Ee1037aS10

Ee1037aS10 - Introduction to Optimization: NonLinear Least...

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Introduction to Optimization: NonLinear Least Squares ,* , Ax b n n LU Factorization 0 fxx x , * ,, , ,() Ax bmnm n least squares Choleski Q RMGS    112 21 2 ,,, , , , 0 n n fx x x 12 0 nn x x EE 103 Slides 7A (SEJ) 1 Newton’s Method: Square Systems   , T kk k k n xx x x * 0 n x 212 , , , 0 0 n x x            11 1 22 2 k k f x f x f x f x f x f x    k n f x f x f x 1 k        2 k k f x f x        0 k f J x x x  1 n k f fx fx         1 1 k f k k f x xJ x f x x x f x   EE 103 Slides 7A (SEJ) 2
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Newton’s Method: Square Systems  1 kk k f xx Jx fx    1 1 , k k f x x J x f x NOT for computation  1       () Nf g J x f x k yx x    f yf x 1 k y x x y EE 103 Slides 7A (SEJ) 3 Newton’s Method: Square Systems Newton’s Algorithm Select an initial vector x D Whil ( i t t i diti h t d DoWhile (some appropriate stopping condition has not occurred) Compute the Jacobian, f Let y solve the system of linear equations f x (assuming f is nonsingular) x x y  End DoWhile variable , '( ) 0 function of one f x Fact: Assume k x x , and f is nonsingular (the n -dimensional analog of '( ) 0 ). Then the rate of convergence is quadratic. EE 103 Slides 7A (SEJ) 4
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Newton’s Method: Square Systems A Simple Example EE 103 Slides 7A (SEJ) 5 Newton’s Method: Square Systems EE 103 Slides 7A (SEJ) 6
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Newton’s Method: Non-Square Systems??  112 ,,, 0 n mn fxx x       11 1 kk k f x f x f xx x    212 , , , 0 0 n f xx x fx x x      22 2 k k f x f x f x fx fx fx xx   12        mm m      0 k f J x xx      , f Jx yf
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This note was uploaded on 06/01/2010 for the course EE EE 103 taught by Professor Jacobsen during the Spring '09 term at UCLA.

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Ee1037aS10 - Introduction to Optimization: NonLinear Least...

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