Verified computation for the geometric mean of two matrices

摘要

An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for the special CARE. By the aid of this special change of variables, the proposed algorithm gives smaller radii, and is more successful than previous approaches. Solutions to the equation are not necessarily Hermitian. We thus establish a theory for verifying that the contained solution is Hermitian. Finally, the positive definiteness of the solution is verified. Numerical results show effectiveness, efficiency, and robustness of the algorithm.