Fields of values and the ADI method for non-normal matrices. (English) Zbl 0784.65026
Generalizing the results of M. Eiermann [Lin. Algebra Appl. 180, 167-197 (1993; Zbl 0784.65022)], the author obtains an upper bound for the error reduction of the ADI method in terms of the fields of values of the matrices which define splitting of the matrix in the iterations. Then it is shown that if \(A\) is an irreducible tridiagonal matrix and if \(B\) is the symmetric matrix similar to \(A\), then the field of values of \(B\) is included in that of \(A\), and the result is generalized to Kronecker sums of tridiagonal matrices. This result may be considered as an alternative to preconditioning for non-normal matrices.
The estimate is tested on a model boundary value problem. Numerical results demonstrate the power of analysis based on the field of values for considered problems and indicate that the resulting estimates may be useful for an a priori prediction of the behaviour of the process in the finite stage of computation.
The estimate is tested on a model boundary value problem. Numerical results demonstrate the power of analysis based on the field of values for considered problems and indicate that the resulting estimates may be useful for an a priori prediction of the behaviour of the process in the finite stage of computation.
Reviewer: Z.Dostal (Ostrava)
MSC:
| 65F10 | Iterative numerical methods for linear systems |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
Keywords:
numerical results; error reduction; ADI method; fields of values; irreducible tridiagonal matrix; Kronecker sums of tridiagonal matrices; preconditioningReferences:
| [1] | Bonsall, F. F.; Duncan, J., Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras (1971), Cambridge U.P: Cambridge U.P New York · Zbl 0207.44802 |
| [2] | Chin, R. C.Y.; Manteuffel, T. A., An analysis of block successive overrelaxation for a class of matrices with complex spectra, SIAM J. Numer. Anal., 25, 564-585 (1988) · Zbl 0655.65060 |
| [3] | Chin, R. C.Y.; Manteuffel, T. A.; de Pillis, J., ADI as a preconditioning for solving the convection-diffusion equation, SIAM J. Sci. Statist. Comput., 5, 281-299 (1984) · Zbl 0549.65068 |
| [4] | Eiermann, M., Fields of values and iterative methods, Linear Algebra Appl., 180, 167-197 (1993) · Zbl 0784.65022 |
| [5] | Eiermann, M.; Varga, R. S.; Niethammer, W., Iterationsverfahren für nicht-symmetrische Gleichungssysteme und Approximationsmethoden im Komplexen, Jahresber. Deutsch. Math.-Verein, 89, 1-32 (1987) · Zbl 0632.65031 |
| [6] | Elman, H. C.; Golub, G. H., Iterative methods for cyclically reduced non-self-adjoint linear systems, Math. Comp., 54, 671-700 (1990) · Zbl 0699.65021 |
| [7] | Gantmacher, F. R., Matrizentheorie (1986), Springer-Verlag: Springer-Verlag New York |
| [8] | Hausdorff, F., Der Wertevorrat einer Bilinearform, Math. Z., 3, 314-316 (1919) · JFM 47.0088.02 |
| [9] | Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge U.P: Cambridge U.P New York · Zbl 0576.15001 |
| [10] | Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge U.P: Cambridge U.P New York · Zbl 0729.15001 |
| [11] | Kato, T., Some mapping theorems for the numerical range, Proc. Japan Acad., 41, 652-655 (1965) · Zbl 0143.36702 |
| [12] | Marcus, M.; Minc, H., A Survey of Matrix Theory and Matrix Inequalities (1964), Allyn and Bacon: Allyn and Bacon Boston · Zbl 0126.02404 |
| [13] | Pearcy, C., An elementary proof of the power inequality for the numerical radius, Michigan Math. J., 13, 289-291 (1966) · Zbl 0143.16205 |
| [14] | Price, H. S.; Varga, R. S., Recent Numerical Experiments Comparing Successive Overrelaxation Iterative Methods with Implicit Alternating Direction Methods, (Gulf Research and Development Co. Report (1962), Harmarville, Pa) |
| [15] | Starke, G., Optimal ADI parameters for nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 28, 1431-1445 (1991) · Zbl 0739.65029 |
| [16] | Trefethen, L. N., Approximation theory and numerical linear algebra, (Algorithms for Approximation II (1990), Chapman & Hall), pp. 336-360 · Zbl 0747.41005 |
| [17] | Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall New York · Zbl 0133.08602 |
| [18] | Wachspress, E. L., Extended application of alternating direction implicit iteration model problem theory, J. Soc. Indust. Appl. Math., 11, 994-1016 (1963) · Zbl 0244.65045 |
| [19] | Wachspress, E. L., Optimum alternating direction implicit iteration parameters for rectangular regions in the complex plane, (Preprint (1990), Australian National Univ: Australian National Univ Canberra, Australia) · Zbl 0111.31401 |
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