Summary
Given a nonsingular linear systemA x=b, a splittingA=M−N leads to the one-step iteration (1)x m =T X m−1 +c withT:=M −1N andc:=M −1 b. We investigate semiiterative methods (SIM's) with respect to (1), under the assumption that the eigenvalues ofT are contained in some compact set Ω of ℂ, with 1≠Ω. There exist SIM's which are optimal with respect to Ω, but, except for some special sets Ω, such optimal methods are not explicitly known in general. Using results about “maximal convergence” of polynomials and “uniformly distributed” nodes from approximation and function theory, we describe here SIM's which are asymptotically optimal with respect to Ω. It is shown that Euler methods, extensively studied by Niethammer-Varga [NV], are special SIM's. Various algorithms for SIM's are also derived here. A 1-1 correspondence between Euler methods and SIM's, generated by generalized Faber polynomials, is further established here. This correspondence gives that asymptotically optimal Euler methods are quite near the optimal SIM's.
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Dedicated to Professor Karl Zeller (Universität Tübingen) on the occasion of his sixtieth birthday (December 28, 1984)
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Eiermann, M., Niethammer, W. & Varga, R.S. A study of semiiterative methods for nonsymmetric systems of linear equations. Numer. Math. 47, 505–533 (1985). https://doi.org/10.1007/BF01389454
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DOI: https://doi.org/10.1007/BF01389454