Generating equations approach for quadratic matrix equation (Q2760359)

The author gives an algorithm for the numerical solution of a quadratic matrix equation with the Hamiltonian matrix. The algorithm transforms the Hamiltonian matrix into a skew-Hamiltonian one. This is then transformed in several steps into a block diagonal matrix with the left upper block having again a block-diagonal structure with blocks of order 1 or 2. The eigenproblem for this matrix is solved. NEWLINENEWLINENEWLINEThe symplectic-orthogonal similarity transformations of a skew-Hamiltonian matrix into a block triangular form with Hessenberg blocks on the diagonal are studied. These transformations use a vector belonging to an invariant subspace of order \(n\) and give a generalization of a result of \textit{C. F. Van Loan} [Linear Algebra Appl. 61, 233-251 (1984; Zbl 0565.65018)] NEWLINENEWLINENEWLINEThe method is applied to a Pfaffian system of partial differential equations, to the similarity transformation of a complex matrix into a symmetric and three-diagonal one and to the eigenproblem for Hermitian matrices.