p103_5_S10

p103_5_S10 - EE103 Lecture Notes, Spring 2010, Prof S....

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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 5 © Copyright Stephen E Jacobsen, 2010 ....................................................................................................... i SECTION 5: ERROR ANALYSIS FOR SYSTEMS OF LINEAR EQUATIONS ............................... 104 Vector and Matrix Norms ..................................................................................................................... 104 Error Bounds ......................................................................................................................................... 106 Examples (Hilbert Matrix of Various Dimensions): . ........................................................................... 109 Condition and Numerical Stability: . ........................................................................................................ 111 Examples of Numerical Instability; and the Effect of Ill-Conditioning; . ............................................. 111 © Copyright Stephen E Jacobsen, 2010
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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 5 104 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   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
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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 5 105 Ex: l norm (“sup norm”)   12 max , , , n x xx x Ex:  1,1 T x  then 1 2 2 2 1 x x x Ex: 1, 2, 2 T x then 1 2 5 3 2 x x x We need to develop a similar notion for matrices. Definition: The function for n x n matrices is a matrix norm if 1. 0 A nnA  2. 00 AA   3. 1 , R A   4. A BA B  5. , AB A B A B n n  Definition: 1. , 1 1 1 max n ij jn i Aa     Max of 1 l norms of the columns of A 2. , 1 1 max n in j Max of 1 l norms of the rows of A The analog of the Euclidean norm, the 2 l norm, is not as simply written as the two above. 3. We define the matrix 2-norm as follows: 2 2 2 0| | | | 1 2 || max max Ax x x  
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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 5 106 This latter norm is not generally easily calculated. Its value turns out to be the largest singular value of the matrix A (we haven’t discussed singular values to this point); this norm is often referred to as the “spectral norm”. For the purpose of numerical calculations in these notes, we will only use the 1 || || , || AA matrix norms.
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p103_5_S10 - EE103 Lecture Notes, Spring 2010, Prof S....

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