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New Word 2007 Document (12)

# New Word 2007 Document (12) - we know that the root lies in...

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The Bisection Idea The idea of the bisection method is very simple. We assume that we are given two values and ( ), and that the function is positive at one of these values (say at ) and negative at the other. (When this occurs, we say that is a ``change-of-sign interval'' for the function.) Assuming is continuous; we know that there must be at least one root in the interval. Intuitively speaking, if you divide a change-of-sign interval in half, one or the other half must end up being a change-of-sign interval, and it is only half as big. Keep dividing in half until the change-of-sign interval is so small that any point in it is within the specified tolerance. More precisely, consider the point . If we are done. (This is pretty unlikely). Otherwise, depending on the sign of
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Unformatted text preview: , we know that the root lies in or . In any case, our change-of-sign interval is now half as large as before. Repeat this process with the new change of sign interval until the interval is sufficiently small and declare victory. We are guaranteed to converge. We can even compute the maximum number of steps this will take, because if the original change-of-sign interval has length then after one bisection the current change-of-sign interval is of length , etc . We know in advance how well we will approximate the root . These are very powerful facts, which make bisection a robust algorithm--that is, it is very hard to defeat it....
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