Bisection

# Bisection - 6.3 Finding Roots 1 6.3 Finding Roots Problem...

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6.3. Finding Roots 1 6.3 Finding Roots Problem Compute the all the roots of the function T ( x ) = 512 x 10 1280 x 8 + 1120 x 6 400 x 4 + 50 x 2 1 that are in the inteval [0 , 1]. The roots should be accurate through six decimals. Program Development Let’s see what T looks like across the interval on interest: 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 512x 10 - 1280x 8 + 1120x 6 - 400x 4 + 50x 2 -1 We observe that there are fi ve zeros in [0,1]. Our plan is to systematically use the method of bisection to fi nd these roots. Bisection exploits the following fact: If a continuous function f changes sign on the interval [ a, b ] , then f must have a root somewhere in [ a, b ] . If f ( a ) f ( b ) 0, then we say that [ a, b ] is a bracketing interval . It is easy to see why a bracketing interval contains a root: you cannot graph the function from ( a, f ( a )) to ( b, f ( b )) without crossing the x -axis. Note that f can have a zero in a non-bracketing interval [ a, b ]. (Consider the func- tion T ( x ) on the interval [0,.6].) Also, f can have multiple roots in a bracketing interval.

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