Lecture 26 - Intersection of Two Curves Intersection of a...

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Intersection of Two Curves Intersection of a circle and a parabola = - = = - + = = - = = - + = 0 x x x x f 0 1 x x x x f or 0 y x y x g 0 1 y x y x f 2 2 1 2 1 2 2 2 2 1 2 1 1 2 2 2 ) , ( ) , ( ) , ( ) , (
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- = - = - + = 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
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Newton-Raphson Method = = = = = ) ( ) ( ) ( ) ( ) ,.... , , , ( ) ,.... , , , ( ) ,.... , , , ( ) ,.... , , , ( x f x f x f x f ) x ( F 0 x x x x f 0 x x x x f 0 x x x x f 0 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 , 0 x F = = ) ( n nonlinear equations in n unknowns
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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 = - ≡ ∆ ≡ -  
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function x = Newton_sys(F, JF, x0, tol, maxit) % Solve the nonlinear system F(x) = 0 using Newton's method
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Fall '09 term at University of Florida.

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Lecture 26 - Intersection of Two Curves Intersection of a...

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