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Improving iterative improvement

This paper describes a scheme for iteratively improving the decomposition of a given matrix A into the product of two matrices B and C. Although B and C may be completely general matrices, the scheme is most practically applicable in the LU and QR decompositions of a matrix. The proposed scheme is first shown to improve perturbed L and U factors of a given matrix A. Next, it is shown that this process will allow for the solution of a badly conditioned system of linear equations when other methods, such as direct triangular decomposition, fail miserably. Finally even when a method such as direct triangular decomposition works, but iterative improvement of the initial solution must be made, the proposed scheme may achieve comparable results and prove to be more economical under certain conditions.