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On the numerical radius of matrices and its application to iterative solution methods. (English) Zbl 0813.15021
The authors present some general results on the numerical radius and its relation with the spectral radius. Estimates of the rate of convergence of basic iterative methods are also presented. The use of the numerical radius in the estimate of the rate of convergence of the successive overrelaxation method for quasi-Hermitian positive definite matrices which are, in general not consistently ordered, are found. Finally the numerical radius is used for the derivation of an upper bound of the largest eigenvalue of a symmetric positive definite matrix preconditioned by a block incomplete factorization method.
Reviewer: P.Narain (Bombay)

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
65F10 Iterative numerical methods for linear systems
15A42 Inequalities involving eigenvalues and eigenvectors
15A12 Conditioning of matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
Full Text: DOI
[1] Anderson L., Department of Computer Science (1976)
[2] DOI: 10.1007/BF01932955 · Zbl 0256.65046
[3] Axelsson O, CERN Technical Report 74-10 (1974)
[4] Axelsson O., in Sparse Matrix Techniques (1977)
[5] Axelsson O, On eigenvalue estimates for block incomplete factorization methods (1977)
[6] DOI: 10.1016/0024-3795(93)90530-2 · Zbl 0784.65022
[7] DOI: 10.1016/0024-3795(82)90155-0 · Zbl 0479.47002
[8] DOI: 10.1016/0024-3795(75)90042-7 · Zbl 0318.15006
[9] Horn R. A., Matrix Analysis (1985) · Zbl 0576.15001
[10] DOI: 10.1017/CBO9780511840371
[11] DOI: 10.1137/0715039 · Zbl 0389.65018
[12] DOI: 10.1016/0024-3795(90)90232-2 · Zbl 0712.15027
[13] DOI: 10.1016/0024-3795(87)90309-0 · Zbl 0624.15012
[14] DOI: 10.1307/mmj/1031732779 · Zbl 0143.16205
[15] DOI: 10.1016/0024-3795(93)90531-R · Zbl 0784.65026
[16] Varga R., Pacific J. Math. 9 pp 925– (1959)
[17] Varga R., Matrix Iterative Analysis (1962) · Zbl 0133.08602
[18] Young D. M., Ph.D. thesis (1950)
[19] young D. M., Iterative Solution of Large Systems (1971)
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