Lecture 26 - Intersection of Two Curves ❚ Intersection of a circle and a parabola = = = = = = = = x x x x f 1 x x x x f or y x y x g 1 y x y x f

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Unformatted text preview: Intersection of Two Curves ❚ Intersection of a circle and a parabola =- = =- + = =- = =- + = x x x x f 1 x x x x f or y x y x g 1 y x y x f 2 2 1 2 1 2 2 2 2 1 2 1 1 2 2 2 ) , ( ) , ( ) , ( ) , ( - = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ - =- + = 1 x 2 x 2 x 2 x f x f x f x f x x x x f 1 x x x x f 1 2 1 2 2 1 2 2 1 1 1 2 2 1 2 1 2 2 2 2 1 2 1 1 ) , ( ) , ( -- = ∆ ∆ - = ∆ i 2 i 1 i 2 i 1 i 1 i 2 i 1 f f x x 1 x 2 x 2 x 2 x J , , , , , , , } ]{ [ Intersection of Two Curves { } -- =- = ∆ ∆ = ∆ + + i 2 1 i 2 i 1 1 i 1 old new i 2 i 1 x y x x x x x x x , , , , , , Solve for x new Newton-Raphson Method = = = = = ) ( ) ( ) ( ) ( ) ,...., , , ( ) ,...., , , ( ) ,...., , , ( ) ,...., , , ( x f x f x f x f ) x ( F x x x x f x x x x f x x x x f x x x x f n 3 2 1 n 3 2 1 n n 3 2 1 3 n 3 2 1 2 n 3 2 1 1 [ ] n 3 2 1 x x x x x , x F = = ) ( ❚ n nonlinear equations in n unknowns ❚ Jacobian (matrix of partial derivatives) ❚ Newton’s iteration Newton-Raphson Method 1 1 1 1 1 2 3 2 2 2 2 1 2 3 3 3 3 3 1 2 3 1 2 3 ( ) n n n n n n n n f f f f x x x x f f f f x x x x J x f f f f x x x x f f f f x x x x ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ O ( ) ( ); x old old new old J y x x x y F x = - ≡ ∆ ≡- function x = Newton_sys(F, JF, x0, tol, maxit) % Solve the nonlinear system F(x) = 0 using Newton's method % Vectors x and x0 are row vectors (for display purposes) % function F returns a column vector, [f1(x), ..fn(x)]' % stop if norm of change in solution vector is less than tol % solve JF(x) y = - F(x) using Matlab's "backslash operator"...
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This note was uploaded on 11/15/2010 for the course EGM 5403 taught by Professor Mei during the Spring '10 term at University of Florida.

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Lecture 26 - Intersection of Two Curves ❚ Intersection of a circle and a parabola = = = = = = = = x x x x f 1 x x x x f or y x y x g 1 y x y x f

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