1979
DOI: 10.1109/tac.1979.1102170
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A Hessenberg-Schur method for the problem AX + XB= C

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Cited by 749 publications
(358 citation statements)
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“…These methods are based on Krylov subspace methods and solve a small SE per iteration of order m p 2 , where m is the restart value. These internal SE are solved using the algorithm proposed by Golub, Nash and Van Loan in [9]. The BAS( m) and BGS( m) can only be used when n >> p. When n < p we can always transpose SE and then apply the same techniques.…”
Section: Numerical Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…These methods are based on Krylov subspace methods and solve a small SE per iteration of order m p 2 , where m is the restart value. These internal SE are solved using the algorithm proposed by Golub, Nash and Van Loan in [9]. The BAS( m) and BGS( m) can only be used when n >> p. When n < p we can always transpose SE and then apply the same techniques.…”
Section: Numerical Resultsmentioning
confidence: 99%
“…For details see again the book by Datta [6]. The best-known and very widely used numerical method for small and dense problems is the Hessenberg-Schur method by Golub, Nash, and Van Loan [9]. The method is based on reduction of the largest of two matrices to Hessenberg form and the other to real Schur form.…”
Section: Block Linear Methods For Large Scale Sylvester Equationsmentioning
confidence: 99%
“…, 0 s T ∈ R s×ms . The low-dimensional Lyapunov equations (3.3) and (3.4) could be solved by direct methods such those described in [5,18,24]. In the sequel, we assume that the eigenvalues λ i (T m ) of the block tridiagonal matrix T m constructed by the nonsymmetric block Lanczos process satisfy λ i (T m ) +λ j (T m ) = 0, for i, j = 1, .…”
Section: Solving the Coupled Lyapunov Matrix Equations By The Block Lmentioning
confidence: 99%
“…In this case, the equations (1.3) have unique solutions [26]. Direct methods for solving the Lyapunov matrix equations (1.3) such as those proposed in [5,18,24] are attractive if the matrices are of moderate size. These methods are based on the Schur or the Hessenberg decomposition.…”
Section: Introductionmentioning
confidence: 99%
“…Caution should be taken when completing this ML estimate from (27) since it is often slow and ill-conditioned. Other standard solution methods for (26) are the Bartels-Stewart method, [7], and the Hessenberg-Schur method, [8].…”
mentioning
confidence: 99%