p103_3_S10

p103_3_S10 - EE103 Lecture Notes, Spring 2010, Prof S....

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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 3 i © COPYRIGHT STEPHEN E JACOBSEN, 2010. ............................................................................................ I SECTION 3: ROOTS OF AN EQUATION OF A SINGLE VARIABLE. .............................................................. 31 FIXED POINT APPROACH (METHOD OF SUCCESSIVE APPROXIMATIONS): ................................................................ 31 NEWTON'S METHOD: .................................................................................................................................... 34 THE SECANT METHOD: .................................................................................................................................. 39 NEWTON'S METHOD AND THE ROOTS OF POLYNOMIALS: .................................................................................... 42 © Copyright Stephen E Jacobsen, 2010
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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 3 31 SECTION 3: ROOTS OF AN EQUATION OF A SINGLE VARIABLE In this section we concern ourselves with an introduction to finding a root of the equation () 0 fx where we assume that f is a continuous function and x is a scalar. We've already seen the bisection algorithm that, generally, is excellent for finding an interval of reasonably small length in which we are assured that a root is present. Of course, the word "root" is nothing more than an expression for a "solution" of the equation. That is, x is said to be a solution or root of the above equation if . Fixed Point Approach (Method of Successive Approximations): The fixed point or successive approximation method is one that's important to present because it provides a method of analysis for other methods, including Newton's method, one of the best. The fixed point method assumes that the equation to be solved is () x gx where g is a continuous function. Note that if we take ( ) ( ) f xx g x , the problem is one of finding a root of the equation ( ) 0 . Example: Assume 5 2 1 f x  . We can think of finding a root of this polynomial as a fixed point problem by writing it as 5 1 2 x x where, of course, 5 () ( 1 ) /2 x  . The idea of the fixed point method, or method of successive approximations, is nothing more than to successively apply the following operation x That is, we select, say, 0 x and we compute 10 x ,
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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 3 32 21 () x gx 32 x …………. 1 kk x Therefore, if the sequence { } k x converges to say * x , i.e., * k x x , we have, by continuity of the function g *1 * lim lim ( ) (lim ) ( ) k k x x g xg x     That is, if the sequence * k x x it must be the case that * x is a solution of the equation x and hence * x is a fixed point. Figure 1: Graphical Example of Fixed-Point Iteration (Note: method cannot converge to the root near x0 (why?) Exercise: Construct a graphical example where there’s one fixed point, but fixed-point iteration will not find that root. Therefore, we have examples that demonstrate that the fixed point method may produce solutions, and may not, even when there are fixed points. Moreover, as seen above in Figure 1, the fixed point method, when it does find a fixed point, may not find a fixed point that is the nearest to the starting point. The reasons will become clear, below,
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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 3 33 where we address the rate of convergence issue. As we’ll see, when convergence does occur we can develop results for the rate of convergence .
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This note was uploaded on 06/01/2010 for the course EE EE 103 taught by Professor Jacobsen during the Spring '09 term at UCLA.

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p103_3_S10 - EE103 Lecture Notes, Spring 2010, Prof S....

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