Lecture 4.pdf - MATH 472 Numerical Methods with Financial...

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MATH 472: Numerical Methods with Financial Applications Lecture 5 Course Basics Chapter 0 Fundamentals Prakash Chakraborty University of Michigan 1 / 18
1 Solving equations Problem: solve the equation f ( x ) = 0 for x . Definition 1.1 The functionf(x)has arootatx=riff(r) = 0.Why we study it? 2 / 18
1.1 The bisection method I Main idea of the bisection method : Bracketing I Think of how you would look up a word in a dictionary I How to make sure that a root exists? Figure: f ( x ) = x 3 + x - 1 Theorem 1.2 (by Intermediate Value Theorem) Let f be a continuous function on [ a, b ] , satisfying f ( a ) f ( b ) < 0 . Then, f has a root in ( a, b ) , i.e. there exists a < r < b such that f ( r ) = 0 . 3 / 18
1.1 The bisection method The Bisection Method: I Assume we are given an interval [ a, b ] where f ( a ) f ( b ) < 0 . I Bisection : Let c = ( a + b ) / 2 . If f ( c ) 6 = 0 , then either (i) f ( c ) f ( a ) < 0 in which case the root must be in [ a, c ] ; or (ii) f ( c ) f ( b ) < 0 in which case the root is in [ c, b ] .

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