Low Rank ADI Solution of Sylvester Equation via Exact Shifts

(Ξ1 Ξ 2 . . . Ξ j−1 (J B − λ pj I) −1) j−1= 1 ∏ µ i − µ qs(i,i) µ i − λ pj s=1µ i − λ psτ(i, j − 1)= for i = 1, . . . , l B , (4.19)µ i − λ pjand for i = l B + 1, . . . , k B it is(I 2 −=(I 2 −⎛)j−11N 2⎜1 ∏µ i − λ pj⎝µ i − λ pjs=1µ i − µ qsµ i − λ psI 2 +1 ∑j−1µ i − λ pjl=1j−1∏t=1t≠l(µ i − λ pt ) 2 N ⎟2⎠µ i − µ qlµ i − λ plµ qt − λ pt) ( )1 τ(i, j − 1) ν(i, j − 1)N 2 I 2 + N 2 , (4.20)µ i − λ pj µ i − λ pj µ i − λ pjwhere τ(i, j − 1) and ν(i, j − 1) are defined as in (4.10)–(4.11).Now, from (2.5) it follows thatn 0∑X = (µ qj − λ pj )Z (j) Y (j)∗= Sj=1⎛∑n 0⎝j=1(µ qj − λ pj )(J A − µ qj I) −1 Γ j−1 Γ j−2 . . . Γ 1 Ĝ ˆF ∗ Ξ 1 Ξ 2 . . . Ξ j−1 (J B − λ pj I) −1 ⎞(4.21)Finally, (4.3) simply follows by taking the norms at the both sides of the aboveequation and taking into consideration the definitions (4.4)–(4.11). ✷⎞⎠ T −1 .4.1 Error BoundTheorem 4.2 Assume A and B having Jordan canonical decompositions (4.1)and (4.2). Let X k be the kth approximation obtained by (2.3) – (2.5) withthe set of ADI parameters corresponding to subsets of exact eigenvalues ofA and B, i.e. {α 1 , α 2 , . . . , α k } = {λ p1 , λ p2 , . . . , λ pk } and {β 1 , β 2 , . . . , β k } ={µ q1 , µ q2 , . . . , µ qk }. Then the following inequality holds‖X−X k ‖ ≤ ‖S‖‖T −1 ‖n 0 ∑j=k+1|µ qj −λ pj |k A ∑i=1||η(i, j − 1)|| · ‖ĝ i ‖|λ i − µ qj |k B ∑l=1||ϑ(l, j − 1)|| ‖ ˆf l ‖|µ l − λ pj |,where η(i, j) and ϑ(l, j) are defined as in (4.4)–(4.11), respectively, and ĝ i , ˆf lare defined as in Theorem 4.1, and {λ pj , j > k} and {λ qj , j > k} are theeigenvalue subsets of A and B complement to the ones already used as shiftsfor obtaining X k .21