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Unformatted text preview: Cholesky Factorization Parallel Dense Cholesky Parallel Sparse Cholesky Parallel Numerical Algorithms Chapter 7 – Cholesky Factorization Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign CSE 512 / CS 554 Michael T. Heath Parallel Numerical Algorithms 1 / 52 Cholesky Factorization Parallel Dense Cholesky Parallel Sparse Cholesky Outline 1 Cholesky Factorization 2 Parallel Dense Cholesky 3 Parallel Sparse Cholesky Michael T. Heath Parallel Numerical Algorithms 2 / 52 Cholesky Factorization Parallel Dense Cholesky Parallel Sparse Cholesky Cholesky Factorization Computing Cholesky Cholesky Algorithm Cholesky Factorization Symmetric positive definite matrix A has Cholesky factorization A = LL T where L is lower triangular matrix with positive diagonal entries Linear system Ax = b can then be solved by forwardsubstitution in lower triangular system Ly = b , followed by backsubstitution in upper triangular system L T x = y Michael T. Heath Parallel Numerical Algorithms 3 / 52 Cholesky Factorization Parallel Dense Cholesky Parallel Sparse Cholesky Cholesky Factorization Computing Cholesky Cholesky Algorithm Computing Cholesky Factorization Algorithm for computing Cholesky factorization can be derived by equating corresponding entries of A and LL T and generating them in correct order For example, in 2 × 2 case a 11 a 21 a 21 a 22 = ‘ 11 ‘ 21 ‘ 22 ‘ 11 ‘ 21 ‘ 22 so we have ‘ 11 = √ a 11 , ‘ 21 = a 21 /‘ 11 , ‘ 22 = q a 22 ‘ 2 21 Michael T. Heath Parallel Numerical Algorithms 4 / 52 Cholesky Factorization Parallel Dense Cholesky Parallel Sparse Cholesky Cholesky Factorization Computing Cholesky Cholesky Algorithm Cholesky Factorization Algorithm for k = 1 to n a kk = √ a kk for i = k + 1 to n a ik = a ik /a kk end for j = k + 1 to n for i = j to n a ij = a ij a ik a jk end end end Michael T. Heath Parallel Numerical Algorithms 5 / 52 Cholesky Factorization Parallel Dense Cholesky Parallel Sparse Cholesky Cholesky Factorization Computing Cholesky Cholesky Algorithm Cholesky Factorization Algorithm All n square roots are of positive numbers, so algorithm well defined Only lower triangle of A is accessed, so strict upper triangular portion need not be stored Factor L is computed in place, overwriting lower triangle of A Pivoting is not required for numerical stability About n 3 / 6 multiplications and similar number of additions are required (about half as many as for LU) Michael T. Heath Parallel Numerical Algorithms 6 / 52 Cholesky Factorization Parallel Dense Cholesky Parallel Sparse Cholesky Parallel Algorithm Loop Orderings ColumnOriented Algorithms Parallel Algorithm Partition For i,j = 1 ,...,n , finegrain task ( i,j ) stores a ij and computes and stores ‘ ij , if i ≥ j ‘ ji , if i < j yielding 2D array of n 2 finegrain tasks Zero entries in upper triangle of L need not be computed or stored, so for convenience in using 2D mesh network, ‘ ij can be redundantly computed as both task...
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 Summer '09
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 Algorithms, Numerical Analysis, Cholesky factorization, Parallel Numerical Algorithms, parallel sparse cholesky, Parallel Dense Cholesky

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