Journal of Computational Physics
Parallel solution of large-scale eigenvalue problem for master equation in protein folding dynamics
Journal of Parallel and Distributed Computing
Anasazi software for the numerical solution of large-scale eigenvalue problems
ACM Transactions on Mathematical Software (TOMS)
Tall and skinny QR factorizations in MapReduce architectures
Proceedings of the second international workshop on MapReduce and its applications
A communication-avoiding thick-restart lanczos method on a distributed-memory system
Euro-Par'11 Proceedings of the 2011 international conference on Parallel Processing
Amesos2 and Belos: Direct and iterative solvers for large sparse linear systems
Scientific Programming
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First, we consider the problem of orthonormalizing skinny (long) matrices. We propose an alternative orthonormalization method that computes the orthonormal basis from the right singular vectors of a matrix. Its advantages are that (a) all operations are matrix-matrix multiplications and thus cache efficient, (b) only one synchronization point is required in parallel implementations, and (c) it is typically more stable than classical Gram--Schmidt (GS). Second, we consider the problem of orthonormalizing a block of vectors against a previously orthonormal set of vectors and among itself. We solve this problem by alternating iteratively between a phase of GS and a phase of the new method. We provide error analysis and use it to derive bounds on how accurately the two successive orthonormalization phases should be performed to minimize total work performed. Our experiments confirm the favorable numerical behavior of the new method and its effectiveness on modern parallel computers.