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

Abstract

In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions Xj to the solution X of large-scale continuous-time algebraic Riccati equations (CAREs) 0=R(X):=AHX+XA+CHC−XBBHX generate the same sequence Xj when used with the same parameters. The Hermitian low-rank approximations Xj are of the form Xj=ZjYjZjH, where Zj is a matrix with only few columns and Yj is a small square Hermitian matrix. Each Xj generates a low-rank Riccati residual R(Xj) 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 Xj of CAREs is proposed. Each member of this family of algorithms proposed here generates the same sequence of Xj 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 AH and CH. 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.