Lecture 7A

Lecture 7A - Introduction to Optimization: NonLinear Least...

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EE 103 Slides 7A (SEJ) 1 Introduction to Optimization: NonLinear Least Squares ( ) () 112 21 2 12 ,,, 0 , , , 0 0 n n nn fxx x fx x x x x = = = # ,* , ,* , , , , Ax b n n LU Factorization Ax b m n m n least squares Choleski QR = => EE 103 Slides 7A (SEJ) 2 Newton’s Method: Square Systems ( ) ,, , T kk k k n xx x x = 212 * 0 , , , 0 0 n n x x x = = = # () ( ) ( )( ) () () () ( ) 11 1 22 2 k k k n f x f x f x f x f x f x f x f x f x ≈+ # () () 1 2 k k k n k f f x f x fx fx J x x x Δ ⎡⎤ ⎢⎥ ⎣⎦ # ( ) ( )( ) 0 k f +− = ( ) 1 1 1 k f k k f Jx x xJ x f x + −= =−
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EE 103 Slides 7A (SEJ) 3 Newton’s Method: Square Systems () ( ) 1 1 1 , kk k f k k f xx Jx fx x x J x f x NOT for computation + −= =− ( ) 1 Nf g xxJ x f x k y ( )( ) f yf x 1 k y x x y + = + EE 103 Slides 7A (SEJ) 4 Newton’s Method: Square Systems Newton’s Algorithm Select an initial vector x 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 xy ←+ End DoWhile Fact: Assume k x x , and f is nonsingular (the n -dimensional analog of '( ) 0 ). Then the rate of convergence is quadratic.
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EE 103 Slides 7A (SEJ) 5 Newton’s Method: Square Systems A Simple Example EE 103 Slides 7A (SEJ) 6 Introduction to Non-Linear Least Squares , Linear Case A xb mn =≥ 2 2 m i n () m i n() () min( ) ( ) T xx T x
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Lecture 7A - Introduction to Optimization: NonLinear Least...

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