2. Newton's Method for Solving Equations

by M. Bourne

Computers use iterative methods to solve equations. The process involves making a guess at the true solution and then applying a formula to get a better guess and so on until we arrive at an acceptable approximation for the solution.

If we wish to find x so that `f(x) = 0` (a common type of problem), then we guess some initial value x0 which is close to the desired solution and then we get a better approximation using Newton's Method:

`x_1=x_0-(f(x_0))/(f'(x_0)`

[This is just based on the point-slope form of a straight line].

Example 1

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Find the root of

2x2x − 2 = 0

between `1` and `2`.

This example has another root, which is negative.

You can explore this example further in Newton's Method Interactive Graph.

Non-polynomial Functions with Multiple Roots

When using a computer to find roots of more complicated functions it's best to understand what is going on and give the computer a guess close to your desired answer.

Example 2

Solve 1− t2 + 2t = 0

[Certain math software is not able to find the solution directly for us. We need to know how to properly use the tool to get the solution, either with graphs or setting up Newton's Method. This could involve giving an initial estimate for the root.]

Further Examples

Example 3

Solve 3x3 − 9x2 + 5x + 2 = 0 using Newton's Method.

Example 4

Solve x2 = 0 using Newton's Method.

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