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Numerical Linear Algebra in Signals, Systems and Control pp 109–122Cite as

Perturbation Analysis of the Mixed-Type Lyapunov Equation

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Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 80))

Abstract

This paper concerns the mixed-type Lyapunov equation \(X=A^*XB+B^*XA+Q,\) where \(A,B,\) and \(Q\) are \(n\times n\) complex matrices and \(A^*\) the conjugate transpose of a matrix \(A.\) A perturbation bound for the solution to this matrix equation is derived, an explicit expression of the condition number is obtained, and the backward error of an approximate solution is evaluated by using the techniques developed in Sun (Linear Algebra Appl 259:183–208, 1997), Sun and Xu (Linear Algebra Appl 362:211–228, 2003). The results are illustrated by using some numerical examples.

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References

  1. Cheng MS (2004) Theory and methods of nonlinear matrix equations \(X\pm A^*X^{-2}A=I\). Ph.D. Dissertation, Peking University

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Acknowledgments

This research was supported in part by NSFC under grant 10571007.

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Authors and Affiliations

  1. Department of Applied Mathematics, Dalian University of Technology, 116024, Dalian, China

    Mingsong Cheng

  2. LMAM, School of Mathematical Sciences, Peking University, 100871, Beijing, China

    Shufang Xu

Authors
  1. Mingsong Cheng
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  2. Shufang Xu
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Corresponding author

Correspondence to Mingsong Cheng .

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Editors and Affiliations

  1. CESAME, Batiment Euler, Universite Catholique de Louvain, 4, avenue Georges Lemaitre, Louvain la Neuve, 1348, Belgium

    Paul Van Dooren

  2. Department of Electrical and Computer, Engineering, Texas A&M University, Zachry Engineering Center, College Station, 77843-3128, Texas, USA

    Shankar P. Bhattacharyya

  3. Dept. Mathematics, Chinese University Hong Kong, Shatin, New Territories, Hong Kong/PR China

    Raymond H. Chan

  4. Dept. Mathematics, University of Connecticut, Auditorium Road 196, Storrs, 06269-3009, Connecticut, USA

    Vadim Olshevsky

  5. , Electrical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

    Aurobinda Routray

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Cheng, M., Xu, S. (2011). Perturbation Analysis of the Mixed-Type Lyapunov Equation. In: Van Dooren, P., Bhattacharyya, S., Chan, R., Olshevsky, V., Routray, A. (eds) Numerical Linear Algebra in Signals, Systems and Control. Lecture Notes in Electrical Engineering, vol 80. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0602-6_6

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  • DOI: https://doi.org/10.1007/978-94-007-0602-6_6

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-0601-9

  • Online ISBN: 978-94-007-0602-6

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