References
Arms, R. J., L. D. Gates andB. Zondek: A method of block iteration. S.I.A.M.4, 220–229 (1956).
Courant, R., K. O. Friedrichs andH. Lewy: Über die partiellen Differenzgleichungen der mathematischen Physik. Math. Annalen100, 32–74 (1928).
Courant, R., andD. Hilbert: Methods of Mathematical Physics, vols. 1 and 2.
— Variational methods for the solutions of problems of equilibrium and vibrations. Bull. Amer. Math. Soc.49, 1–23 (1943).
Forsythe, G. E.: Asymptotic lower bounds for the fundamental frequency of certain polygonal membranes. Pacific J. Math.4, 467–480 (1954).
—: Asymptotic lower bounds for the fundamental frequency of convex membranes. Pacific J. Math.5, 691–702 (1955).
Frankel, S.: Convergence rates of iterative treatments of partial differential equations. M.T.A.C.4, 65–76 (1950).
Friedman, B.: The iterative solution of elliptic partial difference equations. A.E.C. Research and development Report N.Y.U.-7698, 1957.
Friedrichs, K. O.: Die Randwert- und Eigenwertprobleme aus der Theorie der elastischen Platten. Math. Annalen96, 205–229 (1927).
—: Spektraltheorie halbbeschrankter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren. Math. Annalen109, 465–487, 685–713 (1933-34).
—: On differential operators in Hilbert spaces. Amer. J. Math.61, 523–544 (1939).
Heller, J.: Simultaneous, successive and alternating direction iteration schemes. S.I.A.M.8, 150–173 (1960).
Hersch, J.: Equations differentielles et fonctions des cellules. C. R. Acad. Sci. Paris240, 1602–1604 (1955).
Keller, H.: On some iterative methods for solving elliptic difference equations. Quart. Appl. Math.16, 209–226 (1958).
Parter, S. V.: On “two-line” iterative methods for the Laplace and biharmonic difference equations. Numer. Math.1, 240–252 (1959).
—: Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations. Trans. Amer. Math. Soc.99, 153–192 (1961).
Polya, G., andG. Szegö: Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, No. 27. Princeton: University Press 1951.
Schecter, S.: Quasi-tridiagonal matrices and type-insensitive difference equations. Quart. Appl. Math.3, 285–295 (1960).
Varga, R. S.: Factorization and normalized iterative methods. In Boundary Problems in Differential Equations,R. E. Langer (Editor), 121–142. The University of Wisconsin Press, Madison, Wisconsin (1960).
Voigt, W.: Bemerkungen zu dem Problem der transversalen Schwingungen rechteckiger Platten. Nachrichten, Königl. Gesellschaft Göttingen 1893, S. 225–230.
Wienberger, H. F.: Upper and lower bounds for eigenvalues by finite difference methods. Comm. Pure Appl. Math.9, 613–623 (1956).
—: Lower bounds for higher eigenvalues by finite difference methods. Pacific J. Math.8, 339–368 (1958).
Young, D.: Iterative methods for solving partial difference equations of elliptic type. Trans. Amer. Math. Soc.76, 92–111 (1954).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Parter, S.V. “Multi-line” iterative methods for elliptic difference equations and fundamental frequencies. Numer. Math. 3, 305–319 (1961). https://doi.org/10.1007/BF01386031
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01386031