Newton's Method for Finding Roots of FunctionsDate: 06/03/2003 at 17:21:56 From: Melissa Subject: Newton's method for finding roots Hi Dr. Math, I was recently assigned the task of teaching the class about Newton's Method for Finding Roots; however, I am having problems learning it myself. Is there an easy way to learn and understand this method or an example that would allow me to understand the steps taken to work through the problem? Your help is greatly appreciated. Date: 06/04/2003 at 08:39:03 From: Doctor George Subject: Re: Newton's method for finding roots Hi Melissa, Thanks for writing to Doctor Math. Newton's method starts with a guess at the correct solution for the root. Let's say you have f(x) and our initial guess is that f(x) = 0 when x = x_0. The process is to use this initial x_0 to find a better estimate that we will call x_1. The way we do this is to draw a line tangent to f(x) at the point (x_0, f(x_0)). Call the line g(x). To find g(x) we need to compute f'(x) at x = x_0, or f'(x_0). Using the point-slope equation we can write g(x) like this. g(x) - f(x_0) = f'(x_0) (x - x_0) Now we ask where g(x) = 0. This happens where x = x_0 - f(x_0) / f'(x_0). This x is our x_1. Now repeat the process by constructing the tangent to f(x) at x_1, and finding x_2 where g(x) = 0 for this new tangent line. Try drawing these tangent lines to see what is happening. In general x_(i+1) = x_i - f(x_i) / f'(x_i) We repeat this equation until we converge. Each time we use this equation is called an iteration. Now let's do an example. Let's say that f(x) = x^2 - 4, and pretend that we do not know that the roots are x = 2 and -2. By checking f(1.8) and f(2.2) we can tell that there must be a root between 1.8 and 2.2. Let's set x_0 = 2.1 as our guess and compute f'(x) = 2x. The equation that we interate is x_(i+1) = x_i - f(x_i) / f'(x_i). = x_i - [(x_i)^2 - 4] / [2(x_i)] From this we compute x_1 = 2.1 - [(2.1)^2 - 4] / (2*2.1) = 2.00238 x_2 = 2.00238 - [(2.00238)^2 - 4] / (2*2.00238) = 2.00119 x_3 = 2.00119 - [(2.00119)^2 - 4] / (2*2.00119) = 2.00000... If we had started with a guess of -2.1 we would have found the other root of the equation. As long as the function is well-behaved in the region between the initial guess and the root, Newton's method will converge nicely. Coming up with the initial guess is an important step in getting Newton's method to work correctly. That's a start on Newton's method. Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/ |
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