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

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

In the previous slide 2

In this slide 3

3.3 4 Vector and matrix norms

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

Vector norm 6

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

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

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

Eigenvalue review 20 later

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Eigenvalue review 22

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24 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

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Any Questions? 27 3.3 Vector and matrix norms

3.4 28 Error estimates and condition number

Error estimation 29

Any Questions? 30

31 hint#3 hint#2 hint#1

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Condition number 41

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Perturbations (skipped) 43......

Any Questions? 44 3.4 Error estimates and condition number

3.5 45 LU decomposition

LU decomposition Motivation 46

LU decomposition 47

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Any Questions? 52

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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

Any Questions? 59 3.5 LU decomposition

3.6 Direct Factorization 60

Direct factorization 62

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Any Questions? 67 3.6 Direct factorization