Optimal interval enclosing of certain sets of matrices, with application to monotone enclosing of square roots

Abstract

In this paper, two kinds of partial ordering for symmetric matrices are related to each other, namely, the natural partial ordering ≤ generated by the coneK of elementwise nonnegative matrices, and the definite partial ordering $$ \leqslant \cdot$$ generated by the coneK D of nonnegative definite matrices. The main result of this paper shows how a matrix interval in the sense of the definite partial ordering can be enclosed between optimal bounds with respect to the natural partial ordering. By means of this result, it is possible to compute a numerically practicable inclusion based on the natural partial ordering from a given inclusion of some matrix with the definite partial ordering. In this way, an always and moreover quadratically convergent method of elementwise enclosing the square root of a positive definite, symmetric matrix can be constructed.