L9_NonlinearSystems

# L9_NonlinearSystems - MA2213 Lecture 9 Nonlinear Systems...

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Unformatted text preview: MA2213 Lecture 9 Nonlinear Systems Midterm Test Results Topics Calculus Review : Intermediate Value Theorem, Mean Value Theorems for Derivatives and Integrals Roots of One Nonlinear Equation in One Variable Newton’s Method pages 79-89 Secant Method pages 90-97 Roots of Nonlinear Systems (n Equations, n Variables) Newton’s Method pages 352-360 Applications to Eigenvalue-Eigenvector Calculation Applications to Optimization ean Value Theorem for Derivatives 2 heorem 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 here there is at least one point b) (a, c ∈ such that 5 . ) 2 ( ) 5 . ( 2 ) 2 ( ' − = = − y y Newton’s Method ) ( f x y = ewton’s method is based on approximating the graph f y = f(x) with a tangent line and on then using a root of his 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 Newton’s Method ewton’s iteration for finding a root ]) , ([ C f 2 b a ∈ of α ean Value Theorem Æ ξ ∃ between )) ( f ( / ) ( f ' 1 n n n n x x x x − = + )) ( f ( / ) ( f ' 1 n n n n x x x x + − = − + α α )) ( f ( / )] ( f ) ( f ( ) ( f ) [( ' ' n n n n x x x x − − − = α α )) ( f ( / )] ( f ) ( ) ( f ) [( ' ' ' n n n n x x x x ξ α α − − − = Æ the error satisfies α and n x )) ( f ( / )] ( f ) ( f [ ) ( ' ' ' n n n x x x ξ α − − = ean Value Theorem Æ η ∃ between ξ and n x )) ( f ( / ) ( f ) )( ( ' ' ' n n n x x x η ξ α − − Æ | ) ( f | min / | ) ( f | max , | ) ( | | ) ( | ' ' ' 2 1 x x B x B x B n n ≡ − ≤ − − α α uestion Compare B with estimate in slides 33,34 Lect 1 MATLAB for Newton’s 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 = 6*x(n)^5 – 1; x(n+1) = x(n) – f(n) / S; f(n+1) = x(n+1)^6 – x(n) – 1; nd 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 xample 3.3.1 pages 91-92 Secant Method ) ( f x y = is based on approximating the graph of y = f(x) with a ecant line and on then using a root of this straight line s 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 ς ξ α α − + − − − = − t can be shown, using methods from calculus that we sed to derive error bounds for Newton’s method, that he sequence of estimates computed using the secant ethod satisfy equation 3.28 on page 92ethod satisfy equation 3....
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L9_NonlinearSystems - MA2213 Lecture 9 Nonlinear Systems...

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