Low Rank ADI Solution of Sylvester Equation via Exact Shifts

η(i, j) = σ(i, j) for i = 1, . . . l A , (4.4)()1η(i, j) = I 2 − N 2 [σ(i, j)I 2 − µ(i, j) N 2 ] for i = l A + 1, . . . k A ,λ i − µ qj+1(4.5)j−1 ∏ λ i − λ psσ(i, j − 1) =s=1λ i − µ qsand σ(i, 0) = 1, for all i, (4.6)j∑ j∏µ(i, j) =l=1t=1t≠lλ i − λ plλ i − µ qlλ pt − µ qt(λ i − µ qt ) 2 for i = l A + 1, . . . k A , (4.7)ϑ(i, j) = τ(i, j) for i = 1, . . . l B , (4.8)()1ϑ(i, j) = I 2 − N 2 [τ(i, j)I 2 − ν(i, j) N 2 ] for i = l B + 1, . . . k B ,µ i − λ pj+1(4.9)j−1 ∏ µ i − µ qsτ(i, j − 1) =s=1µ i − λ psand τ(i, 0) = 1, for all i, (4.10)j∑ j∏ν(i, j) =l=1t=1t≠lµ i − µ qlµ i − λ plµ qt − λ pt(µ i − λ pt ) 2 for i = l B + 1, . . . k B , (4.11)and ĝ i (for i = 1, . . . , l A ) denotes the ith 1 × r submatrix and 2 × r (fori = l A + 1, . . . , k A ) submatrix of the matrix Ĝ = S−1 G, respectively. Similarly,ˆf j denotes the jth, r × 1 (for j = 1, . . . , l B ) submatrix and r × 2 (for j =l B + 1, . . . , k B ) submatrix of the matrix ˆF ∗ = F ∗ T , respectively.Proof. Similarly to (3.3), one getsZ (j) = S(J A − λ pj−1 I)(J A − µ qj I) −1 S −1 Z (j−1)= S(J A − λ pj−1 I)(J A − µ qj I) −1 (J A − λ pj−2 I)(J A − µ qj−1 I) −1 . . .. . . (J A − λ p1 I)(J A − µ q2 I) −1 (J A − µ q1 I) −1 S −1 G= S(J A − µ qj I) −1 Γ j−1 Γ j−2 . . . Γ 1 S −1 G, (4.12)whereΓ j = (J A − λ pj I)(J A − µ qj I) −1 .Similarly to (3.4), one getsY (j)∗ = Y (j−1)∗ T (J B − λ pj I) −1 (J B − µ qj−1 I)T −1= F ∗ T (J B − λ p1 I) −1 (J B − λ p2 I) −1 (J B − µ q1 I) · · ·· · · (J B − λ pj I) −1 (J B − µ qj−1 I)T −1= F ∗ T Ξ 1 Ξ 2 . . . Ξ j−1 (J B − λ pj I) −1 T −1 ,19