On a family of low-rank algorithms for large-scale algebraic Riccati equations.

In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions X j to the solution X of large-scale continuous-time algebraic Riccati equations (CAREs) 0 = R (X) : = A H X + X A + C H C − X B B H X generate the same sequence X j when used with the same parameters. The Hermitian low-rank approximations X j are of the form X j = Z j Y j Z j H , where Z j is a matrix with only few columns and Y j is a small square Hermitian matrix. Each X j generates a low-rank Riccati residual R (X j) such that the norm of the residual can be evaluated easily allowing for an efficient termination criterion. Here a new family of methods to generate such low-rank approximate solutions X j of CAREs is proposed. Each member of this family of algorithms proposed here generates the same sequence of X j as the four previously known algorithms. The approach is based on a block rational Arnoldi decomposition and an associated block rational Krylov subspace spanned by A H and C H. Two specific versions of the general algorithm will be considered; one will turn out to be a rediscovery of the RADI algorithm, the other one allows for a slightly more efficient implementation compared to the RADI algorithm (in case the Sherman-Morrision-Woodbury formula and a direct solver is used to solve the linear systems that occur). Moreover, our approach allows for adding more than one shift at a time. [ABSTRACT FROM AUTHOR]