Summary
We present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations, say of the formAx=b, whereA ∈ ℝN, N, withA nonsingular, andb ∈ ℝN are given. This hybrid method begins with a limited number of steps of the Arnoldi method to obtain some information on the location of the spectrum ofA, and then switches to a Richardson iterative method based on Faber polynomials. For a polygonal domain, the Faber polynomials can be constructed recursively from the parameters in the Schwarz-Christoffel mapping function. In four specific numerical examples of non-normal matrices, we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid GMRES or restarted versions of the GMRES algorithm. It is, however, sensitive (as other hybrid methods also are) to the amount of information on the spectrum ofA acquired during the first (Arnoldi) phase of this procedure.
Similar content being viewed by others
References
Arnoldi, W.E. (1951): The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math.9, 17–29
Carathéodory, C., Fejér, L. (1911): Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz. Rend. Circ. Mat. Palermo32, 218–239
Eiermann, M. (1989): On semiiterative methods generated by Faber polynomials. Numer. Math.56, 139–156
Eiermann, M. (1992): Fields of values and iterative methods. Linear Algebra Appl. (to appear)
Eiermann, M., Li, X., Varga, R.S. (1989): On hybrid semi-iterative methods. SIAM J. Numer. Anal.26, 152–168
Eiermann, M., Niethammer, W. (1983): On the construction of semiiterative methods. SIAM J. Numer. Anal.20, 1153–1160
Eiermann, M., Niethammer, W., Varga, R.S. (1985): A study of semiiterative methods for non-symmetric systems of linear equations. Numer. Math.47, 505–533
Eiermann, M., Starke, G. (1990): The near-best solution of a polynomial minimization problem by the Carathéodory-Fejér method. Constr. Approx.6, 303–319
Ellacott, S.W. (1983): Computation of Faber series with application to numerical polynomial approximation in the complex plane. Math. Comp.40, 575–587
Elman, H.C., Saad, Y., Saylor, P.E. (1986): A hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations. SIAM J. Sci. Stat. Comput.7, 840–855
Elman, H.C., Streit, R.L. (1986): Polynomial iteration for nonsymmetric indefinite linear systems. In: Numerical Analysis, Lecture Notes in Mathematics 1230, pp. 103–117. Springer, Berlin Heidelberg New York
Faber, G. (1903): Über polynomische Entwicklungen. Math. Ann.57, 389–408
Farkova, N.A. (1988): The use of Faber polynomials to solve systems of linear equations. U.S.S.R. Comput. Maths. Math. Phys.28, 22–32
Fischer, B., Freund, R.W. (1991): Chebyshev polynomials are not always optimal. J. Approx. Theory65, 261–272
Gaier, D. (1987): Lectures on complex approximation. Birkhäuser, Boston Basel Stuttgart
Golub, G.H., Loan, C.F.V. (1989): Matrix computations, 2nd edn. Johns Hopkins University Press, Baltimore London
Golub, G.H., Varga, R.S. (1961): Chebyshev semiiterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods. Numer. Math.3, 147–168
Gutknecht, M.H. (1986): An iterative method for solving linear equations based on minimum norm Pick-Nevanlinna interpolation. In: CKC et al., eds. Approximation theory V. pp. 371–374. New York, Academic Press
Henrici, P. (1974): Applied and computational complex analysis I. Wiley, New York London Sydney Toronto
Henrici, P. (1986): Applied and computational complex analysis III. Wiley, New York London Sydney Toronto
Horn, R.A., Johnson, C.R. (1991): Topics in matrix analysis. Cambridge University Press, Cambridge New York Port Chester Melbourne Sydney
Kövari, T., Pommerenke, C. (1967): On Faber polynomials and Faber expansions. Math. Z.99, 193–206
Kublanovskaja, V.N. (1959): Applications of analytic continuation in numerical analysis by means of change of variables. Trudy Mat. Inst. Steklov53, 145–185
Li, X. (1989): An adaptive method for solving nonsymmetric linear systems involving application of SCPACK. PhD thesis, Kent State University
Manteuffel, T.A. (1978): Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration. Numer. Math.31, 183–208
Nachtigal, N.M., Reichel, L., Trefethen, L.N. (1992): A hybrid GMRES algorithm for nonsymmetric matrix iterations. SIAM J. Matrix Anal. Appl.13, 796–825
Nehari, Z. (1952): Conformal mapping. McGraw-Hill, New York
Niethammer, W., Varga, R.S. (1983): The analysis ofk-step iterative methods for linear systems from summability theory. Numer. Math.41, 177–206
Pommerenke, C. (1965): Konforme Abbildung und Fekete-Punkte. Math. Z.89, 422–438
Rivlin, T.J., Shapiro, H.S. (1961): A unified approach to certain problems of approximation and minimization. J. Soc. Indust. Appl. Math.9, 670–699
Saad, Y. (1980): Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl.34, 269–295
Saad, Y. (1987): Least squares polynomials in the complex plane and their use for solving nonsymmetric linear systems. SIAM J. Numer. Anal.24, 155–169
Saad, Y., Schultz, M.H. (1986): GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput.7, 856–869
Saylor, P.E., Smolarski, D.C. (1991): Implementation of an adaptive algorithm for Richardson's method. Linear Algebra Appl.154–156, 615–646
Smirnov, V.I., Lebedev, N.A. (1968): Functions of a complex variable: Constructive theory. M.I.T. Press, Cambridge, MA
Sonneveld, P. (1989): CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput.10, 36–52
Trefethen, L.N. (1980): Numerical computation of the Schwarz-Christoffel transformation. SIAM J. Sci. Stat. Comput.1, 82–102
Trefethen, L.N. (1981): Near-circularity of the error curve in complex Chebyshev approximation. J. Approx. Theory31, 344–367
Trefethen, L.N. (1990): Approximation theory and numerical linear algebra. In: Algorithms for approximation II, 336–360. Chapman & Hall, London
Varga, R.S. (1957): A comparison of the successive overrelaxation method and semi-iterative methods using Chebyshev polynomials. J. Soc. Indust. Appl. Math.5, 39–46
Walsh, J.L. (1956): Interpolation and approximation by rational functions in the complex domain, 2nd edn. American Mathematical Society, Rhode Island
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Starke, G., Varga, R.S. A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations. Numer. Math. 64, 213–240 (1993). https://doi.org/10.1007/BF01388688
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01388688