Abstract
We consider the problem of computing a scaling \(\alpha \) such that the solution \({\varvec{x}}\) of the scaled linear system \({\varvec{Tx}} = \alpha {\varvec{b}}\) can be computed without exceeding an overflow threshold \(\varOmega \). Here \({\varvec{T}}\) is a non-singular upper triangular matrix and \({\varvec{b}}\) is a single vector, and \(\varOmega \) is less than the largest representable number. This problem is central to the computation of eigenvectors from Schur forms. We show how to protect individual arithmetic operations against overflow and we present a robust scalar algorithm for the complete problem. Our algorithm is very similar to xLATRS in LAPACK. We explain why it is impractical to parallelize these algorithms. We then derive a robust blocked algorithm which can be executed in parallel using a task-based run-time system such as StarPU. The parallel overhead is increased marginally compared with regular blocked backward substitution.
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References
Anderson, E.: LAPACK working note no. 36: robust triangular solves for use in condition estimation. Technical report CS-UT-CS-91-142, University of Tennessee, Knoxville, TN, USA, August 1991
Gates, M., Haidar, A., Dongarra, J.: Accelerating computation of eigenvectors in the dense nonsymmetric eigenvalue problem. In: Daydé, M., Marques, O., Nakajima, K. (eds.) VECPAR 2014. LNCS, vol. 8969, pp. 182–191. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-17353-5_16
Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1989)
Kjelgaard Mikkelsen, C.C., Karlsson, L.: NLAFET working note no. 10: towards highly parallel and compute-bound computation of eigenvectors of matrices in schur form. Technical report 17–10, Umeå University, Umeå, Sweden, May 2017
Kjelgaard Mikkelsen, C.C., Karlsson, L.: NLAFET working note no. 9: robust solution of triangular linear systems. Technical report 17–9, Umeå University, Umeå, Sweden, May 2017
Moon, T., Poulson, J.: Accelerating Eigenvector and Pseudospectra Computation using Blocked Multi-Shift Triangular Solves, July 2016. http://arxiv.org/abs/1607.01477
Acknowledgment
This work is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 671633.
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Kjelgaard Mikkelsen, C.C., Karlsson, L. (2018). Blocked Algorithms for Robust Solution of Triangular Linear Systems. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2017. Lecture Notes in Computer Science(), vol 10777. Springer, Cham. https://doi.org/10.1007/978-3-319-78024-5_7
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DOI: https://doi.org/10.1007/978-3-319-78024-5_7
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