Pivot tightening for some direct methods for solving 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 consider direct methods for the enclosure of the solution set of such a system. The algorithms are obtained from the ordinary elimination proce-dures by replacing the real numbers by the related intervals and the real operations by the respective interval operations. We report on methods by which the breakdown of the interval variants of the Gaussian elimination and the Neville elimination caused by division of an interval containing zero can be avoided for some classes of matrices with identically signed inverses. These include the non-singular totally nonnegative matrices. The approach consists of a tightening of the interval pivot by determining the exact range of the pivot over the matrix interval. By means of the interval Cholesky method an enclo-sure of the solution set for symmetric matrices can be found. 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 interval pivots and other entries of the Cholesky factor is possible.