Model reduction by truncated balanced realization computes a projection of the state space realization by truncating a balanced realization. In the square root form, this is achieved using the SVD of the product of the Cholesky factors of the observability and controllability gramians. These gramians satisfy well-known Lyapunov equations.
Here, the use of the CF-ADI (Cholesky Factor Alternating Direction Implicit) method is proposed to compute rank \(k\) approximations for the Cholesky factors directly in \(O(k)\) operations. Systems of equations need to be solved whose matrices depend on several shift parameters. An improvement is given that reduces these systems to systems with the same matrix but with different right-hand sides. In this way the Krylov vectors can be shared when solved iteratively. Next it is proposed to project onto the sum of the dominant gramian eigenspaces, which corresponds to matching moments of the transfer function at the points corresponding to the negatives of the CF-ADI shift parameters. In general however, there is no moment matched at \(s=0\) (the DC response). The remedy proposed is to add some extra vectors to the subspace on which the system is projected. This gives an approximate matching of the DC component while not changing the approximation in the rest of the spectrum.