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10-Newtons-Method-for-Systems-4UP

# 10-Newtons-Method-for-Systems-4UP - Nonlinear systems...

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Nonlinear systems Newton’s method Implementation Examples Newton’s Method for Nonlinear Systems Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Newton’s Method for Nonlinear Systems MATH 2070U 1 / 21 Nonlinear systems Newton’s method Implementation Examples Newton’s Method for Nonlinear Systems 1 Systems of nonlinear equations 2 Newton’s method for systems of nonlinear equations 3 Implementation details 4 Examples c D. Aruliah (UOIT) Newton’s Method for Nonlinear Systems MATH 2070U 2 / 21 Nonlinear systems Newton’s method Implementation Examples Nonlinear system of equations Determine all pairs ( x , y ) such that xy = 12 x 2 + y 2 = 25 Nonunique solutions ( 3, 4 ) , ( 4, 3 ) , ( - 3, - 4 ) , ( - 4, - 3 ) Simple interpretation in 2D In n dimensions, require n equations & n unknowns x 2 + y 2 = 25 xy = 12 c D. Aruliah (UOIT) Newton’s Method for Nonlinear Systems MATH 2070U 4 / 21 Nonlinear systems Newton’s method Implementation Examples Zeros of nonlinear systems of equations f : D R n R n is a vector field (vector-valued function of vector variable) f ( x ) = f 1 ( x ) f 2 ( x ) . . . f n ( x ) components of f ( x ) are scalar fields f k i.e., f k : D R n R ( k = 1: n ) Goal: Given f : D R n R n , find x * R n such that f ( x * ) = 0 Any system of nonlinear equations can be rewritten as f ( x ) = 0 xy = 12 x 2 + y 2 = 25 f ( x ) = 0 , f ( x ) : = x 1 x 2 - 12 x 2 1 + x 2 2 - 25 , x 1 x 2 : = x y c D. Aruliah (UOIT) Newton’s Method for Nonlinear Systems MATH 2070U 5 / 21

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Nonlinear systems Newton’s method Implementation Examples Reminder: Newton’s method for scalar equations Newton’s method for scalar equation f ( x * ) = 0 Input : f , f , x ( 0 ) for k = 0, 1, 2, . . . until convergence r ( k ) f x ( k ) (evaluate nonlinear residual) δ x ( k ) ← - f x ( k ) - 1 r ( k ) (compute Newton step) x ( k + 1 ) x ( k ) + δ x ( k ) (compute next iterate) end for Iterative framework to find zero x * R such that f ( x * ) = 0, where f : D R R
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