Abstract
Relaxation methods of H.R. Schwarz and A. Ruhe are used for the computation of the spectral norm σ1 of an arbitrary m×n-matrix A (i.e., σ1 is the maximal singular value of A). σ1 is eigenvalue and spectral radius of a symmetric weakly two-cyclic matrix Â. Using this fact, results on the optimal (asymptotic) relaxation factor ωo are derived. Further it is shown that using ωo from the beginning of the algorithm often prevents a rapid convergence. A strategy for the choice of ω is presented.
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
Forsythe, G.E., C.B. Moler: Computer solution of linear algebraic systems, Prentice Hall, Englewood Cliffs, 1967.
Golub, G.H., Kahan, W.: Calculating the singular values and pseudoinverse of a matrix, J. SIAM Numer. Anal., Ser. B2, 205–224 (1965).
Golub, G.H., C. Reinsch: Singular value decomposition and least squares solutions, in "Handbook for Automatic Computation", Vol. II Linear Algebra, 134–151, Springer, Berlin-Heidelberg-New York, 1971.
Kolb, G.: Relaxationsmethoden zur Berechnung der Spektralnorm beliebiger Matrizen, Dissertation, Universität Mannheim, 1980.
Ruhe, A.: SOR-methods for the eigenvalue problem with large sparse matrices, Math. of Comp. 28, 695–710 (1974).
Schwarz, H.R.: The method of coordinate overrelaxation for (A−λB)x=0, Numer. Math. 23, 135–151 (1974).
Schwarz, H.R.: The eigenvalue problem (A−λB)x=0 for symmetric matrices of high order, Comp. Math. in Appl. Mech. and Eng. 3, 11–28 (1974).
Varga, R.S.: Matrix iterative analysis, Prentice Hall, Englewood Cliffs, 1962.
Young, D.M.: Iterative solution of large linear systems, Academic Press, New York-San Francisco-London, 1971.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 Springer-Verlag
About this paper
Cite this paper
Kolb, G., Niethammer, W. (1982). Relaxation methods for the computation of the spectral norm. In: Ansorge, R., Meis, T., Törnig, W. (eds) Iterative Solution of Nonlinear Systems of Equations. Lecture Notes in Mathematics, vol 953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069380
Download citation
DOI: https://doi.org/10.1007/BFb0069380
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11602-8
Online ISBN: 978-3-540-39379-5
eBook Packages: Springer Book Archive