Transcription of Lecture 8 : Fixed Point Iteration Method, Newton’s Method
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Lecture 8 : Fixed Point Iteration Method, Newton’s Method
1 Lecture 8 : Fixed Point Iteration Method , Newton s MethodIn the previous two lectures we have seen some applications of the mean value theorem. We nowsee another this Lecture we discuss the problem of finding approximate solutions of the equationf(x) = 0.(1)In some cases it is possible to find the exact roots of the equation (1), for example, whenf(x) isa quadratic or cubic polynomial. Otherwise, in general, one is interested in finding approximatesolutions using some (numerical) methods . Here, we will discuss a Method called Fixed pointiteration Method and a particular case of this Method called Newton s Point Iteration Method :In this Method , we first rewrite the equation (1) in the formx=g(x)(2)in such a way that any solution of the equation (2), which is a Fixed Point
point then the chance of convergence of the iterative process is high. Remark : If g is invertible then l0 is a flxed point of g if and only if l0 is a flxed point of g¡1: In view of this fact, sometimes we can apply the flxed point iteration method for g¡1 instead of g. For understanding, consider g(x) = 4x¡12 then j g0(x) j= 4 for all x ...
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