Solving a Polynomial of Any Degree
Date: 06/02/2003 at 14:43:27 From: David Subject: What method must be used to solve a polynomial of any degree? How to solve a polynomial of any degree? I want to make a computer program that is capable of solving any polynomial. Newton's method can solve many polynomials, but it does not always give correct answers: sometimes the answers oscillate and at other times it might never converge. Is there a better way of solving polynomials?
Date: 06/04/2003 at 08:55:58 From: Doctor George Subject: Re: What method must be used to solve a polynomial of any degree? Hi David, Thanks for writing to Doctor Math. Using Newton's method successfully can be a challenge. There are some very complex ways to solve your problem. A simpler way is to combine Newton's method with the Bisection method. Let's say that you know that f(a) < 0 and f(b) > 0. There is therefore a root of f(x) in the interval (a,b). Now follow this procedure. 1. Make a guess x_0 in (a,b) and compute x_1 using Newton's method. 2. If x_1 is in (a,b) then continue with Newton's method. 3. If x_1 is not in (a,b), then set x_1 = (a+b)/2. If f(x_1) < 0 the new interval is (x_1,b). Otherwise the new interval is (a,x_1). 4. Now go back to step 1 using x_1 in place of x_0 to get x_2, and so on until you converge. Let's say that you find that 'c' is a root. Then there is a g(x) such that f(x) = (x-c)g(x) g(x) = f(x)/(x-c) Now do Newton's method on g(x) to find the next root of f(x). Does this make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
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