Abstract
The computation of bestR-orders of sequences produced by iterative methods leads to the determination of the cone radius of certain concave operators. The main result of the paper is the representation of the cone radius as the infimum of spectral radii of all linear operators majorizing the concave operator. This characterization is of numerical interest.
Zusammenfassung
Die Berechnung der bestenR-Ordnung von Folgen aus iterativen Näherungsverfahren führt auf die Ermittlung des hier eingeführten Kegelradius bestimmter konkaver Operatoren. Das Hauptergebnis der Arbeit ist die Darstellung des Kegelradius als Infimum der Spektralradien aller den konkaven Operator majorisierenden linearen Operatoren. Diese Charakterisierung ist von numerischem Intersse.
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Bachman, G., Narici, L.: Functional analysis. New York-London: Academic Press 1966.
Barker, G. P., Schneider, H.: Algebraic Perron-Frobenius theory. Linear Algebra Appl.11, 219–233 (1975).
Bohl, E.: Monotonie: Lösbarkeit und Numerik bei Operatorgleichungen. Berlin-Heidelberg-New York: Spring 1974.
Burmeister, W.: Linear supporting operators for order-convex mappings. (In preparation.)
Burmeister, W., Schmidt, J. W.: On theR-order of coupled sequences II. Computing29, 73–81 (1982).
Burmeister, W., Schmidt, J. W.: On theR-order of coupled sequences III. Computing30, 157–169 (1983).
Burmeister, W., Schmidt, J. W.: Über Kegel in endlichdimensionalen Räumen. Beiträge Numer. Math.12, 29–32 (1984).
Fan, K.: Topological proof for certain theorems on matrices with nonnegative elements. Monatsh. Math.62, 219–237 (1958).
Krasnoselskij, M. A.: Positive solutions of operator equations. Groningen: Noordhoff 1964.
Krein, M. G., Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space. Uspehi mat. Nauk SSSR3, 3–95 (1948) and Amer. Math. Soc. Transl.26 (1950).
Marek, I.: Frobenius theory of positive operators: comparison theorems and applications. SIAM J. Appl. Math.19, 607–628 (1970).
Massabo, I., Stuart, C. A.: Positive eigenvectors ofk-set contractions. Nonlinear Anal.3, 35–44 (1979).
Mewborn, A. C.: Generalizations of some theorems on positive matrices to completely continuous linear transformations on a normed linear space. Duke Math. J.27, 273–281 (1960).
Milovanović, G. V., Petković, M. S.: On the convergence order of a modified method for simultaneous finding polynomial zeros. Computing30, 171–178 (1983).
Nussbaum, R. D.: Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem. In: Fixed point theory (Fadell, E., Fournier, G., eds.), (Lecture Notes in Mathematics, Vol. 886), pp. 309–330. Berlin-Heidelberg-New York: Springer 1981.
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. Comp.27, 139–145 (1973).
Schmidt, J. W.: On theR-order of coupled sequences I. Computing26, 333–342 (1981).
Schmidt, J. W., Leder, D.: Ableitungsfreie Verfahren ohne Auflösung linearer Gleichungen. Computing5, 71–81 (1970).
Schneider, H., Turner, R. E. L.: Positive eigenvectors of order-preserving maps. J. Math. Anal. Appl.37, 506–515 (1972).
Varga, R. S.: Matrix iterative analysis. Englewood Cliffs, N.J.: Prentice-Hall 1962.
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Burmeister, W., Schmidt, J.W. Determination of the cone radius for positive concave operators. Computing 33, 37–49 (1984). https://doi.org/10.1007/BF02243074
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DOI: https://doi.org/10.1007/BF02243074