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Chapter 11 Gaussian Elimination (I) Speaker: Lung-Sheng Chien Reference book: David Kincaid, Numerical Analysis

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OutLine Basic operation of matrix - representation - three elementary matrices Example of Gaussian Elimination (GE) Formal description of GE MATLAB usage

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Matrix notation in MATLAB

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x1 x2 x3 x4 6 x1 + (-2) x2 + 2 x3 + 4 x4 12 x1 + (-8) x2 + 6 x x4 3 x1 + (-13) x2 + 9 x3 + 3 x4 (-6) x1 + 4 x2 + 1 x3 + (-18) x x1 x2 x3 x x x x x4 Matrix-vector product Inner-product based outer-product based

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Inner-product based outer-product based Matrix-vector product: MATLAB implementation matvec.m Question: which one is better

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Logical index : 2D physical index : 1D Column-major nature in MATLAB Question: how does column-major affect inner-product based matrix-vector product and outer-product based matrix-vector product?

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is outer-product representationwhere Matrix representation: outer-product

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Elementary matrix [1] (1) The interchange of two rows in A: Define permutation matrix Question 1: whyQuestion 2: how to easily obtain How to explain?

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Question: such that implies Direct calculation Concatenation of permutation matrices

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(2) Multiplying one row by a nonzero constant: Elementary matrix [2] Define scaling matrix How to explain? since

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(3) Adding to one row a multiple of another: Elementary matrix [3] Define GE (Gaussian Elimination) matrix How to explain? outer-product representation

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Use MATLAB notation

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such that Concatenation of GE matrices Suppose Question:

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OutLine Basic operation of matrix Example of Gaussian Elimination (GE) - forward elimination to upper triangle form - backward substitution Formal description of GE MATLAB usage

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x1 x2 x3 x x1 x2 x3 x x1 x2 x3 x

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x1 x2 x3 x x1 x2 x3 x First row does not change thereafter x1 x2 x3 x

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x1 x2 x3 x x1 x2 x3 x x1 x2 x3 x

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x1 x2 x3 x x1 x2 x3 x x1 x2 x3 x x1 x2 x3 x4 12 Backward substitution: inner-product-based

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x1 x2 x3 x x1 x2 x x x1 x2 x x1 x2 126 Backward substitution: outer-product-based

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OutLine Basic operation of matrix Example of Gaussian Elimination (GE) Formal description of GE - component-wise and column-wise representation - recursive structure MATLAB usage

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x1 x2 x3 x x1 x2 x3 x x1 x2 x3 x x1 x2 x3 x

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x1 x2 x3 x x1 x2 x3 x Exercise: checkby MATLAB Eliminate first column: component-wise where

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Exercise: check by MATLAB Eliminate first column: column-wise defineand then Notation:

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x1 x2 x3 x x1 x2 x3 x x1 x2 x3 x Eliminate second column: component-wise

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define Exercise: check, why? Give a simple explanation. Notation: Eliminate second column: column-wise

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x1 x2 x3 x x1 x2 x3 x Eliminate third column: component-wise define then

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Notation: Exercise: check LU-decomposition Question: Do you have any effective method to write down matrix LU-decomposition Question: why don’t we care about right hand side vector

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Question 3: How to measure “goodness of LU-decomposition“? Question 2: does any invertible matrix has LU-decomposition? Question 4: what is order of performance of LU-decomposition (operation count)? Question 5: How to parallelize LU-decomposition? Question 6: How to implement LU-decomposition with help of GPU? Problems about LU-decomposition Question 1: what condition does LU-decomposition fail?

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OutLine Basic operation of matrix Example of Gaussian Elimination (GE) Formal description of GE MATLAB usage - website resource - “help” command - M-file

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MATLAB website

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MATLAB: create matrix

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MATLAB: LU-factorization

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Start MATLAB 2008

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Command “help” : documentation != in C

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Loop in MATLAB

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Decision-making in MATLAB

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IO in MATLAB

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LU-factorization in MATLAB

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Inner-product based matvec.m M-file in MATLAB Description of function matvec, write specificaiton of input parameter Consistency check

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Move to working directory working directory M-file

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Execute M-file Construct matrix Execute M-file, like function call in C-language

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Forward substitutionsince backward substitutionsince Write MATLAB code to do forward substitution and backward substitution Exercise: forward / backward substitution

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LU-decomposition Exercise: LU-decomposition You should find recursive structure of the decomposition

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