Abstract
We develop probabilistic upper bounds for the matrix two-norm, the largest singular value. These bounds, which are true upper bounds with a user-chosen high probability, are derived with a number of different polynomials that implicitly arise in the Lanczos bidiagonalization process. Since these polynomials are adaptively generated, the bounds typically give very good results. They can be computed efficiently. Together with an approximation that is a guaranteed lower bound, this may result in a small probabilistic interval for the matrix norm of large matrices within a fraction of a second.
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Notes
www.netlib.org/svdpack/
Available via www.math.uri.edu/\(\sim \)jbaglama/
We hereby would like to make a case for the replacement of normest in Matlab by a procedure based on Lanczos bidiagonalization.
References
Baglama, J., Reichel, L.: Augmented implicitly restarted Lanczos bidiagonalization methods. SIAM J. Sci. Comput. 27, 19–42 (2005)
Baglama, J., Reichel, L.: Restarted block Lanczos bidiagonalization methods. Numer. Algorithms 43, 251–272 (2007)
Bischof, C.H.: Incremental condition estimation. SIAM J. Matrix Anal. Appl. 11, 312–322 (1990)
Bischof, C.H., Lewis, J.G., Pierce, D.J.: Incremental condition estimation for sparse matrices. SIAM J. Matrix Anal. Appl. 11, 644–659 (1990)
Golub, G.H., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. Soc. Indust Appl. Math. Ser. B Numer. Anal. 2, 205–224 (1965)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996)
Hochstenbach, M.E.: A Jacobi-Davidson type SVD method. SIAM J. Sci. Comput. 23, 606–628 (2001)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Kuczyński, J., Woźniakowski, H.: Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start. SIAM J. Matrix Anal. Appl. 13, 1094–1122 (1992)
Parlett, B.N.: The Symmetric Eigenvalue Problem. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1998)
Stoll, M.: A Krylov-Schur approach to the truncated SVD. Linear Algebra Appl. 436, 2795–2806 (2012)
The Matrix Market. http://math.nist.gov/MatrixMarket, a repository for test matrices
Van Dorsselaer, J.L.M., Hochstenbach, M.E., Van der Vorst, H.A.: Computing probabilistic bounds for extreme eigenvalues of symmetric matrices with the Lanczos method. SIAM J. Matrix Anal. Appl. 22, 837–852 (2000)
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The author gratefully acknowledges support by an NWO Vidi grant. Helpful comments from a referee were greatly appreciated.
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Hochstenbach, M.E. Probabilistic Upper Bounds for the Matrix Two-Norm. J Sci Comput 57, 464–476 (2013). https://doi.org/10.1007/s10915-013-9716-x
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DOI: https://doi.org/10.1007/s10915-013-9716-x
Keywords
- Matrix two-norm
- Probabilistic bound
- SVD
- Singular value problem
- Singular value decomposition
- Subspace method
- Lanczos bidiagonalization
- Lanczos polynomial
- Ritz polynomial
- Condition number
- Large (sparse) matrix