Low Rank ADI Solution of Sylvester Equation via Exact Shifts

J A,i = λ i for i = 1, . . . , l A ,⎛ ⎞⎜J A,i =λ i 1 ⎟⎝ ⎠ ≡ λ i I 2 + N 2 , for i = l A + 1, . . . , k A ,0 λ i2(k A − l A ) + l A = m, I 2 is the 2 × 2 identity matrix, and N 2 =Similarly, let the Jordan canonical form of B be⎛ ⎞⎜ 0 1 ⎟⎝ ⎠ .0 0B = T J B T −1 , T ∈ C n×n , J B = J B,1 ⊕ . . . ⊕ J B,kB , (4.2)J B,i = µ i for i = 1, . . . , l B ,⎛ ⎞⎜J B,i =µ i 1 ⎟⎝ ⎠ ≡ µ i I 2 + N 2 , for i = l B + 1, . . . , k B ,0 µ iwhere 2(k B − l B ) + l B = n.Theorem 4.1 Assume A and B have Jordan canonical decompositions (4.1)and (4.2). Let X be the solution of the Sylvester equation (2.1), obtained by(2.3) – (2.5) (LR-ADI) with the set of ADI parameters{α 1 , α 2 , . . . , α m } = {λ p1 , λ p2 , . . . , λ pm }and{β 1 , β 2 , . . . , β n } = {µ q1 , µ q2 , . . . , µ qn },where each eigenvalue appears as many times as its algebraic multiplicity.Thenn 0‖X‖ ≤ ‖S‖‖T −1 ∑‖ |µ qj − λ pj |j=1where n 0 = min{m, n},k A ∑i=1||η(i, j − 1)|| · ‖ĝ i ‖|λ i − µ qj |k B ∑s=1||ϑ(s, j − 1)|| ‖ ˆf s ‖,|µ s − λ pj |(4.3)18