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 value x1 which is close to the desired solution and then we get a better approximation using Newton's Method:

math

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

 

Example 1

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Find the root of 2x2 − x − 2 = 0 between 1 and 2.


Answer


Functions with Multiple Roots

Many functions have multiple roots, so you need to understand what is going on and give the computer a guess close to your desired answer.

Let's see a LiveMath explanation of this.

LIVEMath

 

Example 2

Solve 1− t2 + 2t = 0

[Scientific Notebook cannot 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..]


Answer




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