Root Finding
5. Use Newton's method to find the positive...

Question

Root Finding

5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and find x_2.

Accepted Answer

Welcome back, everyone. In this problem using Newton's method, estimate the one real solution of the function X the 4th minus 3X minus 1 equals 0. Start with X0 equals 1.6, and then find X2. Now in this problem, it tells us that we need to apply Newton's method to estimate the one real solution of the function. What does Newton's method say? We'll recall that Newton's method basically tells us that a term from our function XN is going to be equal to X, or sorry. XN + 1, let's say it's XN + 1 is gonna be equal to XN minus F of XN divided by the derivative of F of XN. So what this means is that first we need to find our function, then starting with X not, calculate X1 and then X2. Essentially, we're going to apply Newton's method twice. To do that, we'll need FFX and the derivative of FFX. So now, if we let FFX be equal to X 4 minus 3X minus 1. Then the derivative of F of X is going to be equal to 4 x cubed minus 3. If that's the case, then give X not, OK, equals 1.6, we can use it to help us find X1, because this would imply. That X1 is going to be equal to X minus F of X0 divided by the derivative of F of X0. So if we apply that, that's gonna be equal to 1.6 minus F of 1.6 divided by the derivative of F of 1.6. And to find both of those, we're simply going to substitute 1.6 into the function and its derivative. So now by applying that. That means X1 is gonna be equal to 1.6 minus. Uh, of 1.6, so that's 1.6 to the fourth minus 3 multiplied by 1.6. OK, 3 multiplied by 1.6 minus 1. And now that is going to be divided by F of 1.6, so that's 4 multiplied by 1.6 cubed minus 3. No, in that case, this is gonna be equal to 1.6 minus 0.7536. That's our numerator divided by 13.384 or denominator. And when we calculate here, we get X1 to be approximately equal to 1.54369. No, we're just going to repeat the process for X2. That's why we're, we're iterating, right? So now, in this case, for a second iteration, then X2 is gonna be equal to X1 minus F of X1. Divided by F of X1. So that will be 1.54369 minus F of 1.54369 and by now we realize we're just gonna substitute it into our uh we, we can just put it into our expression. So that's 1.54369 to the 4th, minus 3 multiplied by 1.54369 minus 1, OK. All divided. By 4 multiplied by 1.54369 cubed. -3. Now, in this case, then we're gonna have 1.54369 minus in our numerator 0.04752 divided by 11.71432 in our denominator. And this is going to be approximately equal to 1.53963. Thus, by Newton's method, then the one real solution of the given function X2 is 1.53963 or approximately 1.53963. Thanks a lot for watching everyone. I hope this video helped.

Root Finding
5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and find x_2.

Key Concepts

Newton's Method

Derivative

Convergence of Iterative Methods

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