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# Chapter03 - CHAPTER 3 ROOT FINDING This chapter is about...

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3-1 CHAPTER 3 ROOT FINDING This chapter is about different root searching techniques. Given a function f ( x ), we are interested to look for its root ξ (i.e. f ( ξ ) = 0). The general idea is to revise the value of x through a repeated process. The goal is for x to asymptotically approach ξ as the number of iterations increases. Four iteration methods are considered: bisection , fixed-point iteration , Newton’s method , and secant method . Their basic concepts, algorithms, and implementations in Matlab are presented. In order to illustrate the relevancy of root finding to electrical engineering, we consider a diode circuit problem throughout this chapter. V 0 R i voltage x The diode is governed by the diode equation: ) 1 ( / = th V x S e I i where I S and V th are known parameters of the device. We are interested to find the voltage x across the diode. Since EE 3108 Semster B 2007/2008 S C Chan 3-2 iR x V + = 0 , we have (by combining the two equations): ) 1 ( / 0 = th V x S e R I x V 0 2 10 5 026 . 0 / 15 = + × x e x ------(*) Thus, the problem is a root finding problem of the function 2 10 5 ) ( 026 . 0 / 15 + × = x e x f x . The function is plotted below. 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 2 1 0 1 2 3 4 5 We can observe from the graph that the root is 8596 . 0 = ξ (4 s.f.). However, we need better methods because it is not effective to plot a graph for every problem.

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