Abstract
In an eigenvalue problem defined by one or two matrices with block-tridiagonal structure, if only a few eigenpairs are required it is interesting to consider iterative methods based on Krylov subspaces, even if matrix blocks are dense. In this context, using the GPU for the associated dense linear algebra may provide high performance. We analyze this in an implementation done in the context of SLEPc, the Scalable Library for Eigenvalue Problem Computations. In the case of a generalized eigenproblem or when interior eigenvalues are computed with shift-and-invert, the main computational kernel is the solution of linear systems with a block-tridiagonal matrix. We explore possible implementations of this operation on the GPU, including a block cyclic reduction algorithm.
This work was partially supported by the Spanish Ministry of Economy and Competitiveness under grant TIN2013-41049-P. Alejandro Lamas was supported by the Spanish Ministry of Education, Culture and Sport through grant FPU13-06655.
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Notes
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Thrust is a C++ template library included in the CUDA software development toolkit that makes common CUDA operations concise and readable. CUSP is an open source library based on Thrust that targets sparse linear algebra.
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The authors are grateful for the computing resources provided by the Spanish Supercomputing Network (RES).
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Lamas Daviña, A., Roman, J.E. (2016). GPU Implementation of Krylov Solvers for Block-Tridiagonal Eigenvalue Problems. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2015. Lecture Notes in Computer Science(), vol 9573. Springer, Cham. https://doi.org/10.1007/978-3-319-32149-3_18
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