Low Rank ADI Solution of Sylvester Equation via Exact Shifts

p(λ) def = det(λI − A) and q(B) ≡ 0 for q(λ) def = det(λI − B). Choose parameterssuch thateither α i = λ pi for i = 1, . . . , m, or β j = µ qj for j = 1, . . . , n,where {p i } and {q j } denote some permutations of indices 1, . . . , m and 1, . . . , n,respectively. Then the solution X of the Sylvester equation (2.1) can be writtenas⎛⎞∑n 0X ≡ X n0 = S ⎝ (µ qj − λ pj )Φ (j) Ψ (j) ⎠ T −1 , n 0 = min{m, n}, (3.5)j=1whereandΦ (j) = diagσ(i, j − 1) =( )σ(1, j − 1) σ(m, j − 1), . . . , S −1 G, (3.6)λ 1 − µ qj λ m − µ qjj−1 ∏s=1Ψ (j) = F ∗ T diagτ(i, j − 1) =j−1 ∏s=1λ i − λ psλ i − µ qs, σ(i, 0) = 1, i = 1, . . . , m, (3.7)( )τ(1, j − 1) τ(n, j − 1), . . . , , (3.8)µ 1 − λ pj µ n − λ pjµ i − µ qsµ i − λ ps, τ(i, 0) = 1, i = 1, . . . , n.Before we state our first result, we like to emphasize that equation (3.5) is aproper generalization of the similar equation for the solution of the Lyapunovequation from [48]. It is obtained as a by-product of the ADI Method forSylvester Equations from [11] or [47]. A similar equation to (3.5) can alsobe obtained using the eigenvalue decompositions of A and B and the LUdecomposition of the corresponding Cauchy matrix presented in [13].Now, we can state our first result.Theorem 3.1 Assume A and B have eigen-decompositions (3.1) and (3.2).Let X be the solution of the Sylvester equation (2.1). Then the following inequality‖X‖ ≤ ‖S‖‖T −1 ∑n 0m∑ |σ(i, j − 1)| · ‖ĝ i ‖n∑ |τ(l, j − 1)| ‖‖ |µ qj − λ pj |ˆf l ‖,j=1i=1|λ i − µ qj |l=1|µ l − λ pj |(3.9)holds, where σ(·, ·) and τ(·, ·) are defined as in (3.6) and (3.8), respectively,and ĝ i , ˆf l denote the ith row of Ĝ = S−1 G and the lth column of ˆF ∗ = F ∗ T ,respectively.7