The interval Lyapunov matrix equation: analytical results and an efficient numerical technique for outer estimation of the united solution set. (English) Zbl 1255.15002

Summary: This note tries to propose an efficient method for obtaining outer estimations for the so-called united solution set of the interval Lyapunov matrix equation \(\mathbf AX+X\mathbf A^T=\mathbf F\), where \(\mathbf A\) and \(\mathbf F\) are known real interval matrices while \(X\) is the unknown matrix; all of dimension \(n \times n\). We first explore the equation in the more general setting of AE-solution sets, and show that only a small part of Shary’s results on the AE-solution sets of interval linear systems can be generalized to the interval Lyapunov matrix equation. Then, we propose our modification of Krawczyk operator which enables us to reduce the computational complexity of obtaining an outer estimation for the united solution set to cubic, provided that the midpoint of \(\mathbf A\) is diagonalizable.


15A06 Linear equations (linear algebraic aspects)
65F30 Other matrix algorithms (MSC2010)
65G40 General methods in interval analysis
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[1] Antoulas, A., ()
[2] Datta, B.N., Numerical methods for linear control systems, (2004), Academic Press San Diego
[3] D. Kressner, V. Mehrmann, T. Penzl, CTDSX—a collection of benchmark examples for state-space realizations of continuous-time dynamical systems, Tech. Report SLICOT Working Note 1998-9, 1998. http://www.slicot.org/REPORTS/SLWN1998-9.ps.gz.
[4] Penzl, T., A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. sci. comput., 21, 1401-1418, (2000) · Zbl 0958.65052
[5] Saad, Y., Numerical solution of large Lyapunov equations, (), 503-511
[6] Bartels, R.H.; Stewart, G.W., Algorithm 432: solution of the matrix equation \(A X + X B = C\), Commun. ACM, 15, 820-826, (1972) · Zbl 1372.65121
[7] CODATA Internationally recommended values of the fundamental physical constants, The Physics Laboratory of the National Institute of Standards. Available online at: http://physics.nist.gov/cuu/Constants.
[8] Kearfott, R.B.; Kreinovich, V., Applications of interval computations, (1996), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0841.65031
[9] Rump, S.M., INTLAB—interval laboratory, (), 77-105 · Zbl 0949.65046
[10] J. Rohn, Verification Software in MATLAB/INTLAB. Available online at: http://uivtx.cs.cas.cz/ rohn/matlab.
[11] Seif, N.P.; Hussein, S.A.; Deif, A.S., The interval Sylvester matrix equation, Computing, 52, 233-244, (1994) · Zbl 0807.65046
[12] Shashikhin, V.N., Robust stabilization of linear interval systems, J. appl. math. mech., 66, 393-400, (2002) · Zbl 1090.93552
[13] Shashikhin, V.N., Robust assignment of poles in large-scale interval systems, Autom. remote control, 63, 200-208, (2002) · Zbl 1090.93552
[14] Rohn, J.; Kreinovich, V., Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard, SIAM J. matrix anal. appl., 16, 415, (1995) · Zbl 0824.65011
[15] Kreinovich, V.; Lakeyev, A.; Rohn, J.; Kahl, P., Computational complexity and feasiility of data processing and interval computations, (1998), Kluwer Dordrecht · Zbl 0945.68077
[16] Rohn, J., Enclosing solutions of linear interval equations is NP-hard, Computing, 53, 365-368, (1994) · Zbl 0809.65019
[17] Kearfott, R.B.; Nakano, M.T.; Neumaier, A.; Rump, S.M.; Shary, S.P.; van Hentenryck, P., Standardized notation in interval analysis, Reliab. comput., 15, 7-13, (2010) · Zbl 1196.65088
[18] Alefeld, G.; Herzberger, J., Introduction to interval computations, (1983), Academic Press New York
[19] Moore, R.E.; Kearfott, R.B.; Cloud, M.J., Introduction to interval analysis, (2009), SIAM Philadelphia · Zbl 1168.65002
[20] Neumaier, A., Interval methods for systems of equations, (1990), Cambridge University Press Cambridge · Zbl 0706.15009
[21] Shary, S.P., A new technique in systems analysis under interval uncertainty and ambiguity, Reliab. comput., 8, 321-418, (2002) · Zbl 1020.65029
[22] Hashemi, B.; Dehghan, M., Results concerning interval linear systems with multiple right-hand sides and the interval matrix equation \(A X = B\), J. comput. appl. math., 235, 2969-2978, (2011) · Zbl 1221.65087
[23] ()
[24] Kearfott, R.B., Interval analysis: interval Newton methods, (), 76-78
[25] Jansson, C.; Rump, S.M., Rigorous solution of linear programming problems with uncertain data, ZOR, methods models oper. res., 35, 87-111, (1991) · Zbl 0735.90043
[26] Rump, S.M., Verification methods for dense and sparse systems of equations, (), 63-136 · Zbl 0813.65072
[27] Frommer, A.; Hashemi, B., Verified computation of square roots of a matrix, SIAM J. matrix anal. appl., 31, 1279-1302, (2009), Preprint available as technical report BUW-SC 09/2, Bergische Universität Wuppertal. www-ai.math.uni-wuppertal.de/SciComp/preprints/SC0902.pdf · Zbl 1194.65069
[28] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press · Zbl 0729.15001
[29] Rump, S.M., A note on epsilon-inflation, Reliab. comput., 4, 371-375, (1998) · Zbl 0920.65031
[30] Hodel, A.S.; Poolla, K.P.; Tenison, B., Numerical solution of the Lyapunov equation by approximate power iteration, Linear algebra appl., 236, 205-230, (1996) · Zbl 0848.65033
[31] Shary, S.P., Solving the interval tolerance problem, Math. comput. simul., 39, 53-85, (1995)
[32] Shary, S.P., Algebraic approach to the interval linear static identification, tolerance and control problems, or one more application of kaucher arithmetic, Reliab. comput., 2, 3-33, (1996) · Zbl 0853.65048
[33] Shary, S.P., Controllable solution set to interval static systems, Appl. math. comput., 86, 185-196, (1997) · Zbl 0908.65037
[34] Popova, E., Quality of the solution sets of parameter-dependent interval linear systems, Z. angew. math. mech., 82, 723-727, (2002) · Zbl 1013.65042
[35] Popova, E.; Krämer, W., Inner and outer bounds for the solution set of parametric interval linear systems, J. comput. appl. math., 199, 310-316, (2007) · Zbl 1108.65027
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