L9_NonlinearSystems - MA2213 Lecture 9 Nonlinear Systems...

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Unformatted text preview: MA2213 Lecture 9 Nonlinear Systems Midterm Test Results Topics Roots of One Nonlinear Equation in One Variable Secant Method pages 90-97 Roots of Nonlinear Systems (n Equations, n Variables) Newtons Method pages 352-360 Calculus Review : Intermediate Value Theorem, Newtons Method pages 79-89 Mean Value Theorems for Derivatives and Integrals Applications to Eigenvalue-Eigenvector Calculation Applications to Optimization Mean Value Theorem for Derivatives 2 Theorem A.4 p. 494 Let x y = y x . b)) ((a, C f b]), C([a, f 1 a)- (b (c) f ) ( f ) ( f ' =- a b ) , ( ) , ( 1 x There there is at least one point b) (a, c such that 5 . ) 2 ( ) 5 . ( 2 ) 2 ( '- = =- y y Newtons Method ) ( f x y = Newtons method is based on approximating the graph of y = f(x) with a tangent line and on then using a root of this straight line as an approximation to the root of f(x) )) ( f , ( x x y x S x x x / ) ( f 1- = ) ( f ) ( ) ( f ' x x x x y +- = ) , ( ) , ( 1 x ) ( f ' x S = Error of Newtons Method Newtons iteration for finding a root ]) , ([ C f 2 b a of Mean Value Theorem 5 between )) ( f ( / ) ( f ' 1 n n n n x x x x- = + )) ( f ( / ) ( f ' 1 n n n n x x x x +- =- + )) ( f ( / )] ( f ) ( f ( ) ( f ) [( ' ' n n n n x x x x--- = )) ( f ( / )] ( f ) ( ) ( f ) [( ' ' ' n n n n x x x x --- = the error satisfies and n x )) ( f ( / )] ( f ) ( f [ ) ( ' ' ' n n n x x x -- = Mean Value Theorem 5 between and n x )) ( f ( / ) ( f ) )( ( ' ' ' n n n x x x -- = | ) ( f | min / | ) ( f | max , | ) ( | | ) ( | ' ' ' 2 1 x x B x B x B n n - -- Question Compare B with estimate in slides 33,34 Lect 1 MATLAB for Newtons Method )) ( ( f / )) ( ( f ) ( ) 1 ( ' n x n x n x n x- = + MATLAB implementation of formula 3.27 on page 91 Start with one estimate ) 1 ( x >> x(1)=2; f(1) = x(1)^6-x(1)-1 >> for n = 1:10 S = 6*x(n)^5 1; x(n+1) = x(n) f(n) / S; f(n+1) = x(n+1)^6 x(n) 1; end For n = 1:nmax end >> x' ans = 2.00000000000000 1.68062827225131 1.43073898823906 1.25497095610944 1.16153843277331 1.13635327417051 1.13473052834363 1.13472413850022 1.13472413840152 1.13472413840152 Example 3.3.1 pages 91-92 Secant Method ) ( f x y = is based on approximating the graph of y = f(x) with a secant line and on then using a root of this straight line as an approximation to the root of f(x) )) ( f , ( x x y x S x x x / ) ( f 1 1 2- = ) ( f ) ( 1 1 x x x S y +- = ) , ( ) , ( 2 x 1 1 ) ( f ) ( f x x x x S-- = )) ( f , ( 1 1 x x Error of Secant Method )) ( f 2 /( ) ( f ) ( ) ( ' ' ' 1 1 n n n n n x x x - +--- =- It can be shown, using methods from calculus that we used to derive error bounds for Newtons method, that the sequence of estimates computed using the secant method satisfy equation 3.28 on page 92 where ) ( f = where n is between...
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L9_NonlinearSystems - MA2213 Lecture 9 Nonlinear Systems...

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