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.