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Iterative solution of the Lyapunov matrix equation. (English) Zbl 0631.65037
Iterative solution of the Lyapunov matrix equation $$AX+XB=C$$ using ADI theory described by N. S. Ellner and the author [New ADI model problem application, Proc. FJCC, Dallas, Texas, 528-534 (1968)] is reviewed here. A procedure for implementing this technique when A and B are sparse is introduced in this paper.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 65F10 Iterative numerical methods for linear systems 15A24 Matrix equations and identities
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##### References:
 [1] Ellner, N.S.; Wachspress, E.L., New ADI model problem applications, Proc. FJCC, 528-534, (1968), Dallas, Texas [2] Golub, G.H.; Nash, S.; vanLoan, C., A Hessenberg-shur method for solution of the problem AX + XB = C, IEEE trans. automat. control, AC24, 909-913, (1979) · Zbl 0421.65022 [3] Rothschild, D.; Jameson, A., Comparison of four numerical algorithms for solving the Lyapunov matrix equation, Int. J. control, 11, No. 2, 181-198, (1970) · Zbl 0185.40102 [4] Saltzman, N., ADI parameters for some complex spectra, Univ. of tennessee Master’s thesis, (1987) [5] Smith, R.A., Matrix equation XA + BX = C, SIAM J. appl. math., 16, No. 1, 198-201, (1968) · Zbl 0157.22603 [6] Wachspress, E.L., Iterative solution of elliptic systems, (1966), Prentice Hall · Zbl 0161.12203 [7] Young, D.M., Iterative solution of large elliptic systems, (1971), Academic Press
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