p1035

# p1035 - SECTION 5 ERROR ANALYSIS FOR SYSTEMS OF LINEAR...

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SECTION 5: ERROR ANALYSIS FOR SYSTEMS OF LINEAR EQUATIONS ................................ 81 Vector and Matrix Norms ...................................................................................................................... 81 Error Bounds .......................................................................................................................................... 83

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EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 5 81 SECTION 5: ERROR ANALYSIS FOR SYSTEMS OF LINEAR EQUATIONS Let A be an n x n matrix. We have seen examples where solving A xb = may lead to solutions that look good but are actually poor. We now develop an analysis of how such may occur. Consider Ax b = Let e x denote the exact solution and let c x denote a computed solution. Let c rA x b =− be the residual vector. Then 1 ce c e x A x x x xA r ⎛⎞ ⎜⎟ ⎝⎠ =−= ⇒−= [We are assuming that 1 A exists.] Therefore, even though r may be “small,” it may be that 1 A is “big” and thus it may be that 1 Ar is “big”. So, we need to develop what we mean by “big” and “small” for vectors and matrices. Vector and Matrix Norms Definition: A vector norm, , is a function with the following properties. () 12 ,,, T n xx x x = 1. 0 n x xR ≥∀ 2. 00 =⇔= 3. , x α αα =∀ 4. x yxy +≤ + Ex: 22 2 21 2 || T n x x x x =+ + + = " , the Euclidean norm or measure of distance. Ex: 1 l norm (“street walking”) 1 1 n j j x x = =
EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen Section 5 82 Ex: l norm (“sup norm”) { } 12 max , , , n x xx x =

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p1035 - SECTION 5 ERROR ANALYSIS FOR SYSTEMS OF LINEAR...

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