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cs240a-denseGE

# cs240a-denseGE - CS 240A Solving Ax = b in parallel Dense A...

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CS267 Dense Linear Algebra I.1 Demmel Fa 2001 CS 240A: Solving Ax = b in parallel ° Dense A: Gaussian elimination with partial pivoting Same flavor as matrix * matrix, but more complicated ° Sparse A: Iterative methods – Conjugate gradient etc. Sparse matrix times dense vector ° Sparse A: Gaussian elimination – Cholesky, LU, etc. Graph algorithms ° Sparse A: Preconditioned iterative methods and multigrid Mixture of lots of things

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CS267 Dense Linear Algebra I.2 Demmel Fa 2001 CS 240A: Solving Ax = b in parallel ° Dense A: Gaussian elimination with partial pivoting Same flavor as matrix * matrix, but more complicated ° Sparse A: Iterative methods – Conjugate gradient etc. Sparse matrix times dense vector ° Sparse A: Gaussian elimination – Cholesky, LU, etc. Graph algorithms ° Sparse A: Preconditioned iterative methods and multigrid Mixture of lots of things
CS267 Dense Linear Algebra I.3 Demmel Fa 2001 Dense Linear Algebra (Excerpts) James Demmel http://www.cs.berkeley.edu/~demmel/cs267_221001.ppt

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CS267 Dense Linear Algebra I.4 Demmel Fa 2001 Motivation ° 3 Basic Linear Algebra Problems Linear Equations: Solve Ax=b for x Least Squares: Find x that minimizes Σ r i 2 where r=Ax-b Eigenvalues: Find λ and x where Ax = λ x Lots of variations depending on structure of A (eg symmetry) ° Why dense A, as opposed to sparse A? Aren’t “most” large matrices sparse? Dense algorithms easier to understand Some applications yields large dense matrices - Ax=b: Computational Electromagnetics - Ax = λ x: Quantum Chemistry Benchmarking - “How fast is your computer?” = “How fast can you solve dense Ax=b?” Large sparse matrix algorithms often yield smaller (but still large) dense problems
CS267 Dense Linear Algebra I.5 Demmel Fa 2001 Review of Gaussian Elimination (GE) for solving Ax=b ° Add multiples of each row to later rows to make A upper triangular ° Solve resulting triangular system Ux = c by substitution … for each column i … zero it out below the diagonal by adding multiples of row i to later rows for i = 1 to n-1 … for each row j below row i for j = i+1 to n … add a multiple of row i to row j for k = i to n A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k)

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CS267 Dense Linear Algebra I.6 Demmel Fa 2001 Refine GE Algorithm (1) ° Initial Version ° Remove computation of constant A(j,i)/A(i,i) from inner loop … for each column i … zero it out below the diagonal by adding multiples of row i to later rows for i = 1 to n-1 … for each row j below row i for j = i+1 to n … add a multiple of row i to row j for k = i to n A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k) for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i to n A(j,k) = A(j,k) - m * A(i,k)
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cs240a-denseGE - CS 240A Solving Ax = b in parallel Dense A...

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