Download - Numerical Analysis
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Numerical Analysis
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EE, NCKUTien-Hao Chang (Darby Chang)
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In the previous slide Accelerating convergence
– linearly convergent
– Newton’s method on a root of multiplicity
– (exercise 2)
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 equations
– Crout decomposition
– Dollittle decomposition
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3.3
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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
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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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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
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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 Euclidean norm, unfortunately, is 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
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3.4
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Error estimates and condition number
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Error estimation A linear system , and is an
approximate solution The error, , cannot be directly
computed ( is never known) The residue vector, , can be easily
computed
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Any Questions?
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Is a good estimation of ? Construct the relationship between
and From the definition
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hint#3
hint#2
hint#1
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Is a good estimation of ? Construct the relationship between
and From the definition
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answerhint#4hint#3
hint#2
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Is a good estimation of ? Construct the relationship between
and From the definition
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answerhint#4hint#3
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Is a good estimation of ? Construct the relationship between
and From the definition
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answerhint#4
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Is a good estimation of ? Construct the relationship between
and From the definition
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answer
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Is a good estimation of ? Construct the relationship between
and From the definition
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Condition number
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Perturbations (skipped)
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.
.
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Any Questions?
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3.4 Error estimates and condition number
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3.5
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LU decomposition
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LU decomposition
Motivation Gaussian elimination solve a linear system,
, with unknowns
– 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 lower triangular matrix and an upper triangular matrix for which are said to form an LU decomposition of
Here we replace mathematical descriptions with an example to show how Gaussian elimination is used 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)•
•
Direct factorization (3.6)– as an systems of equations
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answer
hint
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Is there any other LU decompositions in addition to using modified Gaussian elimination?– degree of freedoms (number of
unknowns)•
•
Direct factorization (3.6)– as an systems of equations
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answer
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Is there any other LU decompositions in addition to using modified Gaussian elimination?– degree of freedoms (number of
unknowns)•
•
Direct factorization (3.6)– as an systems of equations
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Solving a linear system
When a new RHS comes
– with , actually to solve and • 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
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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)•
•
Direct factorization (3.6)– as an systems of equations
61Recall thathttp://www.dianadepasquale.com/ThinkingMonkey.jpg
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Direct factorization Just add more equations
– ex: diagonal must be
Crout decomposition– for each
Dollittle decomposition– for each
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Any Questions?
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3.6 Direct factorization