Pivot tightening for direct methods for solving symmetric positive definite systems of linear interval equations

Abstract

The paper considers systems of linear interval equations, i.e., linear systems where the coefficients of the matrix and the right hand side vary between given bounds. We focus on symmetric matrices and consider direct methods for the enclosure of the solution set of such a system. One of these methods is the interval Cholesky method, which is obtained from the ordinary Cholesky decomposition by replacing the real numbers by the related intervals and the real operations by the respective interval operations. We present a method by which the diagonal entries of the interval Cholesky factor can be tightened for positive definite interval matrices, such that a breakdown of the algorithm can be prevented. In the case of positive definite symmetric Toeplitz matrices, a further tightening of the diagonal entries and also of other entries of the Cholesky factor is possible. Finally, we numerically compare the interval Cholesky method with interval variants of two methods which exploit the Toeplitz structure with respect to the computing time and the quality of the enclosure of the solution set.