Computing enclosures for the inverse square root and the sign function of a matrix

We study computational methods for obtaining rigorous a posteriori error bounds for the inverse square root and the sign function of an n×nn×n matrix A. Given a computed approximation for the inverse square root of A  , our methods work by using interval arithmetic to obtain a narrow interval matrix which, with mathematical certainty, is known to contain the exact inverse square root. Particular emphasis is put on the computational efficiency of the method which has complexity O(n3)O(n3) and which uses almost exclusively matrix–matrix operation, a key to the efficient use of available software for interval computations. The standard formulation of the method assumes that A can be diagonalized and that the eigenvector matrix of A is well-conditioned. A modification relying on a stable similarlity transformation to block diagonal form is also developed.