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young d iterative solution large linear systems 1971

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Textbook in PDF format The availability of very high-speed computers with large, fast memories has made it possible to obtain accurate numerical solutions of mathematical problems which, although algorithms for handling them were well known previously, could not be used in practice because the number of calculations required would have been prohibitive. A problem for which this is particularly true is that of solving a large system of linear algebraic equations where the matrix of the system is very "sparse," that is, most of its elements vanish. Such systems frequently arise in the numerical solution of elliptic partial differential equations by finite difference methods. These problems, in turn, arise in studies in such areas as neutron diffusion, fluid flow, elasticity, steady-state heat flow, and weather prediction. The solution of a large system with a sparse matrix is usually obtained by iterative methods instead of by direct methods such as the Gauss elimination method. An extensive theory has been developed which is interesting mathematically while at the same time is of practical use. The purpose of the book is to provide a systematic development of a substantial portion of the theory of iterative methods for solving large linear systems. The emphasis is on practical techniques, and the treatment has been made as elementary as possible without unnecessary generalizations. Every effort has been made to give a mathematically rigorous and self-contained treatment with the exception of some basic theorems of matrix theory and analysis which are stated without proof in Chapter 2 . Only in later chapters has it been necessary to depart from this objective to any appreciable extent. The material includes published and unpub- lished work of the author as well as results of others as indicated. Particular reference has been made to the works of Richard Varga and Eugene Wachspress. In order to make the book as widely useful as possible a minimum amount of mathematical background is assumed. In addition to a knowledge of the fundamentals of matrix theory, at a level indicated by Section 2.1, the reader should be familiar with the elementary aspects of real analysis as covered in a good course in "advanced calculus" or "elementary analysis." He should also be acquainted with some elementary complex variable theory. In addition, a general background in numerical analysis, especially matrix methods and finite difference methods for solving partial differential equations, as well as in computer programming would be highly desirable. Matrix Preliminaries Linear Stationary Iterative Methods Convergence of the Basic Iterative Methods Eigenvalues of the SOR Method for Consistently Ordered Matrices Determination of the Optimum Relaxation Factor Norma OR the SOR Method The Modified SOR Method: Fixed Parameters Nonstationary Linear Iterative Methods The Modified SOR Method: Variable Parameters Semi-Iterative Methods Extensions or the SOR Theory: Stieltjes Matrices Generalized Consistently Ordered Matrices Group Iterative Methods Symmetric SOR Method and Related Methods Second-Degree Methods Alternating Direction Implicit Methods Selection of Iterative Method

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young d iterative solution large linear systems 1971

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