New Word 2007 Document (25)

New Word 2007 Document (25) - "sufficiently small"...

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The Bisection Idea The idea of the bisection method is very simple. We assume that we are given two values A and B , and that the function F(X) is positive at one of these values (say at A ) and negative at the other. (When this occurs, we say that [A,B] is a change-of-sign interval for the function. So we know that (assuming F is continuous) that there must be one (at least one) root in the interval. Consider the point C = ( A + B ) / 2 If F(C)=0 we are done. (This is pretty unlikely). Otherwise, depending on the sign of F(C) , we know that the root lies in [A,C] or [C,B] . 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
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Unformatted text preview: "sufficiently small," then declare victory. We are guaranteed to converge. We can even compute the maximum number of steps this will take. We know in advance how well we will approximate the root X* . These are very powerful facts, which make bisection a robust algorithm - that is, it is very hard to defeat it. Discussion : If we know the start points A and B and the interval size tolerance TOL , we can predict beforehand the number of steps the bisection code will (probably) take. What is this formula? Discussion : Name a function which has a root in the interval [-1, 1], but for which bisection could not be used....
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This document was uploaded on 05/07/2010.

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