LUandCholesky1

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Administrivia: October 7, 2009 Administrivia: October 7, 2009 Homework 2 due next Wednesday (see web site). Reading in Davis: Sections 7.6 and 7.7 (nested dissection) Sections 4.4 – 4.8 and 4.11 (Cholesky)

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Nonsymmetric Ax = b: Nonsymmetric Ax = b: Gaussian elimination (without pivoting) Gaussian elimination (without pivoting) 1. Factor A = LU 2. Solve Ly = b for y 3. Solve Ux = y for x Variations: Pivoting for numerical stability: PA=LU Cholesky for symmetric positive definite A: A = LL T Permuting A to make the factors sparser = x
Left-looking Column LU Factorization Left-looking Column LU Factorization for column j = 1 to n do solve scale : l j = l j / u jj Column j of A becomes column j of L and U L 0 L I ( ) u j l j ( ) = a j for u j , l j L L U A j

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Left-looking sparse LU without pivoting (simple) Left-looking sparse LU without pivoting (simple) L = speye(n); for column j = 1 : n dfs in G(L T ) to predict nonzeros of x; x(1:n) = A(1:n, j); // x is a SPA for i = nonzero indices of x in topological order x(i) = x(i) / L(i, i); x(i+1:n) = x(i+1:n) – L(i+1:n, i) * x(i); U(1:j, j) = x(1:j); L(j+1:n, j) = x(j+1:n); cdiv: L(j+1:n, j) = L(j+1:n, j) / U(j, j);
Nonsymmetric Ax = b: Nonsymmetric Ax = b: Gaussian elimination with partial pivoting Gaussian elimination with partial pivoting At step j, swap the unused row with largest diagonal element into the pivot position. 1. Factor PA = LU 2. Solve Ly = Pb for y 3. Solve Ux = y for x = x P

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Left-looking Column LU Factorization Left-looking Column LU Factorization for column j = 1 to n do solve pivot : swap u jj and an elt of l j scale : l j = l j / u jj Column j of A becomes column j of L and U L 0 L I ( ) u j l j ( ) = a j for u j , l j L L U A j
GPLU Algorithm GPLU Algorithm

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