Lecture 6B

# Lecture 6B - Gram-Schmidt Process Classical Gram-Schmidt...

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EE103 SLIDES 6B (SEJ) 1 Gram-Schmidt Process Classical Gram-Schmidt EE103 SLIDES 6B (SEJ) 2 Ex: Classical Gram-Schmidt T R xQ b =

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EE103 SLIDES 6B (SEJ) 3 Ex: Classical Gram-Schmidt 8 8 8 111 1 10 0 0 0 (,) 01 0 0 1 00 1 0 0 Ab ⎛⎞ ⎜⎟ = ⎝⎠ Poorly “conditioned” problem T Rx Q b = EE103 SLIDES 6B (SEJ) 4 Ex: Choleski Poorly “conditioned” problem 8 8 8 1 10 0 0 0 0 0 1 1 0 0 =
EE103 SLIDES 6B (SEJ) 5 Modified Gram-Schmidt EE103 SLIDES 6B (SEJ) 6 Modified Gram-Schmidt

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EE103 SLIDES 6B (SEJ) 7 Modified Gram-Schmidt T Rx Q b = EE103 SLIDES 6B (SEJ) 8 »disp( '[xy] ');disp([xy]) [x y] 10.0000 0 10.2000 0.0040 10.4000 0.0160 10.6000 0.0360 10.8000 0.0640 11.0000 0.1000 » X=[x.^2 x ones(size(x,1),1)] X = 100.0000 10.0000 1.0000 104.0400 10.2000 1.0000 108.1600 10.4000 1.0000 112.3600 10.6000 1.0000 116.6400 10.8000 1.0000 121.0000 11.0000 1.0000 cond(X'*X) = 1.5598e+010 Intro: Orthogonal Polynomials
EE103 SLIDES 6B (SEJ) 9 [x y] 10.0000 0 10.2000 0.0040 10.4000 0.0160 10.6000 0.0360 10.8000 0.0640 11.0000 0.1000 2 2 21 10 0 () Px a x a x + ϕ 2 2 1 0 Suppose we use, instead: 10 5 0 1166667 10 5 1 xx x ϕ =− () ( .) . . 2 2 1 0 1 x = = EE103 SLIDES 6B (SEJ) 10 11 1 22 12 2 26 16 6 1 1 1 () () x xy x XY x ϕϕ ⎤⎛ ⎢⎥ ⎜⎟ == ⎝⎠ ⎣⎦ """""" # 210 2 2 0 To compute c c c so that Px c x c x c x

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Lecture 6B - Gram-Schmidt Process Classical Gram-Schmidt...

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