Roots and the Bisection MethodDate: 08/01/98 at 21:27:12 From: JCG Subject: Polynomials What is the Bisectional Method? How would I solve 2x^3+x^2+5x+2 = 0 or any other polynomial? Can the method be used by something greater than the 3rd degree? Date: 08/05/98 at 14:30:39 From: Doctor Benway Subject: Re: Polynomials The bisection method is one of my favorite surefire methods for finding roots of polynomials. The great thing about this method is that it works for everything, at least everything normal that you're likely to run into at this point in your mathematical education. Have you ever watched the game show "The Price is Right"? That show actually uses a form of the bisection method, in a way. For example: Bob Barker: This lovely toaster oven/can opener from General Electric can be yours if you can guess the price within thirty seconds. Start the clock. Contestant: Ummm....$20 Bob Barker: Higher Contestant: $100 Bob Barker: Lower Contestant: $60 Bob Barker: Lower Contestant: $40 Bob Barker: Higher Contestant: $50 Bob Barker: Correct! You win! Contestant: Wheeeeeeeee! As you notice, the contestant made two initial guesses, $20 and $100, which were too low and too high respectively. This told her that the price was between these two guesses. She then guessed halfway between, $60, and found that was too high, so it must be between $20 and $60. Her next guess, $40, was too low, so she knew it was between $40 and $60. She guessed $50, halfway between, and she was correct! This example turns out to be very similar to how the bisection method works. The usual method is to first plot a function to get an idea about where the roots are. Say f(x) has a root somewhere between 2 and 3 when you graph it. Your goal is to guess numbers as above until you get a number that turns out to be zero when you plug it in. Your initial "guesses" are 2 and 3. To find out whether you need to go "higher" or "lower," you plug the numbers you are guessing into the function. In this case, f(x) being positive or negative will tell you whether you need to go higher or lower. The best way to do this is to look at the graph to tell which way you should go. If you have an increasing function on that interval, a negative number means you guessed too low, and a positive number means you guessed too high (see below): / + / too high ___________*__________ / - / too / low / If the function is decreasing, then the opposite rule applies. Where "bisection" comes in is that the best way to do this is to pick the midpoint of the interval you know the root to lie in, just as the contestant picked midway between the price range she knew. Eventually you will come closer and closer to the answer, and when you feel you have an x where f(x) is close enough to being zero, then you can put it down as an approximation for the root. This works for all continuous functions. It's not the best method, but it is certainly the easiest. If you take a class such as Numerical Analysis, you'll learn better methods for approximating roots. Hope this helps. - Doctor Benway, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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