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SystemsNonLinearEquations

# SystemsNonLinearEquations - Solutions to systems of...

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Solutions to systems of non-linear multivariable equations We have shown methods to find the root of a nonlinear equation. That is find p such that f(p) = 0. We looked at Newton's method and saw that it has a quadratic rate of convergence. We say that Newton's method is simple a fixed point method with a special g(x) function. ) ( ) ( ) ( ' x f x f x x g - = and the method uses ) ( ) ( ' 1 n n n n p f p f p p - = + to find successive approximations. In this section we find a solution to systems of multivariable equation. Consider: ) , ( ) , ( 2 1 y x f v y x f u = = We want to find ) , ( q p such that ) , ( 0 ) , ( 0 2 1 q p f q p f = = We will derive Newton's method for n variables. Before we do this some preliminary theorems must be understood.

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Definition : Assume that ) , ( 1 y x f and ) , ( 2 y x f are functions of the independent variable x and y. Then their Jacobian matrix ) , ( y x J is f x f y f x f z f y f x f z f y f x f z f y f x f 3 3 3 2 2 2 1 1 1 Definition : A fixed point for a system of two equations ) , ( ) , ( 2 1 y x g y y x g x = = is a point (p, q) such that ) , ( ) , ( 2 1 q p g q q p g p = = Definition : For the function ) , ( ) , ( 2 1 y x g y y x g x = = a fixed point iteration is ) , ( ) , ( 2 1 1 1 k k k k k k q p g q q p g p = = + +
Theorem : Assume the functions 1 g and 2 g

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