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Volumn 24, Issue 1, 2003, Pages 260-280

Low rank solution of Lyapunov equations

Author keywords

Alternating direction implicit iteration; Dominant invariant subspace; Iterative methods; Low rank approximation; Lyapunov equation

Indexed keywords


EID: 0037287244     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479801384937     Document Type: Article
Times cited : (297)

References (42)
  • 2
    • 84976855597 scopus 로고
    • Solution of the matrix equation AX + XB = C
    • R. H. BARTELS AND W. STEWART, Solution of the matrix equation AX + XB = C, Comm. ACM, 15 (1972), pp. 820-826.
    • (1972) Comm. ACM , vol.15 , pp. 820-826
    • Bartels, R.H.1    Stewart, W.2
  • 3
    • 0016580689 scopus 로고
    • A new solution method for the Lyapunov matrix equation
    • A. N. BEAVERS, JR. AND E. D. DENMAN, A new solution method for the Lyapunov matrix equation, SIAM J. Appl. Math., 29 (1975), pp. 416-421.
    • (1975) SIAM J. Appl. Math. , vol.29 , pp. 416-421
    • Beavers A.N., Jr.1    Denman, E.D.2
  • 4
    • 0033412866 scopus 로고    scopus 로고
    • Solving stable generalized Lyapunov equations with the matrix sign function
    • P. BENNER AND E. S. QUINTANA-ORTÍ, Solving stable generalized Lyapunov equations with the matrix sign function, Numer. Algorithms, 20 (1999), pp. 75-100.
    • (1999) Numer. Algorithms , vol.20 , pp. 75-100
    • Benner, P.1    Quintana-Ortí, E.S.2
  • 5
    • 0013343170 scopus 로고
    • Alternating direction implicit methods
    • Academic Press, New York
    • G. BIRKHOFF, R. S. VARGA, AND D. YOUNG, Alternating direction implicit methods, in Advances in Computers, Vol. 3, Academic Press, New York, 1962, pp. 189-273.
    • (1962) Advances in Computers , vol.3 , pp. 189-273
    • Birkhoff, G.1    Varga, R.S.2    Young, D.3
  • 7
    • 0031354134 scopus 로고    scopus 로고
    • A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks
    • San Jose, CA
    • I. ELFADEL AND D. LING, A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks, in Proceedings of the International Conference on Computer-Aided Design, San Jose, CA, 1997, pp. 66-71.
    • (1997) Proceedings of the International Conference on Computer-Aided Design , pp. 66-71
    • Elfadel, I.1    Ling, D.2
  • 8
    • 0026173848 scopus 로고
    • Alternating direction implicit iteration for systems with complex spectra
    • N. S. ELLNER AND E. L. WACHSPRESS, Alternating direction implicit iteration for systems with complex spectra, SIAM J. Numer. Anal., 28 (1991), pp. 859-870.
    • (1991) SIAM J. Numer. Anal. , vol.28 , pp. 859-870
    • Ellner, N.S.1    Wachspress, E.L.2
  • 9
    • 0021441691 scopus 로고
    • ∞-error bounds
    • ∞-error bounds, Internat. J. Control, 39 (1984), pp. 1115-1193.
    • (1984) Internat. J. Control , vol.39 , pp. 1115-1193
    • Glover, K.1
  • 12
    • 0000567621 scopus 로고
    • Numerical solution of the stable, nonnegative definite Lyapunov equation
    • S. J. HAMMARLING, Numerical solution of the stable, nonnegative definite Lyapunov equation, IMA J. Numer. Anal., 2 (1982), pp. 303-323.
    • (1982) IMA J. Numer. Anal. , vol.2 , pp. 303-323
    • Hammarling, S.J.1
  • 13
    • 0030556718 scopus 로고    scopus 로고
    • Numerical solution of the Lyapunov equation by approximate power iteration
    • A. S. HODEL, B. TENISON, AND K. R. POOLLA, Numerical solution of the Lyapunov equation by approximate power iteration, Linear Algebra Appl., 236 (1996), pp. 205-230.
    • (1996) Linear Algebra Appl. , vol.236 , pp. 205-230
    • Hodel, A.S.1    Tenison, B.2    Poolla, K.R.3
  • 14
    • 38249011246 scopus 로고
    • Krylov-subspace methods for the Sylvester equation
    • D. Y. HU AND L. REICHEL, Krylov-subspace methods for the Sylvester equation, Linear Algebra Appl., 172 (1992), pp. 283-313.
    • (1992) Linear Algebra Appl. , vol.172 , pp. 283-313
    • Hu, D.Y.1    Reichel, L.2
  • 15
    • 0013430918 scopus 로고
    • On the third and fourth Zolotarev problems in the complex plane
    • M.-P. ISTACE AND J.-P. THIRAN, On the third and fourth Zolotarev problems in the complex plane, SIAM J. Numer. Anal., 32 (1995), pp. 249-259.
    • (1995) SIAM J. Numer. Anal. , vol.32 , pp. 249-259
    • Istace, M.-P.1    Thiran, J.-P.2
  • 16
    • 0028377922 scopus 로고
    • Krylov subspace methods for solving large Lyapunov equations
    • I. M. JAIMOUKHA AND E. M. KASENALLY, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31 (1994), pp. 227-251.
    • (1994) SIAM J. Numer. Anal. , vol.31 , pp. 227-251
    • Jaimoukha, I.M.1    Kasenally, E.M.2
  • 18
    • 0000321992 scopus 로고
    • Explicit solutions of linear matrix equations
    • P. LANCASTER, Explicit solutions of linear matrix equations, SIAM Rev., 12 (1970), pp. 544-566.
    • (1970) SIAM Rev. , vol.12 , pp. 544-566
    • Lancaster, P.1
  • 19
    • 0027543772 scopus 로고
    • Construction of square root factor for solution of the Lyapunov matrix equation
    • V. B. LARIN AND F. A. ALIEV, Construction of square root factor for solution of the Lyapunov matrix equation, Systems Control Lett., 20 (1993), pp. 109-112.
    • (1993) Systems Control Lett. , vol.20 , pp. 109-112
    • Larin, V.B.1    Aliev, F.A.2
  • 20
    • 0032660354 scopus 로고    scopus 로고
    • An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect
    • New Orleans, LA
    • J.-R. LI, F. WANG, AND J. WHITE, An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect, in Proceedings of the 36th Design Automation Conference, New Orleans, LA, 1999, pp. 1-6.
    • (1999) Proceedings of the 36th Design Automation Conference , pp. 1-6
    • Li, J.-R.1    Wang, F.2    White, J.3
  • 22
    • 0013389326 scopus 로고    scopus 로고
    • Reduction of large circuit models via low rank approximate gramians
    • J.-R. LI AND J. WHITE, Reduction of large circuit models via low rank approximate gramians, Int. J. Appl. Math. Comput. Sci., 11 (2001), pp. 1151-1171.
    • (2001) Int. J. Appl. Math. Comput. Sci. , vol.11 , pp. 1151-1171
    • Li, J.-R.1    White, J.2
  • 23
    • 0002909988 scopus 로고
    • Solution of Lyapunov equations by alternating direction implicit iteration
    • A. LU AND E. L. WACHSPRESS, Solution of Lyapunov equations by alternating direction implicit iteration, Comput. Math. Appl., 21 (1991), pp. 43-58.
    • (1991) Comput. Math. Appl. , vol.21 , pp. 43-58
    • Lu, A.1    Wachspress, E.L.2
  • 24
    • 0031622756 scopus 로고    scopus 로고
    • A mixed nodal-mesh formulation for efficient extraction and passive reduced-order modeling of 3D interconnects
    • San Francisco, CA, June
    • N. MARQUES, M. KAMON, J. WHITE, AND L. SILVEIRA, A mixed nodal-mesh formulation for efficient extraction and passive reduced-order modeling of 3D interconnects, in Proceedings of the 35th ACM/IEEE Design Automation Conference, San Francisco, CA, June 1998, pp. 297-302.
    • (1998) Proceedings of the 35th ACM/IEEE Design Automation Conference , pp. 297-302
    • Marques, N.1    Kamon, M.2    White, J.3    Silveira, L.4
  • 26
    • 0019533482 scopus 로고
    • Principal component analysis in linear systems: Controllability, observability, and model reduction
    • B. C. MOORE, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), pp. 17-32.
    • (1981) IEEE Trans. Automat. Control , vol.26 , pp. 17-32
    • Moore, B.C.1
  • 27
    • 0033295674 scopus 로고    scopus 로고
    • A cyclic low-rank Smith method for large sparse Lyapunov equations
    • T. PENZL, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., 21 (2000), pp. 1401-1418.
    • (2000) SIAM J. Sci. Comput. , vol.21 , pp. 1401-1418
    • Penzl, T.1
  • 28
    • 0001082478 scopus 로고    scopus 로고
    • Eigenvalue decay bounds for solutions of Lyapunov equations: The symmetric case
    • T. PENZL, Eigenvalue decay bounds for solutions of Lyapunov equations: The symmetric case, Systems Control Lett., 40 (2000), pp. 139-144.
    • (2000) Systems Control Lett. , vol.40 , pp. 139-144
    • Penzl, T.1
  • 29
    • 0020114118 scopus 로고
    • Model reduction via balanced state space representations
    • L. PERNEBO AND L. M. SILVERMAN, Model reduction via balanced state space representations, IEEE Trans. Automat. Control, 27 (1982), pp. 382-387.
    • (1982) IEEE Trans. Automat. Control , vol.27 , pp. 382-387
    • Pernebo, L.1    Silverman, L.M.2
  • 30
    • 0019069037 scopus 로고
    • Linear model reduction and solution of the algebraic Riccati equation by use of the sign function
    • J. D. ROBERTS, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Internat. J. Control, 32 (1980), pp. 677-687.
    • (1980) Internat. J. Control , vol.32 , pp. 677-687
    • Roberts, J.D.1
  • 31
    • 0001409002 scopus 로고
    • The rational Krylov algorithm for nonsymmetric eigenvalue problems. III. Complex shifts for real matrices
    • A. RUHE, The rational Krylov algorithm for nonsymmetric eigenvalue problems. III. Complex shifts for real matrices, BIT, 34 (1994), pp. 155-176.
    • (1994) BIT , vol.34 , pp. 155-176
    • Ruhe, A.1
  • 32
    • 0024700032 scopus 로고
    • A Schur method for balanced-truncation model reduction
    • M. G. SAFONOV AND R. Y. CHIANG, A Schur method for balanced-truncation model reduction, IEEE Trans. Automat. Control, 34 (1989), pp. 729-733.
    • (1989) IEEE Trans. Automat. Control , vol.34 , pp. 729-733
    • Safonov, M.G.1    Chiang, R.Y.2
  • 34
    • 0026239062 scopus 로고
    • Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations
    • G. STARKE, Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 28 (1991), pp. 1431-1445.
    • (1991) SIAM J. Numer. Anal. , vol.28 , pp. 1431-1445
    • Starke, G.1
  • 35
    • 38249001860 scopus 로고
    • Fejér- Walsh points for rational functions and their use in the ADI iterative method
    • G. STARKE, Fejér- Walsh points for rational functions and their use in the ADI iterative method, J. Comput. Appl. Math., 46 (1993), pp. 129-141.
    • (1993) J. Comput. Appl. Math. , vol.46 , pp. 129-141
    • Starke, G.1
  • 36
    • 0023439931 scopus 로고
    • Truncated balanced realization of a stable nonminimal state-space system
    • M. S. TOMBS AND I. POSTLETHWAITE, Truncated balanced realization of a stable nonminimal state-space system, Internat. J. Control, 46 (1987), pp. 1319-1330.
    • (1987) Internat. J. Control , vol.46 , pp. 1319-1330
    • Tombs, M.S.1    Postlethwaite, I.2
  • 37
    • 0000871354 scopus 로고
    • Optimum alternating-direction-implicit iteration parameters for a model problem
    • E. L. WACHSPRESS, Optimum alternating-direction-implicit iteration parameters for a model problem, J. Soc. Indust. Appl. Math., 10 (1962), pp. 339-350.
    • (1962) J. Soc. Indust. Appl. Math. , vol.10 , pp. 339-350
    • Wachspress, E.L.1
  • 38
    • 38249030580 scopus 로고
    • Iterative solution of the Lyapunov matrix equation
    • E. L. WACHSPRESS, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 1 (1988), pp. 87-90.
    • (1988) Appl. Math. Lett. , vol.1 , pp. 87-90
    • Wachspress, E.L.1
  • 39
    • 0013434405 scopus 로고
    • ADI iterative solution of Lyapunov equations
    • Brussels, North-Holland, Amsterdam
    • E. L. WACHSPRESS, ADI iterative solution of Lyapunov equations, in Iterative Methods in Linear Algebra (Brussels, 1991), North-Holland, Amsterdam, 1992, pp. 229-231.
    • (1991) Iterative Methods in Linear Algebra , pp. 229-231
    • Wachspress, E.L.1
  • 41
    • 84968520180 scopus 로고
    • On the rate of convergence of an alternating direction implicit method in a noncommutative case
    • O. B. WIDLUND, On the rate of convergence of an alternating direction implicit method in a noncommutative case, Math. Comp, 20 (1966), pp. 500-515.
    • (1966) Math. Comp , vol.20 , pp. 500-515
    • Widlund, O.B.1


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