Ee1036ABS10

# Ee1036ABS10 - Least Squares Ax b mxn m n assume rank A n...

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Least Squares ,, , ( ) Ax b mxn m n assume rank A n  TT Normal Equations AA x Ab () 103 ( ) T Choleski Factorization PD Le e p d A A (, ) T T tmp forw L A b xbar back L tmp ( , ) EE103 SLIDES 6AB (SEJ) 1 Least Squares 1.0 1 1.5 2 y x   To determine a second degree approx polynomial, via least-squares. 32 . 0 1.0 4 55 . 0 X ay 65.5  XXa Xy   EE103 SLIDES 6AB (SEJ) 2

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Least Squares via Choleski EE103 SLIDES 6AB (SEJ) 3 Example of How NOT to Use the Normal Equations and Choleski EE103 SLIDES 6AB (SEJ) 4
One way to Improve the “conditioning” problem EE103 SLIDES 6AB (SEJ) 5 QR Factorization A mn m n Ax b want theleast squares solution ,, , Suppose AQ R Qism n Risn n and upper triangular  with positivediagonal elements d , T nn and QQ I EE103 SLIDES 6AB (SEJ) 6

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TT T T T QR Factorization .. LS A Ax Ab A QR A RQ  R Q QRx R Q b RR RQb R Qb x x (; ? ) T R hasaninverse why Solve by backward substitution ,, T Rx Q b to get x theleast squares solution , EE103 SLIDES 6AB (SEJ) 7 Orthonormal Basis Therefore howdoesone factor A QR , ? is ,and the columns are linearly independent. Am n def Consider 12 { | } {} ( m n R xA x lA      { ,,, }_ ) span a a a column space A EE103 SLIDES 6AB (SEJ) 8
Orthonormal Basis 12 {, , , } n Weattempt to find another setof basisvectors q q q for _( ) column space A 2 10 ii T j sothat df i j || || 2 qa n d qq for ij a so called orthonormal set of 

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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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Ee1036ABS10 - Least Squares Ax b mxn m n assume rank A n...

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