On a computer implementation of the block Gauss–Seidel method for normal systems of equations

• Main Authors: Alexander I. Zhdanov, Ekaterina Yu. Bogdanova
• Format: Article
• Language: English
• Published: Samara State Technical University 2016-12-01
• Series: Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki
• Subjects: normal equations; block Gauss–Seidel method; Cholesky method; linear algebraic systems of large dimension
• Online Access: http://mi.mathnet.ru/eng/vsgtu1496
• ISSN: 1991-8615; 2310-7081

This article focuses on the modification of the block option Gauss-Seidel method for normal systems of equations, which is a sufficiently effective method of solving generally overdetermined, systems of linear algebraic equations of high dimensionality. The main disadvantage of methods based on normal equations systems is the fact that the condition number of the normal system is equal to the square of the condition number of the original problem. This fact has a negative impact on the rate of convergence of iterative methods based on normal equations systems. To increase the speed of convergence of iterative methods based on normal equations systems, for solving ill-conditioned problems currently different preconditioners options are used that reduce the condition number of the original system of equations. However, universal preconditioner for all applications does not exist. One of the effective approaches that improve the speed of convergence of the iterative Gauss–Seidel method for normal systems of equations, is to use its version of the block. The disadvantage of the block Gauss–Seidel method for production systems is the fact that it is necessary to calculate the pseudoinverse matrix for each iteration. We know that finding the pseudoinverse is a difficult computational procedure. In this paper, we propose a procedure to replace the matrix pseudo-solutions to the problem of normal systems of equations by Cholesky. Normal equations arising at each iteration of Gauss–Seidel method, have a relatively low dimension compared to the original system. The results of numerical experimentation demonstrating the effectiveness of the proposed approach are given.