On the stability of uniformly asymptotically diagonal systems

Authors:
R. S. Anderssen and B. J. Omodei

Journal:
Math. Comp. **28** (1974), 719-730

MSC:
Primary 49G20; Secondary 47A50

DOI:
https://doi.org/10.1090/S0025-5718-1974-0425744-X

MathSciNet review:
0425744

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Abstract | References | Similar Articles | Additional Information

Abstract: In a number of papers ([1], [2]), Delves and Mead have derived some useful (though limited) rate of convergence results which can be applied to variational approximations for the solution of linear positive definite operator equations when the coordinate system is uniformly asymptotically diagonal. Independently, Mikhlin [5] has examined the stability of such variational approximations in the case of positive definite operators and concluded that the use of strongly minimal coordinate systems is a necessary and sufficient condition for their stability. Since, in general, the Delves and Mead results will only be applicable to actual variational approximations when their uniformly asymptotically diagonal system is at least strongly minimal, we examine the properties of uniformly asymptotically diagonal systems in terms of the minimal classification of Mikhlin.

We show that

(a) a normalized uniformly asymptotically diagonal system is either nonstrongly minimal or almost orthonormal;

(b) the largest eigenvalue of a normalized uniformly asymptotically diagonal system is bounded above, independently of the size of the system;

(c) the special property of normalized uniformly asymptotically diagonal systems mentioned in (b) is often insufficient to prevent their yielding unstable results when these systems are not strongly minimal.

**[1]**L. M. Delves and K. O. Mead,*On the convergence rates of variational methods. I. Asymptotically diagonal systems*, Math. Comp.**25**(1971), 699–716. MR**0311131**, https://doi.org/10.1090/S0025-5718-1971-0311131-9**[2]**K. O. Mead and L. M. Delves,*On the prediction of the expansion coefficients in a variational calculation*, J. Inst. Math. Appl.**10**(1972), 166–175. MR**0347082****[3]**S. G. Mikhlin,*The problem of the minimum of a quadratic functional*, Translated by A. Feinstein. Holden-Day Series in Mathematical Physics, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. MR**0171196****[4]**S. G. Mikhlin,*Variational methods in mathematical physics*, Translated by T. Boddington; editorial introduction by L. I. G. Chambers. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR**0172493****[5]**S. G. Mikhlin,*The numerical performance of variational methods*, Translated from the Russian by R. S. Anderssen, Wolters-Noordhoff Publishing, Groningen, 1971. MR**0278506****[6]**L. N. Dovbyš,*A remark about minimal systems*, Trudy Mat. Inst. Steklov. 96 (1968), 188–189 (Russian); translated in Proc. Steklov Inst. Math., No. 96 (1968): Automat. Programming, Numer. Methods and Functional Anal., Amer. Math. Soc., Providence, R.I., 1969, pp. 235–237. MR**0273379****[7]**Alfred Brauer,*Limits for the characteristic roots of a matrix*, Duke Math. J.**13**(1946), 387–395. MR**0017728****[8]**S. Gerschgorin, "Über die Abgrenzung der Eigenwerte einer Matrix,"*Izv. Akad. Nauk SSSR, Ser. Fiz.-Mat.*, v. 6, 1931, pp. 749-754.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0425744-X

Keywords:
Ritz-solution,
uniformly asymptotically diagonal systems,
minimal classification,
stability of variational methods

Article copyright:
© Copyright 1974
American Mathematical Society