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Unformatted text preview: Lecture 06 1 September 20, 2007 Outline: 1) The LU factorization: final comments Row swaps: PA=LU Permutation Matrices and an Example A "real" problem: SpringDemo and Matlab Sparse, TriDiagonal matrices LU vs inv(A); 2) One last Operation: The matrix transpose A T symmetric matrices A T =A product Rule (AB) T =B T A T symmetry of R T R and RR T Lecture 06: Systems of Linear Equations #5: Everything else LU Factorization and row exchanges: PA=LU The Problem: A full description of Gaussian Elimination includes Permutation matrices The Fix: In General you can't know the order of permutations before you begin, but you can track permutations as you proceed such that at the end, you can permute A once such that PA=LU. Two Matlab approaches (version 7+) [L,U,P]=lu(A); % such that P*A=L*U-or- [L,U,p]=lu(A,'vector') % such that A(p,:)=L*U LU Factorization and row exchanges: PA=LU For small problems, we'll just permute first then find the LU (and let Matlab handle the hard stuff) Example: A = [ 0 1 2 ; 1 0 1; 0 1 1 ] (several choices for P) LU Factorization and row exchanges: PA=LU...
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- Fall '07