A modification of Schur's method for solving a matrix algebraic Riccati equation (Q1803101)

The Schur method is covered in detail in this paper. The other most important papers which deal with the numerical solution of the algebraic Riccati equation \[ XGX - A^ TX - XA - F = 0;\;A,F = F^ T,\quad G = G^ T \in \mathbb{R}^{n\times n},\tag{1} \] where the accompanying matrix \[ M = \left|{A\atop F}{G\atop -A^ T}\right|,\quad M \in \mathbb{R}^{2n\times 2n}\tag{2} \] has a real simple spectrum, are reviewed here. The author also proposes a modification of the Schur method which uses the Hamiltonian structure of the accompanying matrix (2) -- the method of block reduction of the Hamiltonian matrix \(M\) to Schur form under the condition that \(M\) has no multiple eigenvalues. This modification enables to reduce the number of arithmetic operations (multiplications). Moreover it permits to save the initial matrix coefficients of the equation (1) in order to make an effective iterative refinement of the obtained results if it is necessary (the memory expenses are the same as for the Schur method). Test results are treated. The advantages and shortages of the new method are described. Open questions are also discussed here.