. It is defined as the matrix of all first partial derivatives of a vector function F(v) s.t.
ex:

5. Putting it all Together
Now we have all the pieces we need to make the Taylor expansion: or
We then derive the algorithm by solving the second equation for r
for k = 0,1,2,………

6. Example 𝐷𝐹 𝑣 −1 = 1 2𝑥+4𝑦 − 2𝑦 2𝑥+4𝑦 2 2𝑥+4𝑦 2𝑥 2𝑥+4𝑦
𝐷𝐹 𝑣 −1 = 1 2𝑥+4𝑦 − 2𝑦 2𝑥+4𝑦 2 2𝑥+4𝑦 2𝑥 2𝑥+4𝑦
𝐷𝐹 𝑣 = 2𝑥 2𝑦 −2 1
𝑣 1 =
𝑣 2 =
𝑣 3 =

7. Advantages & Disadvantages
Different solutions can be found with a different starting guess
Fast Convergence
Disadvantages
Will only work if the Jacobian can be computed
If the Jacobian is singular the algorithm breaks
Number of iterations can not be determined before the algorithm begins