Fast inversion algorithms for a class of structured operator matrices.
type: Item
instance of: scholarly article
title: Fast inversion algorithms for a class of structured operator matrices.
zbMATH Open document ID: 1033.65032
DOI: 10.1016/S0024-3795(03)00475-0
author: I. Gohberg
author: Y. Eidelman
published in: Linear Algebra and its Applications
publication date: 2003-09-09T00:00:00Z
review text: Diagonal plus semiseparable operator matrices are considered. The entries of these matrices are linear bounded operators on Hilbert spaces. Linear complexity inversion algorithms for such matrices are studied. For applications it is necessary to seek fast algorithms. Three different fast algorithms for inversion of operator matrices are presented: The diagonal inversion method, the reduction to the shooting method and the method of consecutive inversion. Algorithms for the numerical solution of differential and integral equations are derived as application. For this approach quadrature methods of high accuracy for solution of integral equations on small segments with a small number of nodes are combined with exact formulas for the inversion of operator matrices. The authors compare different algorithms and analyze their complexity and the choice of parameters. Every algorithm has its own area of stability. In each application such an algorithm is to choose which is the most appropriate from the point of view of stability. The results of a first computer experiment for a boundary value problem of differential equations is added. The boundary value problem is transformed into an integral equation. A system of three differential equations of order two is tranformed to a system with six differential equations of order 1 with homogeneous boundary conditions. Finally, the presented algorithms are used to solve algebraic equations with block matrices.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 1978112
zbMATH Keywords: structural matrices
zbMATH Keywords: fast inversion algorithms
zbMATH Keywords: boundary value problem
zbMATH Keywords: integral equation
zbMATH Keywords: differential equation
zbMATH Keywords: numerical examples
zbMATH Keywords: stability
zbMATH Keywords: linear algebraic equations
zbMATH Keywords: operator matrices
zbMATH Keywords: linear bounded operators
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: diagonal inversion method
zbMATH Keywords: shooting method
zbMATH Keywords: method of consecutive inversion
zbMATH Keywords: quadrature methods
zbMATH Keywords: complexity
reviewed by: Werner H. Schmidt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1978112