Numerical Analysis

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EE, NCKUTien-Hao Chang (Darby Chang)

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

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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 n2+n equations

– Crout decomposition

– Dollittle decomposition

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3.3

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Vector and Matrix Norms

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

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Vector norm

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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

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0lim )(

xx k

k

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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

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Matrix norms

Natural matrix norms

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Natural matrix norms

Computing maximum norm

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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

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Eigenvalue review

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later

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Eigenvalue review

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In action

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

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Any Questions?

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3.3 Vector and Matrix Norms

3.4

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Error Estimates and Condition Number

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

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Any Questions?

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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)=Aeanswer

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hint#1

hint#2

hint#4hint#3

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Condition number

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Perturbations (skipped)

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Any Questions?

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3.4 Error Estimates and Condition Number

3.5

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LU Decomposition

LU decomposition

Motivation Gaussian elimination solve a linear system, Ax=b, with n unknowns– (2/3)n3 + (3/2)n2 – (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

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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

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Any Questions?

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Is there any other LU decompositions in addition to using modified Gaussian elimination?– degree of freedoms (number of

unknowns)

– An2, LUn2+n

Direct factorization (3.6)– as an systems of n2+n equations

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hint

answer

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

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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

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Any Questions?

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3.5 LU Decomposition

3.6Direct Factorization

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Is there any other LU decompositions in addition to using modified Gaussian elimination?– degree of freedoms (number of

unknowns)

– An2, LUn2+n

Direct factorization (3.6)– as an systems of n2+n equations

54http://www.dianadepasquale.com/ThinkingMonkey.jpg

Recall that

Direct factorization Just add more n equations

– ex: diagonal must be 1

Crout decomposition– lii=1 for each i=1, 2, …, n

Dollittle decomposition– uii=1 for each i=1, 2, …, n

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Any Questions?

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3.6 Direct Factorization