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Perturbation analysis of the matrix equation X-A * X -p A=Q. (English) Zbl 1173.15006
The authors consider the matrix equation X-A * X -p A=Q with 0<p1, with X, A and Q being n×n-complex matrices, Q being positive definite. They prove (using a new method) existence and uniqueness of a positive definite solution X. Except a generalization of known results for arbitrary p(0,1], a sharper perturbation bound and backward error of an approximation of this solution are evaluated. Explicit expressions of the condition number for the unique positive definite solution are obtained. The results are illustrated by numerical examples.
MSC:
15A24Matrix equations and identities
65F30Other matrix algorithms
15A12Conditioning of matrices
References:
[1]Zhan, X.; Xie, J.: On the matrix equation X+A*X-1A=I, Linear algebra appl. 247, 337-345 (1996) · Zbl 0863.15005 · doi:10.1016/0024-3795(95)00120-4
[2]Zhan, X.: Computing the extremal positive definite solutions of a matrix equations, SIAM J. Sci. comput. 17, 1167-1174 (1996) · Zbl 0856.65044 · doi:10.1137/S1064827594277041
[3]Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X+ATX-1A=I, Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013 · doi:10.1016/0024-3795(93)90115-5
[4]Engwerda, J. C.; Ran, A. C. M.; Rijkeboer, A. L.: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+ATX-1A=Q, Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008 · doi:10.1016/0024-3795(93)90295-Y
[5]Guo, C. H.; Lancaster, P.: Iterative solution of two matrix equations, Math. comp. 68, 1589-1603 (1999) · Zbl 0940.65036 · doi:10.1090/S0025-5718-99-01122-9
[6]Ferrante, A.: Hermitian solutions of the equation X=Q+NX-1N*, Linear algebra appl. 247, 359-373 (1996) · Zbl 0876.15011 · doi:10.1016/0024-3795(95)00121-2
[7]Ivanov, I. G.; El-Sayed, S. M.: Properties of positive definite solution of the equation X+A*X-2A=I, Linear algebra appl. 279, 303-316 (1998) · Zbl 0935.65041 · doi:10.1016/S0024-3795(98)00023-8
[8]Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V.: On matrix equations X±a*X-2A=I, Linear algebra appl. 326, 27-44 (2001) · Zbl 0979.15007 · doi:10.1016/S0024-3795(00)00302-5
[9]Zhang, Y. H.: On Hermitian positive definite solutions of matrix equation X+A*X-2A=I, Linear algebra appl. 372, 295-304 (2003) · Zbl 1035.15017 · doi:10.1016/S0024-3795(03)00530-5
[10]Zhang, Y. H.: On Hermitian positive definite solutions of matrix equation X-A*X-2A=I, J. comput. Math. 23, 408-418 (2005) · Zbl 1087.15020
[11]Liu, X. G.; Gao, H.: On the positive definite solutions of the matrix equations xs±ATX-ta=In, Linear algebra appl. 368, 83-97 (2003) · Zbl 1025.15018 · doi:10.1016/S0024-3795(02)00661-4
[12]Xu, S. F.: Perturbation analysis of the maximal solution of the matrix equation X+A*X-1A=P, Linear algebra appl. 336, 61-70 (2001) · Zbl 0992.15013 · doi:10.1016/S0024-3795(01)00300-7
[13]Sun, J. G.; Xu, S. F.: Perturbation analysis of the maximal solution of the matrix equation X+A*X-1A=P. II, Linear algebra appl. 362, 211-228 (2003) · Zbl 1020.15012 · doi:10.1016/S0024-3795(02)00490-1
[14]Hasanov, V. I.; Ivanov, I. G.; Uhlig, F.: Improved perturbation estimates for the matrix equation X±a*X-1A=Q, Linear algebra appl. 379, 113-135 (2004) · Zbl 1039.15005 · doi:10.1016/S0024-3795(03)00424-5
[15]Hasanov, V. I.; Ivanov, I. G.: On two perturbation estimates of the extreme solutions to the equations X±a*X-1A=Q, Linear algebra appl. 413, 81-92 (2006) · Zbl 1087.15016 · doi:10.1016/j.laa.2005.08.013
[16]Hasanov, V. I.: Positive definite solutions of the matrix equations X±a*X-qa=Q, Linear algebra appl. 404, 166-182 (2005) · Zbl 1078.15012 · doi:10.1016/j.laa.2005.02.024
[17]Li, J.: The Hermitian positive definite solutions and perturbation analysis of the matrix equation X-A*X-1A=Q, Math. numer. Sinica 30, 129-142 (2008) · Zbl 1174.65385
[18]M.C.B. Reurings, Symmetric matrix equation, Ph.D. Thesis, Vrije Universiteit, Amsterdan, 2003.
[19]Rice, J. R.: A theory of condition, SIAM J. Numer. anal. 3, 287-310 (1966) · Zbl 0143.37101 · doi:10.1137/0703023