On the numerical solution of system of linear algebraic equations with ill-conditioned matrices

Abstract

The system of linear algebraic equations (SLAE) is considered. If the matrix of the system is non-degenerate, then there is a unique solution to the system. In a degenerate case, the system may not have a solution or have infinitely many solutions. In this case, the concept of a normal solution is introduced. The case of a non-degenerate square matrix can theoretically be considered good in terms of existence and uniqueness of the solution, but in the theory of computational methods, nondegenerate matrices are divided into two categories: “ill-conditioned” and “well-conditioned”. Badly called matrices for which the solution of the system of equations is practically unstable. One of the important characteristics of practical solution stability A system of linear equations is a condition number. Usually, regularization methods are used to obtain a reliable solution. A common strategy is to use Tikhonov’s stabilizer or his modifications, or the representation of the desired solution in the form of orthogonal sums of two vectors, one of which is determined stably, and for searching the second requires some stabilization procedure. In this article the methods of numerical solution of SLAE are considered with a positive defined symmetric matrix or oscillating matrix type using regularization, leading to SLAEs with a reduced conditionality number.

Keywords:

a system of linear algebraic equations, ill-posed problems, ill-conditioned problems, condition number, regularization method