Perturbation analysis for the periodic generalized coupled Sylvester equation

References

• M. Arioli, M. Baboulin, and S. Gratton, A partial condition number for linear least squares problems, SIAM J. Matrix Anal. Appl. 29(2) (2007), pp. 413–433. doi: 10.1137/050643088
   Web of Science ®Google Scholar
• Y. Cao and L. Petzold, A subspace error estimate for linear systems, SIAM J. Matrix Anal. Appl. 24(3) (2003), pp. 787–801. doi: 10.1137/S0895479801390649
   Web of Science ®Google Scholar
• X. Chen, Solving the (generalized) periodic Sylvester equation with the matrix sign function, Math. Numer. Sin. 34(2) (2012), pp. 153–162.
   Google Scholar
• C. Coll, M. Fullana, and E. Sanchez, Reachability and observability indices of a discrete-time periodic descriptor system, Appl. Math. Comput. 153 (2004), pp. 485–496.
   Web of Science ®Google Scholar
• B. Datta, Numerical Methods for Linear Control Systems: Design and Analysis, Elsevier, London, 2003.
   Google Scholar
• H. Diao, X. Shi, and Y. Wei, Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation, Sci. China Math. 56 (2013), pp. 967–982. doi: 10.1007/s11425-013-4583-3
   Web of Science ®Google Scholar
• H. Diao, H. Xiang, and Y. Wei, Mixed, componentwise condition numbers and small sample statistical condition estimation of Sylvester equations, Numer. Linear Algebra Appl. 19 (2012), pp. 639–654. doi: 10.1002/nla.790
   Web of Science ®Google Scholar
• F. Ding and T. Chen, Iterative least-squares solutions of coupled Sylvester matrix equations, Systems Control Lett. 54 (2005), pp. 95–107. doi: 10.1016/j.sysconle.2004.06.008
   Web of Science ®Google Scholar
• A. Dmytryshyn and B. Kågström, Coupled Sylvester-type matrix equations and block diagonalization, SIAM J. Matrix Anal. Appl. 36(2) (2015), pp. 580–593. doi: 10.1137/151005907
   Web of Science ®Google Scholar
• I. Gohberg and I. Koltracht, Mixed, componentwise, and structured condition numbers, SIAM J. Matrix Anal. Appl. 14 (1993), pp. 688–704. doi: 10.1137/0614049
   Web of Science ®Google Scholar
• G. Golub and W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), pp. 205–224. doi: 10.1137/0702016
   Google Scholar
• A. Graham, Kronecker Products and Matrix Calculus: With Applications, John Wiley, New York, 1981.
   Google Scholar
• R. Granat, B. Kågström, and D. Kressner, Computing periodic deflating subspaces associated with a specified set of eigenvalues, BIT 47 (2007), pp. 763–791. doi: 10.1007/s10543-007-0143-y
   Web of Science ®Google Scholar
• T. Gudmundsson, Small-sample statistical estimates for the sensitivity of eigenvalue problems, SIAM J. Matrix Anal. Appl. 18 (1997), pp. 868–886. doi: 10.1137/S0895479895296021
   Web of Science ®Google Scholar
• M. Hajarian, Developing CGNE algorithm for the periodic discrete-time generalized coupled Sylvester matrix equations, Comp. Appl. Math. 34 (2015), pp. 755–771. doi: 10.1007/s40314-014-0138-7
   Web of Science ®Google Scholar
• N. J. Higham, Perturbation theory and backward error for AX−XB=C, BIT 33 (1993), pp. 124–136. doi: 10.1007/BF01990348
   Web of Science ®Google Scholar
• N. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
   Google Scholar
• M. E. Hochstenbach, Probabilistic upper bounds for the matrix two-norm, J. Sci. Comput. 57 (2013), pp. 464–476. doi: 10.1007/s10915-013-9716-x
   Web of Science ®Google Scholar
• I. Jonsson and B. Kågström, Recursive blocked algorithms for solving triangular systems-Part I: One-sided and coupled Sylvester-type matrix equations, ACM Trans. Math. Softw. 28 (2002), pp. 392–415. doi: 10.1145/592843.592845
   Web of Science ®Google Scholar
• I. Jonsson and B. Kågström, Recursive blocked algorithms for solving triangular systems-Part II: Two-sided and generalized Sylvester and Lyapunov matrix equations, ACM Trans. Math. Softw. 28 (2002), pp. 416–435. doi: 10.1145/592843.592846
   Web of Science ®Google Scholar
• B. Kågström, A perturbation analysis of the generalized Sylvester equation (AR−LB,DR−LE)=(C,F), SIAM J. Matrix Anal. Appl. 15 (1994), pp. 1045–1060. doi: 10.1137/S0895479893246212
   Web of Science ®Google Scholar
• C. S. Kenney and A. J. Laub, Small-sample statistical condition estimates for general matrix functions, SIAM J. Sci. Comput. 15 (1994), pp. 36–61. doi: 10.1137/0915003
   Web of Science ®Google Scholar
• C. S. Kenney, A. J. Laub, and M. S. Reese, Statistical condition estimation for linear systems, SIAM J. Sci. Comput. 19 (1998), pp. 566–583. doi: 10.1137/S1064827595282519
   Web of Science ®Google Scholar
• C. S. Kenney, A. J. Laub, and M. S. Reese, Statistical condition estimation for linear least squares, SIAM J. Matrix Anal. Appl. 19 (1998), pp. 906–923. doi: 10.1137/S0895479895291935
   Web of Science ®Google Scholar
• M. Konstantinov, D. Gu, V. Mehrmann, and P. Petkov, Perturbation Theory for Matrix Equations, Elsevier, Amsterdam, 2003.
   Google Scholar
• A. J. Laub and J. Xia, Applications of statistical condition estimation to the solution of linear systems, Numer. Linear Algebra Appl. 15 (2008), pp. 489–513. doi: 10.1002/nla.570
   Web of Science ®Google Scholar
• Y. Lin and Y. Wei, Condition numbers of the generalized Sylvester equation, IEEE Trans. Automat. Control 52 (2007), pp. 2380–2385. doi: 10.1109/TAC.2007.910727
   Web of Science ®Google Scholar
• J. R. Rice, A theory of condition, SIAM J. Numer. Anal. 3 (1966), pp. 287–310. doi: 10.1137/0703023
   Google Scholar
• A. Varga, On computing minimal realizations of periodic descriptor systems, in Proceedings of IFAC workshop on periodic control systems, St. Petersburg, Russia, 2007
   Google Scholar
• Z-J. Xie, W. Li, and X-Q. Jin, On condition numbers for the canonical generalized polar decomposition of real matrices, Electron. J. Linear Algebra 26 (2013), pp. 842–857. doi: 10.13001/1081-3810.1691
   Web of Science ®Google Scholar