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Numerical Analysis 1 EE, NCKU Tien-Hao Chang (Darby Chang)

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In the previous slide 2

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In this slide 3

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3.3 4 Vector and matrix norms

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Pivoting strategies are designed to reduce the impact roundoff error The size of a vector/matrix is necessary to measure the error 5

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Vector norm 6

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7 The two most commonly used norms in practice

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Vector norm Equivalent 10

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12 The Euclidean norm and the maximum norm are equivalent

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Matrix norms Similarly, there are various matrix norms, here we focus on those norms related to vector norms –natural matrix norms 13

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Matrix norms Natural matrix norms 14

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Natural matrix norms Computing maximum norm 16

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Natural matrix norms Computing Euclidean norm Euclidean norm, unfortunately, is not as straightforward as computing maximum matrix norms Requires knowledge of the eigenvalues of the matrix 19

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Eigenvalue review 20 later

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Eigenvalue review 22

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24 In action

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Any Questions? Vector and matrix norms

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Error estimates and condition number

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Error estimation 29

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Any Questions? 30

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31 hint#3 hint#2 hint#1

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32 answer hint#4hint#3 hint#2

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33 answer hint#4hint#3

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34 answer hint#4

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35 answer

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Condition number 41

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Perturbations (skipped)

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Any Questions? Error estimates and condition number

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LU decomposition

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LU decomposition Motivation 46

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LU decomposition 47

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Any Questions? 52

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53 answer hint

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54 answer

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Solving a linear system 56

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Solving a linear system In summary Anyway, the two-step algorithm (LU decomposition) is superior to Gaussian elimination with back substitution 58

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Any Questions? LU decomposition

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3.6 Direct Factorization 60

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61 Recall that

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Direct factorization 62

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Any Questions? Direct factorization

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