CSC_349_HANDOUT#8 - COMPUTER SCIENCE 349A Handout Number 8...

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1 COMPUTER SCIENCE 349A Handout Number 8 THE BISECTION METHOD (Section 5.2: pp. 116-123 5 th ed; pp. 124-131 6 th ed) -- can be used to compute a zero of any function ) ( x f that is continuous on an interval ] , [ u x x l for which 0 ) ( ) ( < × u x f x f l . Consider u x x and l as two initial approximations to a zero, say t x , of ) ( x f . The new approximation is the midpoint of the interval ] , [ u x x l , which is 2 u r x x x + = l . If 0 ) ( = r x f , then r x is the desired zero of ) ( x f . Otherwise, a new interval ] , [ u x x l that is half the length of the previous interval is determined as follows. If 0 ) ( ) ( < × r x f x f l then ] , [ r x x l contains a zero, so set r u x x . Otherwise, 0 ) ( ) ( < × r u x f x f (necessarily) and ] , [ u r x x contains a zero, so set r x x l . The above procedure is repeated, continually halving the interval ] , [ u x x l , until ] , [ u x x l is sufficiently small, at which time the midpoint 2 u r x x x + = l will be arbitrarily close to a zero of ) ( x f . Convergence criterion As this is an iterative algorithm that computes a sequence of approximations K K , , , , , , 1 3 2 1 i i x x x x x
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