Fast verified computation for the solvent of the quadratic matrix equation

Abstract

Two fast algorithms for numerically computing an interval matrix containing the solvent of the quadratic matrix equation AX^2 + BX + C = 0 with square matrices A, B, C and X are proposed. These algorithms require only cubic complexity, verify the uniqueness of the contained solvent, and do not involve iterative process. Let \ap{X} be a numerical approximation to the solvent. The first and second algorithms are applicable when A and A\ap{X}+B are nonsingular and numerically computed eigenvector matrices of \ap{X}^T and \ap{X} + \inv{A}B, and \ap{X}^T and \inv{(A\ap{X}+B)}A are not ill-conditioned, respectively. The first algorithm moreover verifies the dominance and minimality of the contained solvent. Numerical results show efficiency of the algorithms.