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

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Nonlinear systems Newtons method Implementation Examples Newtons Method for Nonlinear Systems Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Newtons Method for Nonlinear Systems MATH 2070U 1 / 21 Nonlinear systems Newtons method Implementation Examples Newtons Method for Nonlinear Systems 1 Systems of nonlinear equations 2 Newtons method for systems of nonlinear equations 3 Implementation details 4 Examples c D. Aruliah (UOIT) Newtons Method for Nonlinear Systems MATH 2070U 2 / 21 Nonlinear systems Newtons 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) Newtons Method for Nonlinear Systems MATH 2070U 4 / 21 Nonlinear systems Newtons 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 * ) = Any system of nonlinear equations can be rewritten as f ( x ) = xy = 12 x 2 + y 2 = 25 f ( x ) = , f ( x ) : = x 1 x 2- 12 x 2 1 + x 2 2- 25 , x 1 x 2 : = x y c D. Aruliah (UOIT) Newtons Method for Nonlinear Systems MATH 2070U 5 / 21 Nonlinear systems Newtons method Implementation Examples Reminder: Newtons method for scalar equations Newtons method for scalar equation f ( x * ) = Input : f , f , x ( ) for k = 0, 1, 2, . . . until convergence r ( k ) f x ( k ) (evaluate nonlinear residual) x ( k ) - h f x ( k ) i- 1 r ( k ) (compute Newton step) x ( k + 1 ) x ( k ) + x ( k ) (compute next iterate)...
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online