Abstract
We analyze the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities—the lower spectral radius and the largest Lyapunov exponent—are not algorithmically approximable.
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L. Arnold, H. Crauel, and J.-P. Eckmann (eds.),Lyapunov Exponents, Proceedings of a Conference Held in Oberwolfach, Lecture Notes in Mathematics, vol. 1486, Springer-Verlag, Berlin, 1991.
N. E. Barabanov, Lyapunov indicators of discrete inclusions, parts I, II, and III,Automat, i Telemekk,2 (1988), 40–46,3 (1988), 24–29, and5 (1988), 17–24 (Translation inAutomat. Remote Control).
M. Berger and Y. Wang, Bounded semigroup of matrices,Linear Algebra Appl.,166 (1992), 21–27.
V. D. Blondel and J. N. Tsitsiklis, When is a pair of matrices mortal?, Technical report LIDS-P-2314, LIDS, MIT, January 1996. To appear inInformation Processing Letters.
V. D. Blondel and J. N. Tsitsiklis, Complexity of stability and controllability of elementary hybrid systems, preprint (1997).
P. Bougerol, Filtre de Kalman Bucy et exposants de Lyapounov, inLyapunov Exponents (L. Arnold et al, eds.), Lecture Notes in Mathematics, vol. 1486, Springer-Verlag, Berlin, 1991, pp. 112–122.
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, vol. 15, SIAM, Philadelphia, PA, 1994.
R. Brayton and C. Tong, Constructive stability and asymptotic stability of dynamical systems,IEEE Trans. Circuits and Systems,27 (1980), 1121–1130.
J. Cohen, Subadditivity, generalized products of random matrices and operation research,SIAM Rev.,30 (1988), 69–86.
J. Cohen, H. Kesten, and M. Newman (eds.),Random Matrices and Their Applications, Contemporary Mathematics, vol. 50, American Mathematical Society, Providence, RI, 1986.
R. Darling, The Lyapunov exponent for product of infinite-dimensional random matrices, inLyapunov Exponents (L. Arnold et al., eds.), Lecture Notes in Mathematics, vol. 1486, Springer-Verlag, Berlin, 1991, pp. 206–215.
I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge,Linear Algebra Appl.,162 (1992), 227–263.
L. Elsner, The generalized spectral radius theorem: an analytic-geometric proof,Linear Algebra Appl.,220 (1995), 151–159.
M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, New York, 1979.
R. Gharavi and V. Anantharam, An upper bound for the largest Lyapunov exponent of a Markovian random matrix product of nonnegative matrices, preprint (1995).
G. Gripenberg, Computing the joint spectral radius,Linear Algebra Appl.,234 (1996), 43–60.
L. Gurvits, Stability of discrete linear inclusion,Linear Algebra Appl.,231 (1995), 47–85.
J. E. Hopcroft and J. D. Ullman,Formal Languages and Their Relation to Automata, Addison-Wesley, Reading, MA, 1969.
R. A. Horn and C. R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, 1985.
J. Kingman, Subadditive ergodic theory,Ann. Probab.,1 (1976), 883–909.
V. S. Kozyakin, Algebraic unsolvability of problem of absolute stability of desynchronized systems,Avtomat. i Telemekh.,6 (1990), 41–47 (Translation inAutomat. Remote Control).
J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices,Linear Algebra Appl.,214 (1995), 17–42.
R. Lima and M. Rahibe, Exact Lyapunov exponent for infinite products of random matrices,J. Phys. A,27 (1994), 3427–3437.
Y. Matiyasevich and G. Sénizergues, Decision problems for semi-Thue systems with a few rules, preprint (1996).
V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,Trans. Moscow Math. Soc.,19 (1968), 197–231.
C. H. Papadimitriou,Computational Complexity, Addison-Wesley, Reading, MA, 1994.
C. H. Papadimitriou and J. N. Tsitsiklis, The complexity of Markov decision processes,Math. Oper. Res.,12 (1987), 441–450.
M. Paterson, Unsolvability in 3 × 3 matrices,Stud. Appl. Math.,49 (1970), 105–107.
K. Ravishankar, Power law scaling of the top Lyapunov exponent of a product of random matrices,J. Stat. Phys.,54 (1989), 531–537.
J. Roerdink, The biennal life strategy in a random environment,J. Math. Biol.,26 (1988), 199–215.
G.-C. Rota and G. Strang, A note on the joint spectral radius,Indag. Math.,22 (1960), 379–381.
J. N. Tsitsiklis, On the stability of asynchronous iterative processes,Math. Systems Theory,20 (1987), 137–153.
J. N. Tsitsiklis and V. D. Blondel, The spectral radius of a pair of matrices is hard to compute,Proceedings of the 35th Conference on Decision and Control, Kobe, December 1996, pp. 3192–3197.
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This work was completed while Blondel was visiting Tsitsiklis at MIT. This research was supported by the ARO under Grant DAAL-03-92-G-0115.
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Tsitsiklis, J.N., Blondel, V.D. The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate. Math. Control Signal Systems 10, 31–40 (1997). https://doi.org/10.1007/BF01219774
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DOI: https://doi.org/10.1007/BF01219774