Abstract and Applied Analysis

Solutions and Improved Perturbation Analysis for the Matrix Equation X - A * X - p A = Q   ( p > 0 )

Jing Li

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Abstract

The nonlinear matrix equation X - A * X - p A = Q with p > 0 is investigated. We consider two cases of this equation: the case p 1 and the case 0 < p < 1 . In the case p 1 , a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case 0 < p < 1 , a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 575964, 12 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511915

Digital Object Identifier
doi:10.1155/2013/575964

Mathematical Reviews number (MathSciNet)
MR3064406

Zentralblatt MATH identifier
1291.15041

Citation

Li, Jing. Solutions and Improved Perturbation Analysis for the Matrix Equation $X-{A}^{*}{X}^{-p}A=Q  \left(p&gt;0\right)$. Abstr. Appl. Anal. 2013 (2013), Article ID 575964, 12 pages. doi:10.1155/2013/575964. https://projecteuclid.org/euclid.aaa/1393511915


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References

  • W. N. Anderson, Jr., G. B. Kleindorfer, P. R. Kleindorfer, and M. B. Woodroofe, “Consistent estimates of the parameters of a linear system,” Annals of Mathematical Statistics, vol. 40, pp. 2064–2075, 1969.
  • W. N. Anderson, Jr., T. D. Morley, and G. E. Trapp, “The cascade limit, the shorted operator and quadratic optimal control,” in Linear Circuits, Systems and Signal Processsing: Theory and Application, C. I. Byrnes, C. F. Martin, and R. E. Saeks, Eds., pp. 3–7, North-Holland, New York, NY, USA, 1988.
  • R. S. Bucy, “A priori bounds for the Riccati equation,” in Proceedings of the Berkley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory, pp. 645–656, University of California Press, Berkeley, Calif, USA, 1972.
  • B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson's equations,” SIAM Journal on Numerical Analysis, vol. 7, pp. 627–656, 1970.
  • D. V. Ouellette, “Schur complements and statistics,” Linear Algebra and its Applications, vol. 36, pp. 187–295, 1981.
  • W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Reports on Mathematical Physics, vol. 8, no. 2, pp. 159–170, 1975.
  • J. Zabczyk, “Remarks on the control of discrete-time distributed parameter systems,” SIAM Journal on Control and Optimization, vol. 12, pp. 721–735, 1974.
  • A. Ferrante and B. C. Levy, “Hermitian solutions of the equation $X=Q+N{X}^{-1}{N}^{\ast\,\!}$,” Linear Algebra and its Applications, vol. 247, pp. 359–373, 1996.
  • V. I. Hasanov, “Notes on two perturbation estimates of the extreme solutions to the equations $X\pm {A}^{\ast\,\!}{X}^{-1}A=Q$,” Applied Mathematics and Computation, vol. 216, no. 5, pp. 1355–1362, 2010.
  • V. I. Hasanov and I. G. Ivanov, “On two perturbation estimates of the extreme solutions to the equations $X\pm {A}^{\ast\,\!}{X}^{-1}A=Q$,” Linear Algebra and its Applications, vol. 413, no. 1, pp. 81–92, 2006.
  • V. I. Hasanov, I. G. Ivanov, and F. Uhlig, “Improved perturbation estimates for the matrix equations $X\pm {A}^{\ast\,\!}{X}^{-1}A=Q$,” Linear Algebra and its Applications, vol. 379, pp. 113–135, 2004.
  • J. Li and Y. H. Zhang, “The Hermitian positive definite solution and its perturbation analysis for the matrix equation $X-{A}^{\ast\,\!}{X}^{-1}A=Q$,” Mathematica Numerica Sinica, vol. 30, no. 2, pp. 129–142, 2008 (Chinese).
  • I. G. Ivanov, V. I. Hasanov, and B. V. Minchev, “On matrix equations $X\pm {A}^{\ast\,\!}{X}^{-2}A=I$,” Linear Algebra and its Applications, vol. 326, no. 1–3, pp. 27–44, 2001.
  • Y. Zhang, “On Hermitian positive definite solutions of matrix equation $X-{A}^{\ast\,\!}{X}^{-2}A=I$,” Journal of Computational Mathematics, vol. 23, no. 4, pp. 408–418, 2005.
  • V. I. Hasanov and I. G. Ivanov, “On the matrix equation $X-{A}^{\ast\,\!}{X}^{-n}A=I$,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1340–1356, 2005.
  • V. I. Hasanov and I. G. Ivanov, “Solutions and perturbation estimates for the matrix equations $X\pm {A}^{\ast\,\!}{X}^{-n}A=Q$,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 513–525, 2004.
  • X. Liu and H. Gao, “On the positive definite solutions of the matrix equations ${X}^{s}\pm {A}^{T}{X}^{-t}A={I}_{n}$,” Linear Algebra and its Applications, vol. 368, pp. 83–97, 2003.
  • J. C. Engwerda, “On the existence of a positive definite solution of the matrix equation $X+{A}^{T}{X}^{-1}A=I$,” Linear Algebra and its Applications, vol. 194, pp. 91–108, 1993.
  • J. C. Engwerda, A. C. M. Ran, and A. L. Rijkeboer, “Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation $X+{A}^{T}{X}^{-1}A=Q$,” Linear Algebra and its Applications, vol. 186, pp. 255–275, 1993.
  • C.-H. Guo and P. Lancaster, “Iterative solution of two matrix equations,” Mathematics of Computation, vol. 68, no. 228, pp. 1589–1603, 1999.
  • J.-g. Sun and S.-F. Xu, “Perturbation analysis of maximal solution of the matrix equation $X+{A}^{\ast\,\!}{X}^{-1}A=P$. II,” Linear Algebra and its Applications, vol. 362, pp. 211–228, 2003.
  • X. Zhan, “Computing the extremal positive definite solutions of a matrix equation,” SIAM Journal on Scientific Computing, vol. 17, no. 5, pp. 1167–1174, 1996.
  • X. Zhan and J. Xie, “On the matrix equation $X+{A}^{\ast\,\!}{X}^{-1}A=I$,” Linear Algebra and its Applications, vol. 247, pp. 337–345, 1996.
  • I. G. Ivanov and S. M. El-sayed, “Properties of positive definite solutions of the equation $X+{A}^{\ast\,\!}{X}^{-2}A=I$,” Linear Algebra and its Applications, vol. 279, no. 1-3, pp. 303–316, 1998.
  • Y. Zhang, “On Hermitian positive definite solutions of matrix equation $X+{A}^{\ast\,\!}{X}^{-2}A=I$,” Linear Algebra and its Applications, vol. 372, pp. 295–304, 2003.
  • V. I. Hasanov, “On positive definite solutions of the family of matrix equations $X+{A}^{\ast\,\!}{X}^{-n}A=Q$,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 277–301, 2006.
  • J. Cai and G. L. Chen, “On the Hermitian positive definite solution of nonlinear matrix equation ${X}^{s}-{A}^{\ast\,\!}{X}^{-t}A=Q$,” Applied Mathematics and Computation, vol. 217, pp. 2448–2456, 2010.
  • X. Duan and A. Liao, “On the existence of Hermitian positive definite solutions of the matrix equation ${X}^{s}-{A}^{\ast\,\!}{X}^{-t}A=Q$,” Linear Algebra and its Applications, vol. 429, no. 4, pp. 673–687, 2008.
  • X. Yin, S. Liu, and L. Fang, “Solutions and perturbation estimates for the matrix equation ${X}^{s}-{A}^{\ast\,\!}{X}^{-t}A=Q$,” Linear Algebra and its Applications, vol. 431, no. 9, pp. 1409–1421, 2009.
  • D. Zhou, G. Chen, G. Wu, and X. Zhang, “Some properties of the nonlinear matrix equation ${X}^{s}-{A}^{\ast\,\!}{X}^{-t}A=Q$,” Journal of Mathematical Analysis and Applications, vol. 392, no. 1, pp. 75–82, 2012.
  • V. I. Hasanov, “Positive definite solutions of the matrix equations $X\pm {A}^{\ast\,\!}{X}^{-q}A=Q$,” Linear Algebra and its Applications, vol. 404, pp. 166–182, 2005.
  • J. F. Wang, Y. H. Zhang, and B. R. Zhu, “The Hermitian positive definite solutions of the matrix equation $X+{A}^{\ast\,\!}{X}^{-q}A=I(q>0)$,” Mathematica Numerica Sinica, vol. 26, no. 1, pp. 61–72, 2004 (Chinese).
  • X. Yin, S. Liu, and T. Li, “On positive definite solutions of the matrix equation $X+{A}^{\ast\,\!}{X}^{-q}A=Q(0<p \leq 1)$,” Taiwanese Journal of Mathematics, vol. 16, no. 4, pp. 1391–1407, 2012.
  • X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X-\sum _{i=1}^{m}{A}_{i}^{\ast\,\!}{X}^{-1}{A}_{i}=I$,” Linear Algebra and its Applications, vol. 429, no. 1, pp. 110–121, 2008.
  • X. Duan, C. Li, and A. Liao, “Solutions and perturbation analsis for the nonlinear matrix equation $X+\sum _{i=1}^{m}{A}_{i}^{\ast\,\!}{X}^{-1}{A}_{i}=I$,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4458–4466, 2011.
  • Y.-m. He and J.-h. Long, “On the Hermitian positive definite solution of the nonlinear matrix equation $X+\sum _{i=1}^{m}{A}_{i}^{\ast\,\!}{X}^{-1}{A}_{i}=I$,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3480–3485, 2010.
  • X. Yin and S. Liu, “Positive definite solutions of the matrix equations $X\pm {A}^{\ast\,\!}{X}^{-q}A=Q(q\geq 1)$,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3727–3739, 2010.
  • H. Xiao and J. T. Wang, “On the matrix equation $X-{A}^{\ast\,\!}{X}^{-p}A=Q(p > 1)$,” Chinese Journal of Engineering Mathematics, vol. 26, no. 2, pp. 305–309, 2009.
  • J. Li and Y. Zhang, “Perturbation analysis of the matrix equation $X-{A}^{\ast\,\!}{X}^{-q}A=Q$,” Linear Algebra and its Applications, vol. 431, no. 9, pp. 1489–1501, 2009.
  • J. R. Rice, “A theory of condition,” SIAM Journal on Numerical Analysis, vol. 3, pp. 287–310, 1966.
  • R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Posts and Telecom press, Beijing, China, 2005.
  • R. A. Horn and C. R. Johnson, Matrix Analysis, Posts and Telecom press, Beijing, China, 2005.