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Algorithms have two costs: arithmetic and communication. The latterrepresents the cost of moving data, either between levels of a memoryhierarchy, or between processors over a network. Communication often dominatesarithmetic and represents a rapidly increasing proportion of the total cost, sowe seek algorithms that minimize communication. In \cite{BDHS10} lower boundswere presented on the amount of communication required for essentially all$O(n^3)$-like algorithms for linear algebra, including eigenvalue problems andthe SVD. Conventional algorithms, including those currently implemented in(Sca)LAPACK, perform asymptotically more communication than these lower boundsrequire. In this paper we present parallel and sequential eigenvalue algorithms(for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithmsthat do attain these lower bounds, and analyze their convergence andcommunication costs.

Minimizing Communication for Eigenproblems and the Singular Value Decomposition