Regularization of unstable finite-dimensional linear problems based on augmented systems

A new method for solving unstable problems that can be reduced to arbitrary systems of linear algebraic equations (which may not be of full rank or may be inconsistent) is examined. This method is based on the reduction of an arbitrary (in general, inconsistent) linear system to an equivalent consistent augmented system with a symmetric matrix. The proposed approach makes it possible to entirely eliminate the problem of choosing a regularization parameter for arbitrary (in general, inconsistent) linear systems, because this parameter must be coordinated only with a measure of error in the matrix of the original system. Issues related to efficient numerical implementation of the proposed regularizing algorithms are discussed.