Low Rank ADI Solution of Sylvester Equation via Exact Shifts

ThereforeX − X k =n 0 ∑j=k+1⎛∑n 0⎝= S(µ qj − λ pj )Z (j) Y (j)∗j=k+1(µ qj − λ pj )Φ (j) Ψ (j) ⎞⎠ T −1 . (3.11)Inequality (3.10) simply follows after taking the norm from both sides in theabove equality. ✷Remark 3.2 The bound (3.10) can be used as the upper bound for the decayof singular values of the solution X. The similar bounds can be found in [35],[3] and [48]. Especially, in the case of Lyapunov equation it holds‖X − X k ‖ ≤ trace(X) − trace(X k ),thus one can see that bound (3.10) is a generalization of [48, Theorem 3.1].Example 3.1 The following example will illustrate the quality of the bound(3.9). Let A = SΛS −1 with Λ = diag(30, 40, 60, 70, 90) and Didyou⎛⎞10.2028 0.0153 0.4186 0.8381 0.50280.1987 10.7468 0.8462 0.0196 0.7095S =0.6038 0.4451 10.5252 0.6813 0.4289.⎜ 0.2722 0.9318 0.2026 10.3795 0.3046⎟⎝⎠0.1988 0.4660 0.6721 0.8318 10.1897Let B = T ΩT −1 with Ω = diag(31, 41, 61, 71, 91) and⎛⎞20.1934 0.6979 0.4966 0.6602 0.72710.6822 20.3784 0.8998 0.3420 0.3093T =0.3028 0.8600 20.8216 0.2897 0.8385.⎜ 0.5417 0.8537 0.6449 20.3412 0.5681⎟⎝⎠0.1509 0.5936 0.8180 0.5341 20.3704generateAand BusingS, · · ·exactlyasshownhere?If not,youshouldconsiderdoingso.Thenreadercouldrepeatyourexamples.Noneed toprint Aand B.Thisappliesto allexamples.9