Modified Bernoulli Iteration Methods for Quadratic Matrix Equation (Journal of Computational Mathematics, 2007)
We construct a modified Bernoulli iteration method for solving the quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and $C$ are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the Sherman-Morrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.
Keywords: Quadratic matrix equation; Quadratic eigenvalue problem; Solvent; Bernoulli's iteration; Newton's method; Local convergence.