ILU - Column Cholesky Factorization: A=RTR for j = 1 : n...

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Column Cholesky Factorization: Column Cholesky Factorization: A=R A=R T R R for j = 1 : n for k = 1 : j-1 % cmod(j,k) for i = j : n A(i,j) = A(i,j) – A(i,k)*A(j,k); end ; end ; % cdiv(j) A(j,j) = sqrt(A(j,j)); for i = j+1 : n A(i,j) = A(i,j) / A(j,j); end ; end ; T R T R T R A j
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Data structures Data structures Full: 2-dimensional array of real or complex numbers (nrows*ncols) memory 31 0 53 0 59 0 41 26 0 31 41 59 26 53 1 3 2 3 1 Sparse: compressed column storage about (1.5*nzs + .5*ncols) memory D
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Sparse Column Cholesky Factorization: Sparse Column Cholesky Factorization: A=R A=R T R R for j = 1 : n for k < j with A(j,k) nonzero % sparse cmod(j,k) A(j:n, j) = A(j:n, j) – A(j:n, k)*A(j, k); end ; % sparse cdiv(j) A(j,j) = sqrt(A(j,j)); A(j+1:n, j) = A(j+1:n, j) / A(j,j); end ; T R T R T R A j
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Graphs and Sparse Matrices Graphs and Sparse Matrices : Cholesky factorization : Cholesky factorization 10 1 3 2 4 5 6 7 8 9 10 1 3 2 4 5 6 7 8 9 G(A) G + (A) Symmetric Gaussian elimination: for j = 1 to n add edges between j’s higher-numbered neighbors Fill : new nonzeros in factor
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1. Preorder Independent of numerics 1. Symbolic Factorization Elimination tree Nonzero counts Supernodes Nonzero structure of R 1. Numeric Factorization Static data structure Supernodes use BLAS3 to reduce memory traffic 1. Triangular Solves Sparse Cholesky factorization: Sparse Cholesky factorization: A=R A=R T R R
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Incomplete Cholesky factorization (IC, ILU) Incomplete Cholesky factorization (IC, ILU) Compute factors of A by Gaussian elimination, but ignore fill
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ILU - Column Cholesky Factorization: A=RTR for j = 1 : n...

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