Newton-Raphson Method for Nonlinear Systems of Equations

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  • 0:03 What is the…
  • 1:03 Setting Up for this Method
  • 2:07 Calculus & Linear…
  • 3:40 Packaging the Information
  • 4:27 The Newton-Raphson Method
  • 7:47 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Solving a system of nonlinear equations might sound complicated! But anyone who can multiply and add can get a solution by following the Newton-Raphson method. In this lesson, we will explain step-by-step just how this is done.

What is the Newton-Raphson Method?

You and your friend are playing a number guessing game. Your friend is thinking of a number between 1 and 10. You guess 7, and your friend says the number needs to be less. Your friend is totally honest, so the number to determine stays the same during the game. He's also totally helpful. In fact, rather than merely telling you if your number needs to be less or more, your friend suggests what number to try next!

Are you thinking of a number?
Thinking of the number 5.

That's pretty much what happens if your friend is Newton-Raphson. We are presented with the problem of finding a solution to a system of equations. In this case, several unknown numbers have to be determined. We make a guess. Newton-Raphson suggests the next numbers to try. We keep using the suggestions and quickly get to the solution. Amazingly, the Newton-Raphson method doesn't know the solution ahead of time; it can only suggest the next number to try. We have the solution when the number suggested is very close to the last suggestion.

Setting up for This Method

Let's look at an example. We'd like to find the values for x and y that solve the following system of two equations:

A system of two equations.

This example is for two equations, but the work can be extended to larger numbers of equations. Also, these are nonlinear equations where our usual solution methods will not work. Newton-Raphson is an iterative method, meaning we'll get the correct answer after several refinements on an initial guess. We start by writing each equation with all the terms on the same side. Then we make them equal to functions we can call f1 and f2:

The equations written in f = 0 format.

If you can differentiate functions like x to a power, that will take care of the calculus part. If you can find the inverse of a matrix and multiply matrices, we're good for the linear algebra part. Here's a brief review.

Calculus and Linear Algebra Review

If we want the derivative of a function like x^3, we put the exponent 3 in front, and we decrease the exponent by 1. The derivative of x^3 becomes 3x^2. Just for practice, and because we'll use these results, let's find the derivatives of f1 and f2.

First, the derivative of f1 - but hold on! Do we differentiate with respect to x or with respect to y? The answer is, we do both. This is called taking a partial derivative. We differentiate f1 by treating x as the variable and everything else as a constant. When we take the partial derivative of x^2 y with respect to x, we treat y as a constant, giving us 2xy as the derivative. Here is the partial derivative of f1 with respect to x:

Derivative of f1 wrt x.

We can also have a partial derivative with respect to y, and we can do the same with f2. That gives us three more results and three more opportunities to practice with partial derivatives:


Now let's briefly review linear algebra. If we had a matrix with 2 rows and 2 columns, we could find the inverse using this:

Inverse of a 2x2 matrix.

To multiply two matrices together, we would use this:

Multiplying 2 matrices together.

Packaging the Information

To make things more compact and organized, we store those partial derivatives in a special matrix. It's called the Jacobian and is labeled 'J'. In general, this is the Jacobian for two equations:

General expression for a Jacobian.

We can drop those partial derivatives expressions into the Jacobian to get this:

The Jacobian for our example.

We're almost there! We need some matrices with 2 rows and 1 column to store the other information. This type of matrix is called a column vector. Here are those column vectors:

Column vectors.

The F vector contains our two equations, f1 and f2, but evaluated with the current values for x and y. The v vector contains our current x and y values.

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