Fast inversion algorithms for a class of structured operator matrices.
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.
scientific article; zbMATH DE number 1978112