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# p1033 - SECTION 3 ROOTS OF AN EQUATION OF A SINGLE VARIABLE...

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SECTION 3: ROOTS OF AN EQUATION OF A SINGLE VARIABLE ............................................. 31 Fixed Point Approach (Method of Successive Approximations): ....................................................... 31 Newton's Method: ................................................................................................................................... 35 The Secant Method: ............................................................................................................................... 40 Newton's Method and the Roots of Polynomials: ................................................................................ 43

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EE103 Lecture Notes, Fall 2007, 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 = . Ex: 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
EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 3 32 10 () x gx = , 21 x = 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: Example of Fixed Point Method

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EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 3 33 Figure 2: Typical sequence of iterates of FP method Figure 3: Example of nonconvergenc of FP method
EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 3 34 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, 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 when as we address the rate of convergence issue. That is, when convergence does occur we can develop a result for the rate of convergence. Assume k x x and assume '' g , the second derivative, exists and is continuous. Let kk exx =− denote the error at the th k iteration. Then Taylor's theorem states that we can represent, exactly, ( ) k gx by the first three terms of the Taylor expansion about the point x ; the last term is, of course, the so-called remainder term. But 1 () x + = ; therefore 2 1 1 2 ( ) ( ) '( ) ''( ( , )) k k k x g xe g x x e ξ + == + +

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p1033 - SECTION 3 ROOTS OF AN EQUATION OF A SINGLE VARIABLE...

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