Download - Numerical Analysis
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