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p1036 - SECTION 6 NEWTON'S METHOD FOR SYSTEMS OF NONLINEAR...

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SECTION 6: NEWTON’S METHOD FOR SYSTEMS OF NONLINEAR EQUATIONS ................. 87 Fixed-Point Contraction Methods ............................................................................................................. 87 Fixed-Point Methods for Some Specially Structured Systems of Linear Equations ......................... 89 Newton’s Method for Systems of Nonlinear Equations ........................................................................... 91 Taylor’s Theorem ................................................................................................................................... 91 Newton’s Method .................................................................................................................................... 94 NEWTON'S ALGORITHM .............................................................................................................. 95 Quadratic Convergence of Newton's Method ...................................................................................... 96
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EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 6 87 SECTION 6: NEWTON’S METHOD FOR SYSTEMS OF NONLINEAR EQUATIONS In this section we discuss the extension of fixed-point methods to certain types of systems of linear equations. We will then go on to develop Newton’s method for systems of nonlinear equations. As in the single variable case, we introduce the fixed-point approach. Consider the equations ( ) () 11 1 2 22 1 2 12 ,,, n n nn n x gxx x x x x x = = = # which we abbreviate as ( ) x gx = where T n T n xx x x gg g g = = Of course, the fixed point algorithm or method is succinctly stated as 1 kk xg x o r x + = The above is the general schema of the fixed-point approach for solving systems of equations, linear and nonlinear. Of course, much depends upon the particular system of equations, the ones we want to solve, ( ) 112 212 0 , , , 0 0 n n fxx x x x = = = # as to whether they can, or should, be put in the fixed-point form. Fixed-Point Contraction Methods Definition: We say g is contractive if there is a number 1 K < such that ( ) ( ) gx gy Kx y ≤−
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EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 6 88 Example : Scalar Case, () x gx = () () () ( ) () () ' Taylor s Theorem gx gy g x y gx gy g ξ = +− −≤ Therefore, if 1 gK ≤< then g is contractive. Example: Let M be an x nn matrix, and consider finding a solution (a vector x ) for the system of equations: Take x Mx c gx M x c = + = + i.e. ( ) 11 1 T T n gx mxc = + = + # Then () ( ) ( ) Mx y M x y −=− Therefore, if 1 M < , then g is contractive. Fact : Consider = and let x be a solution (fixed point, x = ). Assume g is contractive. Then, the recursion ( ) 1 kk x + = converges to the solution x . Proof: ( ) ( ) 1 1 1 21 10 0 || || K k k k x xx g x g x Kx x Kx x Kxx x xK + + < + −= ≤− #
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EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 6 89 Since 1 K < , 0 k K as k →∞ . Therefore, 0 kk x xx x →⇔ → . Note, we have also shown that x is unique. Exercise: Show 1 1 10 1 and k k K K k K K x x −≤ ∴− Note that this result shows that, given the initial point 0 x and the next point 1 x , we can derive a sufficient number of iterations it would take, for example, to be within 6 10 of the answer x . Fixed-Point Methods for Some Specially Structured Systems of Linear Equations Def: Let A be an x nn matrix. The x matrix C is said to be an approximate inverse for A if || 1 IC A < for some matrix norm. Note: If C is an approximate inverse, then the columns of A are linearly independent (i.e., A is nonsingular). To see this: 00 , 0( ) || ( ) || || || 1 If x Ax then CAx I CA x x x A x A x A ∃≠∋ = =⇒ − = ⇒= −⋅ ⇒− and the latter is a contradiction.
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This note was uploaded on 10/22/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Fall '08 term at UCLA.

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p1036 - SECTION 6 NEWTON'S METHOD FOR SYSTEMS OF NONLINEAR...

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