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Communication efficient Gaussian elimination with partial pivoting using a shape morphing data layout

By Grey Ballard, Oded Schwartz, James Demmel, Sivan Toledo and Benjamin Lipshitz


High performance for numerical linear algebra often comes at the expense of stability. Computing the LU decomposition of a matrix via Gaussian Elimination can be organized so that the computation involves regular and efficient data access. However, maintaining numerical stability via partial pivoting involves row interchanges that lead to inefficient data access patterns. To optimize communication efficiency throughout the memory hierarchy we confront two seemingly contradictory requirements: partial pivoting is efficient with column-major layout, whereas a block-recursive layout is optimal for the rest of the computation. We resolve this by introducing a shape morphing procedure that dynamically matches the layout to the computation throughout the algorithm, and show that Gaussian Elimination with partial pivoting can be performed in a communication efficient and cache-oblivious way. Our technique extends to QR decomposition, where computing Householder vectors prefers a different data layout than the rest of the computation. Categories and Subject Descriptor

Topics: F.2.1 [Analysis of Algorithms and Problem Complexity, Numerical Algorithms and Problems, Computations on matrices General Terms Algorithms
Year: 2013
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