Fast verified computation for the solvent of the quadratic matrix equation

Two fast algorithms for numerically computing an interval matrix containing the solvent of the quadratic matrix equation AX2+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 X̃ be a numerical approximation to the solvent. The first and second algorithms are applicable when A and AX̃ + B are nonsingular and numerically computed eigenvector matrices of X̃T and X̃+A−1B, and X̃T and (AX̃+B)−1A 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.