Accurate approximation of eigenvalues and zeros of selected eigenfunctions of regular Sturm-Liouville problems
Abstract: A method for simultaneously approximating to high accuracy the corresponding eigenvalue and zeros of the $(n + 1)$st eigenfunction of a regular Sturm-Liouville eigenvalue problem is presented. It is based upon equilibrating the minimum eigenvalues of several problems on subintervals that form a partition of the orginal interval. The method is easily derived from classical mini-max variational principles. The equilibration is accomplished iteratively using an approximate Newton Method. Numerical results are given.
- Garrett Birkhoff and George Fix, Accurate eigenvalue computations for elliptic problems, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 111–151. MR 0260199
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- Ali Hasan Nayfeh, Perturbation methods, John Wiley & Sons, New York-London-Sydney, 1973. Pure and Applied Mathematics. MR 0404788
- J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
- Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
- Hans F. Weinberger, Variational methods for eigenvalue approximation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Based on a series of lectures presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, Tenn., June 26–30, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15. MR 0400004
G. Birkhoff & G. Fix, "Accurate eigenvalue computations for elliptic problems," in Numerical Solution of Field Problems in Continuum Physics (G. Birkhoff and R. S. Varga, eds.), Amer. Math. Soc., Providence, 1970, pp. 111-151.
E. A. Coddington & N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
A. H. Nayfeh, Perturbation Methods, Wiley-Interscience, New York, 1973.
J. M. Ortega & W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962.
H. F. Weinberger, Variational Methods for Eigenvalue Approximation, SIAM, Philadelphia, Pa., 1974.
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