Proof of Correctness for Sparse Tiling of Gauss-Seidel

Michelle Mills Strout, Larry Carter and Jeanne Ferrante
April 1, 2003

Gauss-Seidel is an iterative computation used for solving a set of simultaneous linear equations, A\vec{u}=\vec{f}. If the matrix A uses a sparse matrix representation, storing only nonzeros, then the data dependences in the computation arise from A's nonzero structure. We use this structure to schedule the computation at runtime using a technique called {\em full sparse tiling}. The sparse tiled computation exhibits better data locality and therefore improved performance. This paper gives a complete proof that a serial schedule for full sparse tiled Gauss-Seidel generates results equivalent to those that a typical Gauss-Seidel computation produces. We also provide implementation and correctness details for full sparse tiling with reduced worst-case complexity. [NOTE: Dave Wargo told me this techreport would be assigned #CS2003-0741 and I am assuming it will be April 2003 as the date, please let me know if these assumptions are incorrect.]

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