Numerical Solution of a Quadratic Matrix Equation

This paper is concerned with the efficient numerical solution of the matrix equation $AX^2 + BX + C = 0$, where A, B, C and X are all square matrices. Such a matrix X is called a solvent. This equation is very closely related to the problem of finding scalars $\lambda $ and nonzero vectors x such that $(\lambda ^2 A + \lambda B + C)x = 0$. The latter equation represents a quadratic eigenvalue problem, with each $\lambda $ and x called an eigenvalue and eigenvector, respectively. Such equations have many important physical applications.By presenting an algorithm to calculate solvents, we shall show how the eigenvalue problem can be solved as a byproduct. Some comparisons are made between our algorithm and other methods currently available for solving both the solvent and eigenvalue problems. We also study the effects of rounding errors on the presented algorithm, and give some numerical examples.