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A review of infinite matrices and their applications. (English) Zbl 1168.15023

Authors’ summary: Infinite matrices, the forerunner and a main constituent of many branches of classical mathematics (infinite quadratic forms, integral equations, differential equations, etc.) and of the modern operator theory, are revisited to demonstrate their deep influence on the development of many branches of mathematics, classical and modern, replete with applications. This review does not claim to be exhaustive, but attempts to present research by the authors in a variety of applications. These include the theory of infinite and related finite matrices, such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces.
Matrices considered here have special structures like diagonal dominance, tridiagonal, sign distributions, etc. and are frequently nonsingular. Moreover, diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations. The main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems, using techniques like conformal mapping, iterations, truncations, etc. to derive estimates based solutions.
Particular attention is paid to computable precise error estimates, and explicit lower and upper bounds. Topics include Bessel’s, Mathieu equations, viscous fluid flow, simply and doubly connected regions, digital dynamics, eigenvalues of the Laplacian, etc. Also presented are results in generalized inverses and semi-infinite linear programming.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
15A09 Theory of matrix inversion and generalized inverses
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
65L10 Numerical solution of boundary value problems involving ordinary differential equations
35P05 General topics in linear spectral theory for PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
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