Summary
In a recent paper, [4], Csordas and Varga have unified and extended earlier theorems, of Varga in [10] and Woźnicki in [11], on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings. The main purpose of this note is to show a connection between the Csordas-Varga paper and a paper by Beauwens, [1], in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular. Monotonic norms already used in [1] play an important role in our work here.
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References
Beauwens, R.: Factorization Iterative Methods,M-operators andH-operators. Numer. Math.31, 335–357 (1979)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. New York: Academic Press 1979
Collatz, L.: Aufgaben Monotoner Art. Arch. Math.3, 366–376 (1952)
Csordas, G., Varga, R.S.: Comparisons of Regular Splittings of Matrices. Numer. Math.44, 23–35 (1952)
Householder, A.S.: The Theory of Matrices in Numerical Analysis. New York: Blaisdell 1964
Ortega, J.M., Rheinboldt, W.C.: Monotone Iterations for Nonlinear Equations with applications to Gauss-Seidel Methods. SIAM J. Numer. Anal.4, 171–190 (1967)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. New York: Academic Press 1970
Rheinboldt, W.C., Vandergraft, J.S.: A Simple Approach to the Perron-Frobenius Theory for Positive Operators on General Partially Ordered Finite-Dimensional Linear Spaces. Math. Comput.27, 139–145 (1973)
Varga, R.S.: Factorization and Normalized Iterative Methods. In: Boundary Problems in Differential Equations (R.E. Langer, ed.), pp. 121–142
Varga, R.S.: Matrix Iterative Analysis. Englewood Cliffs, N.J.: Prentice-Hall 1962
Woźnicki, Z.: Two-sweep Iterative Methods for Solving Large Linear Systems and Their Application to the Numerical Solution of Multi-group Multi-dimensional Neutron Diffusion Equation. Doctoral Dissertation, Institute of Nuclear Research, Swicrk k/Otwocka, Poland, 1973
Young, D.M.: Iterative Solutions of Large Linear Systems. New York: Academic Press 1971
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Research supported in part by NSF grant number DMS-8400879
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Miller, V.A., Neumann, M. A note on comparison theorems for nonnegative matrices. Numer. Math. 47, 427–434 (1985). https://doi.org/10.1007/BF01389590
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DOI: https://doi.org/10.1007/BF01389590