Lecture 7 Typeset Notes

Lecture 7 Typeset - 1 1 n n n n y y x x n n y y x x y x g y y y g x x x g n n n n − = − ∂ ∂ − ∂ ∂ = = = = Thus Υ Φ Τ ΢ Σ − =

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ME 218: ENGINEERING COMPUTATIONAL METHODS Fall 2011 Notes Set #7 Newton-Raphson method for solving nonlinear equations in more than one variable: For the single variable system, the Newton-Raphson method is given as: x n+1 = x n – f(x n )/f’(x n ) This iterative formulation is obtained by the Taylor series expansion of the nonlinear function f(x) and then assuming the function to be linear at the point of interest. The method works pretty in a similar fashion for a system of nonlinear equations in more than one variable. Let f(x, y) and g(x, y) be two nonlinear equations in the two variables x and y . Then, the Taylor series expansions are given as: .... ) ( ) ( ) , ( ) , ( 0 , 0 , 0 0 0 0 0 0 + + + = = = = = y y y f x x x f y x f y x f y y x x y y x x higher order terms Now, approximating f(x, y) as zero and introducing the iterative scheme: ) , ( ) ( ) ( 1 , 1 , n n n n y y x x n n y y x x y x f y y y f x x x f n n n n = + + = = + = = Similarly, for the other function:
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Unformatted text preview: ) , ( ) ( ) ( 1 , 1 , n n n n y y x x n n y y x x y x g y y y g x x x g n n n n − = − ∂ ∂ + − ∂ ∂ + = = + = = Thus, Υ Φ Τ ΢ Σ Ρ − = Υ Φ Τ ΢ Σ Ρ − − Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + = = ) , ( ) , ( 1 1 , n n n n n n n n y y x x y x g y x f y y x x y g x g y f x f n n Defining the Jacobian, J = n n y y x x y g x g y f x f = = Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , , the algorithm is given as: Υ Φ Τ ΢ Σ Ρ − Υ Φ Τ ΢ Σ Ρ = Υ Φ Τ ΢ Σ Ρ − + + ) , ( ) , ( 1 1 1 n n n n n n n n y x g y x f J y x y x The problem with the algorithm occurs when J becomes singular (akin to the single variable case, where f’(x) becomes zero)....
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This note was uploaded on 12/14/2011 for the course ME 218 taught by Professor Unknown during the Fall '08 term at University of Texas at Austin.

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Lecture 7 Typeset - 1 1 n n n n y y x x n n y y x x y x g y y y g x x x g n n n n − = − ∂ ∂ − ∂ ∂ = = = = Thus Υ Φ Τ ΢ Σ − =

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