Methods for obtaining reliable solutions to systems of linear algebraic equations
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METHODS FOR OBTAINING RELIABLE SOLUTIONS TO SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS I. V. Sergienko,a† A. N. Khimich,a‡ and M. F. Yakovlev a††
UDC 519.6 + 518:512.25
Abstract. The general case of incompatible systems of linear algebraic equations with matrices of arbitrary rank is considered. The estimates for total errors are obtained for all the considered cases under conditions of approximate data. In solving systems by iterative methods, the conditions of completion of iterative processes that provide solutions with a prescribed accuracy are considered in detail. A special attention is given to the solution of incompatible systems with symmetric positive semidefinite matrices by the method of three-stage regularization in which an algorithm for choosing the regularization parameter is proposed that allows finding solutions with the required accuracy. Keywords: approximate initial data, incompatible systems, total error of the solution, iterative methods. INTRODUCTION Various physical processes and phenomena comprehensively studied with the use of their mathematical models to reveal new properties, principles, and qualities, as well as the stress–strain analysis of units, products, and physical fields and synthesis of new materials used to design and construct installations and structures are often reduced in some way to the solution of systems of linear algebraic equations (SLAEs) with various types of matrices. Moreover, numerical modeling of nonlinear processes is usually implemented by a converging series of linear models, whose design also involves the solution of SLAEs. The values of elements of the matrices and components of the vectors of the right-hand sides of the SLAEs are approximate since input data are obtained by measurements or preliminary computations. A characteristic property of the considered problems in such a formulation is that their properties are unknown a priori. Depending on prescribed limits of the error of elements of the matrix and vector of the right-hand side, systems can be compatible or incompatible, problems can be well- or ill-posed, and at the same time well- or ill-conditioned. The task is, in a machine environment, to analyze the properties of the problem being solved, to generate an algorithm for the approximate solution, and to evaluate its reliability. Dependence of the solution error on problem properties was considered in [1–6], etc. Based on certain properties of the problems, various methods were designed to solve them. For example, the study [2] deals with the Tikhonov method of regularization of ill-posed problems, [7] considers an analytic technique to compute pseudoinverses, and [8, 9] proposes direct and iterative methods to find pseudosolutions. We will analyze the accuracy of solutions obtained by direct and iterative methods under approximate initial data, depending on the mathematical properties of the problem being solved, for both well- and ill-posed problems. SLAE WITH NONDEGENERATE MATRICES Let us consider a system of equations with accurate initial data
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