Skeleton decomposition of rectangular matrices and its application for structural regularization of ill-conditioned systems of linear algebraic equations

The new method for regularization of ill-conditioned systems of linear algebraic equations (SLAE) has been considered. The main idea of the method is the correction of structural characteristics of SLAE under consideration. In the paper we mean under "structural characteristics" of a matrix the dimensions of rows and columns of a coefficient matrix. The structural regularization of an ill-conditioned SLAE means here the searching of the values of abovementioned parameters, which provide stability of the solution of a SLAE with respect to variations of input data. The proposed method has been based on the obtaining of skeleton decompositions of matrix by means of Gram-Schmidt procedure of orthogonalization of finite-dimensional vectors.

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