A condensed form for a symplectic pencil and solution of the discrete algebraic Riccati equation

Considers the problem of computing a basis for the stable deflating subspace of a symplectic pencil. An algorithm for computing a “triangular-Hessenberg” condensed form of the pencil is first described. This algorithm uses a combination of orthogonal and non-orthogonal structure preserving transformations. The condensed form is then used to develop an algorithm incorporating a block implementation of multiple shifts to obtain an upper block triangular form of the symplectic pencil. A basis for the stable deflating subspace can then be obtained directly from this block triangular pencil