Root Finding
2. Use Newton's method to estimate the one ...

Question

Root Finding

2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

Accepted Answer

Hello. In this video, we are going to be using Newton's method to determine the one real solution of the given function. We are going to start with X0 equal to 2, and then find X 2. In order to approach this problem, let's just quickly review what Newton's method states. Newton's method is a recursive formula where XN + 1 is equal to XN minus the quantity F of XN divided by F of XN. So what this is saying is that if we want to find the next term in the recursive formula, we take a starting point and subtract the value of a function with that point plugged in, divided by its derivative. Now, in this problem, we are given X cubed minus 2X minus 5 is equal to 0. We are going to allow our F of X for Newton's method to be X cubed minus 2X minus 5. The next thing we want to do is we need to take its derivative. By using the power rule, we get F of X is equal to 3 X2 minus 2. Now, the note that is given to us is saying that our end goal is to find the second term in this recursive formula, but we are going to allow our starting value X 0 to equal to 2. So by setting up Newton's method, using our starting value and the two functions we have, in order to find the first term, we are going to have X1 equal to X0 minus F of X0 divided by F of X0. So let's go ahead and plug in the given values. We will have X1 equal to 2, minus 2 plugged into our FFX function, which is 2 cubed minus 2 multiplied by 2. 5 divided. By 3, multiplied by 2 squared minus 2. Now this is going to simplify to be 2. Plus 1 divided by 10, which is going to give us a value of 2.1. This is going to be our first term in the recursive formula X1. Now, what we want to do is, in order to determine the next term in the recursive formula, which is X2, we are going to take the previous result X1, plug it into Newton's method to solve for X 2. So that is going to give us 2.1 minus. F of 2.1, divided by F of 2.1. That will give us X 2 is equal to 2.1 minus the quantity, 2.1 cubed, minus 2 multiplied by 2.1, minus 5 is divided by 3 multiplied by 2.1 squared minus 2. Now this quantity is going to simplify to be 2.0946, and this is going to be the first real solution. To the given equation by using Newton's method. So I hope this video helps you in understanding how to use this approximation method, and I will go ahead and see you all in the next video.

Root Finding
2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

Key Concepts

Newton's Method

Derivative

Convergence of Iterative Methods

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