Parallel complexity of householder QR factorization
Gaussian Elimination with Partial Pivoting and Householder QR factorization are two very popular methods to solve linear systems. Implementations of these two methods are provided in state-of-the-art numerical libraries and packages, such as LAPACK and MATLAB. Gaussian Elimination with Partial Pivoting was already known to be P-complete. Here we prove that the Householder QR factorization is likely to be inherently sequential as well. We also investigate the problem of speedup vs non degeneracy and accuracy in numerical algorithms.
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- 1.Anderson, E. et al. Lapack User's Guide, (Society for Industrial and Applied Mathematics, Philadelphia, 1992).Google Scholar
- 2.Csanky, L., Fast parallel matrix inversion algorithms, SIAM J. Comput. 5 (1976) 618–623.Google Scholar
- 3.Demmel, J. W., Trading off parallelism and numerical accuracy, Tech. Rep. CS-92-179, Univ. of Tennessee, June 1992 (Lapack Working Note 52).Google Scholar
- 4.Golub, G. H. and C. F. Van Loan, Matrix Computations (The Johns Hopkins University Press, Baltimore, 1989).Google Scholar
- 5.Greenlaw, R., H. J. Hoover, and W. L. Ruzzo, A Compendium of Problems Complete for P, Technical Report 91-05-01, Dept. of Computer Science and Engineering, University of Washington (1991).Google Scholar
- 6.Hoover, H. J., M. M. Klawe, and N. Pippenger, Bounding Fan-out in Logical Networks, J. ACM 31 (1984) 13–18.Google Scholar
- 8.Leoncini, M., How Much Can We Speedup Gaussian Elimination with Pivoting? in: Proc. 6th ACM Symp. on Parallel Algorithms and Architectures (1994) 290–297. Journal of Computer and System Sciences, to appear.Google Scholar
- 9.Pan, V., Complexity of Parallel Matrix Computations, Theoretical Computer Science. 54 (1987) 65–85.Google Scholar
- 10.Sigmon, K., Matlab Primer. The MATH WORKS Inc., 1994.Google Scholar
- 11.Vavasis, S.A., Gaussian Elimination with Pivoting is P-complete, SIAM J. Disc. Math. 2 (1989) 413–423.Google Scholar