Increasing the stability of triangular decomposition of ill-conditioned matrices

An approach to increasing the stability of triangular decomposition of a dense positive definite matrix with a large condition number with the use of the Gauss and the Cholesky methods is considered. It is proposed to introduce additions to standard computational schemes, which consist in the use of an incomplete scalar product of two vectors, which is formed by cutting off the lower digits of the sum of the products of two numbers. Cutting off being performed in the process of factorization leads to an increase in the diagonal elements of triangular matrices to a random number and prevents the appearance of very small numbers during the decomposition according to Gauss and a negative radical expression in the Cholesky method. The number of additional operations required to obtain an accurate solution is estimated. The results of computational experiments are presented.