Ефективність елементарних неунітарних перетворень в поєднанні з віртуальними матрицями в організації обчислень матричної алгебри

Modern applied problems of matrix algebra require algorithms for efficient and reliable operation with a large number of equations. The purpose of such algorithms is to transform these equations with a wide range of numerical values of their coefficients. The transformations are aimed at simplifying the form of matrices in order to make them one of the canonical ones. Many numerical methods, schemes, algorithms have been developed to achieve these goals. The quality of the transformation results is certainly influenced not only by the method itself, but also by the order in which individual operations are performed. In this paper, an acceptable (in our opinion) version of working with known methods and a scheme for their use is proposed when the issues of accuracy and stability, saving RAM and a rational number of operations are consistent with each other. An elementary non-unitary matrix is accepted as the basic one in these transformations.To limit the uncontrolled growth of the coefficients, a certain order of operations is assigned using Virtual matrices, which are based on the management of row / column permutations. Chains of transformations using elementary matrices form cyclical processes and can be universal. The paper gives examples of using the proposed schemes for two important typical matrix transformations.