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Ee1035aS10

# Ee1035aS10 - Error Analysis for System of Linear Equations...

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Error Analysis for System of Linear Equations 12 2 (1 10 ) 2 10 kk xx    (, ) ( 0 , 2 ) 11 2 21 2 2( ) 0 2 10 ((1 10 ) ) 10 k ex x x        (1,1) (0, 2) 1.0  x EE103 SLIDES 5A (SEJ) 1 b=sum(A’)’ n=13 Error Analysis 1, 1/ 2, 1/ 3, 1/ 4 L O [ x,perror,sumresid]=sej(n) xsol = 1.0000 A 1/2, 1/3, 1/4, 1/5 1/3, 1/4, 1/5, 1/6 4 5 6 7 M M M M P P P P 0.9997 1.0117 0.8071 2 7244 1/ 4, 1/5, 1/6, 1/ 7 N Q 2.7244 -8.3249 33.4625 74 1604 -74.1604 117.9244 -119.7725 80.4201 -29.0922 6.0000 sumresid = 1.8097e-014 perror = 5703.46 EE103 SLIDES 5A (SEJ) 2

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Why does the former type of problem occur Error Analysis Why does the former type of problem occur? and nonsingular Ax b n n ec c x xr A x b    1 ce c e rA x A x A x x A  x x A r  EE103 SLIDES 5A (SEJ) 3 Vector and Matrix Norms vector norm  12 ,,, T n n xx x x 1. 0 2. 0 0 x x R   3 . , 4. x xy x y    EE103 SLIDES 5A (SEJ) 4
Examples of Vector Norms 2 22 2 21 2 || T n xx x x l  1 1 1 n j j lx x   12 max , , , n x x x 1 (1 , 1 ) | | 2 T  1 (1, 2, 2) || || 5 T 2 2 1 x x 2 3 2 x x EE103 SLIDES 5A (SEJ) 5 Matrix Norms 0 A nn A  10 0 AA  1 1. 00 2. , A AR A   3. 4 AB A B A B A B n n   . , EE103 SLIDES 5A (SEJ) 6

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Some Matrix Norms 1 1 11 max{|| || } max , 1 n j ij jj A aa n o r m     1 j n j n i  n 1 1 max{|| || max , ii j in j A a a norm 2 2 22 0 1 max max , x x Ax A Ax spectral norm x  2 EE103 SLIDES 5A (SEJ) 7 Example of 3 Matrix Norms L O 001 00 || || 25 A A M M M P P P 11 51 01 5 1 1 2 || || 13.2396 10 A A N M Q P 00 5 1 "compatibility" (| | | | | | , ) T xy x y Cauchy Swchwartz Ineq  1 Ax A x A xA x 2 Ax A x  EE103 SLIDES 5A (SEJ) 8
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Ee1035aS10 - Error Analysis for System of Linear Equations...

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