Method for preconditioning matrix equations based on zero divisors

The article discusses the method of preconditioning of algebraic matrix equations without transforming the matrix of the right-hand side. The method is based on the technique of matrix zero divisors. The presence of the right (left) zero divisor is associated with linear dependence of the columns (rows) of the matrix. In this case, the problem of determining such preconditioning matrices is posed and solved, which, along with a decrease in the condition number of the matrix of coefficients on the left side, leave the matrix of coefficients on the right-hand side unchanged. The transformations consist solely in the rotation of the system around its exact, although still unknown, solution. This makes it possible to further improve the accuracy of determining the solution by eliminating possible computational errors in the transformation of the right-hand side of the matrix equation. It is shown that the choice of preconditioners, which make it possible to reduce the degree of conditionality of the equation, can be made on the basis of the method of simple iteration or taking into account the lower estimate of the condition number of the matrix on the left-hand side by fixing its eigenvalues using well-known and well-developed methods. The advantages of the proposed method are demonstrated by numerical examples. The reasoning given in the article regarding the left-handed matrix equation (the matrix of coefficients is on the left of the unknown matrix) is also valid for the right-handed and two-sided equations.

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