Abstract
The known algorithms for linear systems of equations perform significantly slower where the input matrix is ill conditioned, that is lies near a matrix of a smaller rank. The known methods counter this problem only for some important but special input classes, but our novel randomized augmentation techniques serve as a remedy for a typical ill conditioned input and similarly facilitates computations with rank deficient input matrices. The resulting acceleration is dramatic, both in terms of the proved bit-operation cost bounds and the actual CPU time observed in our tests. Our methods can be effectively applied to various other fundamental matrix and polynomial computations as well.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bunch, J.R.: Stability of Methods for Solving Toeplitz Systems of Equations. SIAM J. Sci. Stat. Comput. 6(2), 349–364 (1985)
Benzi, M.: Preconditioning Techniques for Large Linear Systems: a Survey. J. of Computational Physics 182, 418–477 (2002)
Coppersmith, D., Winograd, S.: Matrix Multiplicaton via Arithmetic Progressions. J. of Symbolic Computation 9(3), 251–280 (1990)
Demmel, J.: The Probability That a Numerical Analysis Problem Is Difficult. Math. of Computation 50, 449–480 (1988)
Demillo, R.A., Lipton, R.J.: A Probabilistic Remark on Algebraic Program Testing. Information Processing Letters 7(4), 193–195 (1978)
Edelman, A.: Eigenvalues and Condition Numbers of Random Matrices, PhD Thesis (106 pages), Math Dept., MIT (1989); SIAM J. on Matrix Analysis and Applications, 9(4), 543–560 (1988)
Gu, M.: Stable and Efficient Algorithms for Structured Systems of Linear Equations. SIAM J. on Matrix Analysis and Applications 19, 279–306 (1998)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)
Pan, V.Y.: On Computations with Dense Structured Matrices. Math. of Computation 55(191), 179–190 (1990)
Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser/Springer, Boston/New York (2001)
Pan, V.Y., Grady, D., Murphy, B., Qian, G., Rosholt, R.E., Ruslanov, A.: Schur Aggregation for Linear Systems and Determinants. Theoretical Computer Science 409(2), 255–268 (2008)
Pan, V.Y., Qian, G.: On Solving Linear System with Randomized Augmentation. Tech. Report TR 2009009, Ph.D. Program in Computer Science, Graduate Center, the City University of New York (2009), http://www.cs.gc.cuny.edu/tr/techreport.php?id=352
Pan, V.Y., Qian, G., Zheng, A.: Randomized Preprocessing versus Pivoting. Tech. Report TR 2009010, Ph.D. Program in Computer Science, Graduate Center, the City University of New York (2009), http://www.cs.gc.cuny.edu/tr/techreport.php?id=352
Sankar, A., Spielman, D., Teng, S.-H.: Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices. SIAM Journal on Matrix Analysis 28(2), 446–476 (2006)
Van Barel, M.: A Supefast Toeplitz Solver (1999), http://www.cs.kuleuven.be/~marc/software/index.html
Van Barel, M., Heinig, G., Kravanja, P.: A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems. SIAM Journal on Matrix Analysis and Applications 23(2), 494–510 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pan, V.Y., Qian, G., Zheng, AL. (2010). Advancing Matrix Computations with Randomized Preprocessing. In: Ablayev, F., Mayr, E.W. (eds) Computer Science – Theory and Applications. CSR 2010. Lecture Notes in Computer Science, vol 6072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13182-0_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-13182-0_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13181-3
Online ISBN: 978-3-642-13182-0
eBook Packages: Computer ScienceComputer Science (R0)