Low Rank ADI Solution of Sylvester Equation via Exact Shifts

∑where γ = n 0j=1|µ j − λ j |τ (j)max σ (j)max and σ (j)max, τ (j)max are defined as in (3.20).Proof. Take the norm of δX ≈ δX 1 + δX 2 + δX 3 and use bounds (3.21),(3.22), (3.23) to get (3.24). ✷Remark 3.3 From (3.17), and similar equalities for δX 2 and δX 3 , we canobtain a sharper boundn‖δX‖ 0‖X‖ ≤ ∑j=1(×|µ j − λ j | ‖S∆ Φ (j)S −1 ‖ ‖T ∆ Ψ (j)T −1 ‖‖δA‖ + ‖δB‖ + ‖S−1 (GδF ∗ + δGF ∗ )T ‖‖X‖)+ O(ɛ 2 ),where diagonal matrices ∆ Φ (j) and ∆ Ψ (j) are defined as in (3.18) and (3.19).Remark 3.4 Using ‖AB‖ F ≤ ‖A‖‖B‖ F and ‖AB‖ F ≤ ‖A‖ F ‖B‖ (see, e.g.,[45, Theorem II. 3.9.]) similarly like in the previous calculations one can obtainfollowing bound for Frobenius norm‖δX‖ F‖X‖ F≤∑where γ = n 0j=1(κ(T )‖S‖‖S −1 δA‖ + κ(S)‖T −1 ‖‖δB T ‖+‖S‖‖T −1 ‖ ‖S−1 (GδF ∗ + δGF ∗ ))T ‖ Fγ + O(ɛ 2 ), (3.25)‖X‖ F|µ j − λ j |τ (j)max σ (j)max and σ (j)max, τ (j)max are defined as in (3.20).As an illustration of the quality of the perturbation bound (3.25) we willpresent a comparison between this bound and two bounds from [23]. The firstone is‖δX‖ F‖X‖ F≤ √ 3Ψɛ + O(ɛ 2 ), (3.26)whereΨ = ‖P −1 [α(X T ⊗I m ) −β(I n ⊗X) −δI mn ]‖/‖X‖ F ,{ ‖δA‖Fand ɛ = max α , ‖δB‖ Fβ, ‖δC‖ Fδas in [23]. The second bound is a weaker version of (3.26):P = I n ⊗A−B T ⊗I m}, while α, β and δ are scaling factors‖δX‖ F‖X‖ F≤ √ 3Φɛ + O(ɛ 2 ), (3.27)13