Low Rank ADI Solution of Sylvester Equation via Exact Shifts

Similarly from (4.19), (4.20), (4.8) and (4.9) we haveϑ (j)max =( ϑ(1, j − 1)∥ diag ,µ 1 − λ pj= maxkϑ(2, j − 1), . . . , ϑ(k )∥B, j − 1) ∥∥∥∥µ 2 − λ pj µ kB − λ pj‖ϑ(k, j − 1)‖. (4.24)|µ k − λ pj |For the solutions of the Sylvester equations (3.15) and (3.16), one can obtainthe following bounds⎛‖δX 2 ‖ ≤ κ(S)‖T −1 ∑n 0‖‖δBT ‖‖X‖ ⎝j=1⎛‖δX 3 ‖ ≤ ‖S‖‖T −1 ‖‖S −1 (GδF ∗ + δGF ∗ ∑n 0)T ‖ ⎝|µ qj − λ pj | η (j)max ϑ (j)j=1max⎞⎠ , (4.25)|µ qj − λ pj | η (j)max ϑ (j)⎞⎠max .(4.26)Theorem 4.3 Assume A and B having Jordan canonical decompositions (4.1)and (4.2). Let X be the solution of the Sylvester equation (2.1) and let X +δXbe the solution of the perturbed Sylvester equation (3.12). Ifis sufficiently small, thenɛ = max{‖δA‖, ‖δB‖, ‖δG‖, ‖δF ‖}‖δX‖‖X‖ ≤ ( κ(T )‖S‖‖S −1 δA‖ + κ(S)‖T −1 ‖‖δBT ‖+‖S‖‖T −1 ‖ ‖S−1 (GδF ∗ + δGF ∗ ))T ‖γ + O(ɛ 2 ), (4.27)‖X‖where γ = ∑ n 0j=1 |µ qj −λ pj |ϑ (j)max η max, (j) and η max (j) and ϑ (j)max are defined as in (4.23)and (4.24).Proof. From ‖δX‖ ≤ ‖δX 1 ‖ + ‖δX 2 ‖ + ‖δX 3 ‖ + O(ɛ 2 ) using previous bounds(4.22), (4.25) and (4.26) for ‖δX 1 ‖, ‖δX 2 ‖ and ‖δX 3 ‖, respectively we obtainassertion of the theorem. ✷Remark 4.1 Again, using ‖AB‖ F ≤ ‖A‖‖B‖ F and ‖AB‖ F ≤ ‖A‖ F ‖B‖ ([45,Theorem II. 3.9.], similarly like in the previous calculations, one can obtainfollowing bound for Frobenius norm23