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

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In the previous slide Accelerating convergence –linearly convergent –Newton’s method on a root of multiplicity >1 –(exercises) Proceed to systems of equations –linear algebra review –pivoting strategies 2

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In this slide Error estimation in system of equations –vector/matrix norms LU decomposition –split a matrix into the product of a lower and a upper triangular matrices –efficient in dealing with a lots of right-hand-side vectors Direct factorization –as an systems of n 2 +n equations –Crout decomposition –Dollittle decomposition 3

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3.3 4 Vector and Matrix Norms

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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

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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 One of the other uses of norms is to establish the convergence Two trivial questions: –converge or diverge in different norms? –converge to different limit values in different norms? The answer to both is no –all vector norms are 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 Is, unfortunately, 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 A linear system Ax=b, and x’ is an approximate solution The error, e=x’-x, cannot be directly computed ( x is never known) The residue vector, r=Ax’-b, can be easily computed – r=0 x’=x e=0 29

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Any Questions? 30

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Is ||r|| a good estimation of ||e|| ? Construct the relationship between r and e From the definition r=Ax’-b=Ax’-Ax=A(x’-x)=Ae answer 31 hint#1 hint#2 hint#4hint#3

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Condition number 36

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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 Gaussian elimination solve a linear system, Ax=b, with n unknowns – (2/3)n 3 + (3/2)n 2 – (7/6)n –with back substitution –the minimum number of operations If there are a lots of right-hand-side vectors –how many operations for a new RHS? –with Gaussian elimination, all operations are also carried out on the RHS 41

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LU decomposition Given a matrix A, a lower triangular matrix L and an upper triangular matrix U for which LU=A are said to form an LU decomposition of A Here we replace mathematical descriptions with an example to show how use Gaussian elimination to obtain an LU decomposition 42

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Any Questions? 47

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Is there any other LU decompositions in addition to using modified Gaussian elimination? –degree of freedoms (number of unknowns) – A n 2, LU n 2 +n Direct factorization (3.6) –as an systems of n 2 +n equations 48 hint answer

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Solving a linear system A LU When a new RHS comes – Ax=b PAx=Pb LUx=Pb –with z=Ux, actually to solve Lz=Pb and Ux=z both steps are easy notice that Pb does not require real matrix- vector multiplication 49

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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 51

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Any Questions? LU Decomposition

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3.6 Direct Factorization 53

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Is there any other LU decompositions in addition to using modified Gaussian elimination? –degree of freedoms (number of unknowns) – A n 2, LU n 2 +n Direct factorization (3.6) –as an systems of n 2 +n equations 54 Recall that

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Direct factorization Just add more n equations –ex: diagonal must be 1 Crout decomposition – l ii =1 for each i=1, 2, …, n Dollittle decomposition – u ii =1 for each i=1, 2, …, n 55

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Any Questions? Direct Factorization

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