Lecture5 - 1 LECTURE 2.1 Overview: (Covers section 2.1)...

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1 LECTURE 2.1 Dr. Vyas Overview: (Covers section 2.1) Fixed point iteration method Algebraic and graphical interpretation Absolute and relative error considerations Examples/programs FIXED POINT ITERATION METHOD: Consider the equation, , sin( ) f eff s s f h l ϕ = occurring in analysis of tire (tyre in UK) strength and dynamics, wherein a tire is modeled as a beam. 1 A cross-section of a typical tire is illustrated as below. For a given value of , f eff f h l , a transcendental equation of the form, sin( ) s s K = results. One way to solve such an equation is to look at it as an intersection of the line s y = with the curve, sin( ) s y = . This leads us to the idea of the fixed point iteration method. Fixed Point Iteration Method 1. FIXED POINT: A fixed point of a function ( ) g x is a real number P such that ( ) P g P = . 2. This means that it is an intersection of a curve ( ) g x with a line y x = . Recall that this line passes thru the origin and has a slope of unity (one). 3. The iteration 1 ( ) n n p g p + = for 0,1, n = K is called fixed point iteration. The fixed point iteration is the simplest case of an iterative procedure with applications ranging from root finding to studying chaos in dynamical systems. For example, 1 Mastinu, G., et al, “ A semi-analytical tyre model for steady and transient state simulations ”, Vehicle System Dynamics Supplement, volume 27 (1997), pp. 2-21, Swets & Zeitlinger (Publishers) Remark : This journal volume is also available in the Morris Library at the University of Delaware.
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2 the equation, ( ) (1 ) g x x x μ = - is a deceptively simple model for discrete dynamical system that illustrates the concept of chaos. ALGEBRAIC INTERPRETATION: 4. This means that given a starting value for p , 0 p p = , and a function ( ) g x , the first iteration is 1 0 1 1 0 ( ) ( ) n n p g p p p g p + + = = = . The next round of iteration corresponds to 1 n = . 5. We take 1 p found from the previous iteration 1 0 ( ) p g p = and use it in the relation 1 1 1 2 1 ( ) ( ) n n p g p p p g p + + = = = to find a new iterated value 2 p . Thereafter, using the 2 p in the relation 1 2 1 3 2 ( ) ( ) n n p g p p p g p + + = = = a new iterated value 3 p is determined. This continues so on and so forth until the new and old values appear to be becoming more and more identical to each other. 6. At some round of iteration (a particular value of n ) the values are practically identical to our satisfaction . This implies that for our purposes we can state that further iterations do not bring forth any significant numerical improvement. This calls for attention to how much error we are prepared to tolerate. 7. While the above description makes the whole idea seem very simple, one must be aware of certain subtleties associated with the process. These are discussed next via Theorems and graphical interpretations. 8.
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This note was uploaded on 04/14/2008 for the course MATH 353 taught by Professor Vyas during the Spring '08 term at University of Delaware.

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Lecture5 - 1 LECTURE 2.1 Overview: (Covers section 2.1)...

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