Abstract
Our weakly random additive preconditioners facilitate the solution of linear systems of equations and other fundamental matrix computations. Compared to the popular SVD-based multiplicative preconditioners, these preconditioners are generated more readily and for a much wider class of input matrices. Furthermore they better preserve matrix structure and sparseness and have a wider range of applications, in particular to linear systems with rectangular coefficient matrices. We study the generation of such preconditioners and their impact on conditioning of the input matrix. Our analysis and experiments show the power of our approach even where we use very weak randomization and choose sparse and/or structured preconditioners.
Supported by PSC CUNY Awards 68291–0037 and 69330–0038.
This is a preview of subscription content, log in via an institution to check access.
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
Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)
Benzi, M.: Preconditioning Techniques for Large Linear Systems: a Survey. J. of Computational Physics 182, 418–477 (2002)
Chen, K.: Matrix Preconditioning Techniques and Applications. Cambridge University Press, Cambridge (2005)
Pan, V.Y., Ivolgin, D., Murphy, B., Rosholt, R.E., Tang, Y., Yan, X.: Additive Preconditioning for Matrix Computations, Technical Report TR 2007003, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York (2007), http://www.cs.gc.cuny.edu/tr/files/TR-2007003.pdf
Pan, V.Y., Ivolgin, D., Murphy, B., Rosholt, R.E., Tang, Y., Yan, X.: Additive Preconditioning and Aggregation in Matrix Computations. Computers and Math. with Applications 55(8), 1870–1886 (2008)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore, Maryland (1996)
Stewart, G.W.: Matrix Algorithms, Vol I: Basic Decompositions; Vol II: Eigensystems. SIAM, Philadelphia (1998)
Demillo, R.A., Lipton, R.J.: A Probabilistic Remark on Algebraic Program Testing. Information Processing Letters 7(4), 193–195 (1978)
Zippel, R.E.: Probabilistic Algorithms for Sparse Polynomials. In: EUROSAM 1979 and ISSAC 1979. LNCS, vol. 72, pp. 216–226. Springer, Berlin (1979)
Schwartz, J.T.: Fast Probabilistic Algorithms for Verification of Polynomial Identities. Journal of ACM 27(4), 701–717 (1980)
Wang, X.: Affect of Small Rank Modification on the Condition Number of a Matrix. Computer and Math (with Applications) 54, 819–825 (2007)
Pan, V.Y.: Null Aggregation and Extensions, Technical Report TR2007009, CUNY Ph.D. Program in Computer Science, Graduate Center, City University of New York (2007), http://www.cs.gc.cuny.edu/tr/files/TR-2007009.pdf
Tao, T., Van Wu,: The Condition Number of a Randomly Perturbed Matrix. In: Proc. Annual Symposium on Theory of Computing (STOC 2007). ACM Press, New York (2007)
Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser, Boston, and Springer, New York (2001)
Dongarra, J.J., Duff, I.S., Sorensen, D.C., van Der Vorst, H.A.: Numerical Linear Algebra for High-Performance Computers. SIAM, Philadelphia (1998)
Vandebril, R., Van Barel, M., Golub, G., Mastronardi, N.: A Bibliography on Semiseparable Matrices. Calcolo 42(3–4), 249–270 (2005)
Pan, V.Y.: Matrix Computations with Randomized Expansion and Aggregation (preprint, 2008)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pan, V.Y., Ivolgin, D., Murphy, B., Rosholt, R.E., Tang, Y., Yan, X. (2008). Additive Preconditioning for Matrix Computations. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_37
Download citation
DOI: https://doi.org/10.1007/978-3-540-79709-8_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79708-1
Online ISBN: 978-3-540-79709-8
eBook Packages: Computer ScienceComputer Science (R0)