On the degree of mixed polynomial matrices

SIAM Journal on Matrix Analysis and Applications

Volumn 20, Issue 1, 1998, Pages 196-227

  Murota, Kazuo ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

Combinatorial matrix theory; Degree of determinant; Mixed matrix; Polynomial matrix; Valuated matroid

Indexed keywords

EID: 0032217345     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479896311438     Document Type: Article

References (68)

1. A. V. AHO, J. E. HOPCROFT, AND J. D. ULLMAN, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.
2. K. E. BRENAN, S. L. CAMPBELL, AND L. R. PETZOLD, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland, New York, 1989.
3. R. A. BRUALDI AND H. J. RYSER, Combinatorial Matrix Theory, Cambridge University Press, London, 1991.
4. P. BUJAKIEWICZ, Maximum Weighted Matching for High Index Differential Algebraic Equations, Doctoral thesis, Delft University of Technology, The Netherlands, 1994.
5. C. COMMAULT AND J.-M. DION, Structure at infinity of linear multivariable systems: A geometric approach, IEEE Trans. Automat. Control, AC-27 (1982), pp. 693-696.
6. C. COMMAULT, J.-M. DION, A. PEREZ, Disturbance rejection for structured systems, IEEE Trans. Automat. Control, AC-36 (1991), pp. 884-887.
7. A. W. M. DRESS AND W. TERHALLE, Well-layered maps and the maximum-degree k × k-subdeterminant of a matrix of rational functions, Appl. Math. Lett., 8 (1995), pp. 19-23.
8. A. W. M. DRESS AND W. TERHALLE, Well-layered maps - A class of greedily optimizable set functions, Appl. Math. Lett., 8 (1995), pp. 77-80.
9. A. W. M. DRESS AND W. TERHALLE, Rewarding maps - On greedy optimization of set functions, Adv. Appl. Math., 16 (1995), pp. 464-483.
10. A. W. M. DRESS AND W. WENZEL, Valuated matroid: A new look at the greedy algorithm, Appl. Math. Lett., 3 (1990), pp. 33-35.
11. A. W. M. DRESS AND W. WENZEL, Valuated matroids, Adv. Math., 93 (1992), pp. 214-250.
12. J. EDMONDS, Systems of distinct representatives and linear algebra, J. Res. Nat. Bur. Stand., 71B (1967), pp. 241-245.
13. J. EDMONDS, Submodular functions, matroids and certain polyhedra, in Combinatorial Structures and Their Applications, R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds., Gordon and Breach, New York, 1970, pp. 69-87.
14. J. EDMONDS, Matroid intersection, Ann. Discrete Math., 14 (1979), pp. 39-49.
15. U. FAIGLE, Matroids in combinatorial optimization, in Combinatorial Geometries, N. White, ed., Cambridge University Press, London, 1987, pp. 161-210.
16. A. FRANK, A weighted matroid intersection algorithm, J. Algorithms, 2 (1981), pp. 328-336.
17. S. FUJISHIGE, A primal approach to the independent assignment problem, J. Oper. Res. Soc. Japan, 20 (1977), pp. 1-15.
18. S. FUJISHIGE, Submodular Functions and Optimization, Ann. Discrete Math. 47, NorthHolland, Amsterdam, 1991.
19. F. R. GANTMACHER, The Theory of Matrices, Chelsea, New York, 1959.
20. C. W. GEAR, Differential-algebraic equation index transformations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 39-47.
21. C. W. GEAR, Differential algebraic equations, indices, and integral algebraic equations, SIAM J. Numer. Anal., 27 (1990), pp. 1527-1534.
22. I. GOHBERG, P. LANCASTER, AND L. RODMAN, Matrix Polynomials, Academic Press, New York, 1982.
23. E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II, Springer-Verlag, Berlin, 1991.
24. M. L. J. HAUTUS, The formal Laplace transform for smooth linear systems, in Mathematical Systems Theory, Lecture Notes in Econom. and Math. Systems 131, G. Marchesini and S. K. Mitter, eds., Springer-Verlag, Berlin, 1976, pp. 29-47.
25. Y. HAYAKAWA, S. HOSOE, AND M. ITO, Dynamical degree and controllability for linear systems with intermediate standard form, Trans. Inst. Electron. Comm. Engrs. Japan, J64A (1981), pp. 752-759 (in Japanese).
26. V. HOVELAQUE, C. COMMAULT, AND J.-M. DION, Analysis of linear structured systems using primal-dual algorithm, Systems Control Lett., 27 (1996), pp. 73-85.
27. M. IRI, Applications of matroid theory, in Mathematical Programming - The State of the Art, A. Bachem, M. Grötschel, and B. Korte, eds., Springer-Verlag, Berlin, 1983, pp. 158-201.
28. M. IRI AND N. TOMIZAWA, An algorithm for finding an optimal "independent assignment," J. Oper. Res. Soc. Japan, 19 (1976), pp. 32-57.
29. S. IWATA AND K. MUROTA, Combinatorial Relaxation Algorithm for Mixed Polynomial Matrices, RIMS Preprint 1113, Kyoto University, Kyoto, Japan, 1996.
30. S. IWATA, K. MUROTA, AND I. SAKUTA, Primal-dual combinatorial relaxation algorithms for the maximum degree of subdeterminants, SIAM J. Sci. Comput., 17 (1996), pp. 993-1012.
31. B. KORTE, L. LOVÁSZ, AND R. SCHRADER, Greedoids, Springer-Verlag, Berlin, 1991.
32. E. L. LAWLER, Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York, 1976.
33. K. MUROTA, Use of the concept of physical dimensions in the structural approach to systems analysis, Japan J. Appl. Math., 2 (1985), pp. 471-494.
34. K. MUROTA, Systems Analysis by Graphs and Matroids - Structural Solvability and Controllability, Algorithms Combin. 3, Springer-Verlag, Berlin, 1987.
35. K. MUROTA, On the irreducibility of layered mixed matrices, Linear and Multilinear Algebra, 24 (1989), pp. 273-288.
36. K. MUROTA, Mixed matrices - Irreducibility and decomposition, in Combinatorial and Graph-Theoretical Problems in Linear Algebra, IMA Vol. Math. Appl. 50, R. A. Brualdi, S. Friedland, and V. Klee, eds., Springer-Verlag, New York, 1993, pp. 39-71.
37. K. MUROTA, Computing the degree of determinants via combinatorial relaxation, SIAM J. Comput., 24 (1995), pp. 765-796.
38. K. MUROTA, Combinatorial relaxation algorithm for the maximum degree of subdeterminants: Computing Smith-McMillan form at infinity and structural indices in Kronecker form, Appl. Algebra Engrg., Comm. Comput., 6 (1995), pp. 251-273.
39. K. MUROTA, Finding optimal minors of valuated bimatroids, Appl. Math. Lett., 8 (1995), pp. 37-42.
40. K. MUROTA, Structural approach in systems analysis by mixed matrices - An exposition for index of DAE, in ICIAM 95 (Proceedings of the Third International Congress on Industrial and Applied Mathematics held in Hamburg, Germany, July 3-7, 1995), Math. Res. 87, K. Kirchgässner, O. Mahrenholtz, and R. Mennicken, eds., Akademie Verlag, Berlin, 1996, pp. 257-279.
41. K. MUROTA, Valuated matroid intersection, I: Optimality criteria, SIAM J. Discrete Math., 9 (1996), pp. 545-561.
42. K. MUROTA, Valuated matroid intersection, II: Algorithms, SIAM J. Discrete Math., 9 (1996), pp. 562-576.
43. K. MUROTA, Convexity and Steinitz's exchange property, Adv. Math., 124 (1996), pp. 272-311. Extended abstract in Integer Programming and Combinatorial Optimization (Proceedings of 5th International IPCO Conference, Vancouver, British Columbia, Canada, June 3-5, 1996), Lecture Notes in Comput. Sci. 1084, W. H. Cunningham, S. T. McCormick, and M. Queyranne, eds., Springer-Verlag, New York, 1996, pp. 260-274.
44. K. MUROTA, Convexity and Steinitz's exchange property, Adv. Math., 124 (1996), pp. 272-311. Extended abstract in Integer Programming and Combinatorial Optimization (Proceedings of 5th International IPCO Conference, Vancouver, British Columbia, Canada, June 3-5, 1996), Lecture Notes in Comput. Sci. 1084, W. H. Cunningham, S. T. McCormick, and M. Queyranne, eds., Springer-Verlag, New York, 1996, pp. 260-274.
45. K. MUROTA, Characterizing a valuated delta-matroid as a family of delta-matroids, J. Oper. Res. Soc. Japan, 40 (1997), pp. 565-578.
46. K. MUROTA AND M. IRI, Structural solvability of systems of equations - A mathematical formulation for distinguishing accurate and inaccurate numbers in structural analysis of systems, Japan J. Appl. Math., 2 (1985), pp. 247-271.
47. K. MUROTA, M. IRI AND M. NAKAMURA, Combinatorial canonical form of layered mixed matrices and its application to block-triangularization of systems of linear/nonlinear equations, SIAM J. Algebraic Discrete Meth., 8 (1987), pp. 123-149.
48. K. MUROTA AND J. W. VAN DER WOUDE, Structure at infinity of structured descriptor systems and its applications, SIAM J. Control Optim., 29 (1991), pp. 878-894.
49. M. NEWMAN, Integral Matrices, Academic Press, London, 1972.
50. A. RECSKI, Matroid Theory and Its Applications in Electric Network Theory and in Statics, Algorithms Combin. 6, Springer-Verlag, Berlin, 1989.
51. K. J. REINSCHKE, Multivariable Control, A Graph-Theoretic Approach, Springer-Verlag, New York, 1988.
52. H. H. ROSENBROCK, State-Space and Multivariable Theory, Nelson, London, 1970.
53. N. SUDA, B. WAN, AND I. UENO, The orders of infinite zeros of structured systems, Trans. Soc. Instrument Control Engrs. Japan, 25 (1989), pp. 1062-1068 (in Japanese).
54. F. SVARICEK, An improved graph theoretic algorithm for computing the structure at infinity of linear systems, in Proceedings of the 29th Conference on Decision and Control, Honolulu, HI, December 1990, pp. 2923-2924.
55. F. SVARICEK, Zuverlässige numerische Analyse linearer Regelungssysteme, Teubner, Stuttgart, 1995.
56. N. TOMIZAWA AND M. IRI, An algorithm for determining the rank of a triple matrix product AXB with applications to the problem of discerning the existence of the unique solution in a network, Trans. Inst. Electron. Comm. Engrs. Japan, 57A (1974), pp. 834-841 (in Japanese). English translation in Electron. Comm. Japan, 57A (1974), pp. 50-57.
57. N. TOMIZAWA AND M. IRI, An algorithm for determining the rank of a triple matrix product AXB with applications to the problem of discerning the existence of the unique solution in a network, Trans. Inst. Electron. Comm. Engrs. Japan, 57A (1974), pp. 834-841 (in Japanese). English translation in Electron. Comm. Japan, 57A (1974), pp. 50-57.
58. N. TOMIZAWA AND M. IRI, An algorithm for solving the "independent assignment" problem with application to the problem of determining the order of complexity of a network, Trans. Inst. Electron. Comm. Engrs. Japan, 57A (1976), pp. 627-629 (in Japanese).
59. J. UNGAR, A. KRÖNER, AND W. MARQUARDT, Structural analysis of differential-algebraic equation systems: Theory and application, Comput. Chemical Engrg., 19 (1995), pp. 867-882.
60. G. C. VERGHESE AND T. KAILATH, Rational matrix structure, IEEE Trans. Automat. Control, AC-26 (1981), pp. 434-439.
61. M. VIDYASAGAR, Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985.
62. D. J. A. WELSH, Matroid Theory, Academic Press, London, 1976.
63. N. WHITE, ed., Theory of Matroids, Cambridge University Press, London, 1986.
64. J. C. WILLEMS, From time series to linear systems, Part I: Finite dimensional linear time invariant systems, Automatica, 22 (1986), pp. 561-580.
65. J. C. WILLEMS, Dynamical systems, controllability, and observability: A post-modern point of view, in Mathematical System Theory, A. C. Antoulas, ed., Springer-Verlag, Berlin, 1991, pp. 17-37.
66. J. W. VAN DER WOUDE, Causality relations in linear structured systems, in Comm. Control Signal Processing, E. Arikan, ed., Elsevier, New York, 1990, pp. 772-778.
67. J. W. VAN DER WOUDE, On the structure at infinity of a structured system, Linear Algebra Appl., 148 (1991), pp. 145-169.
68. J. W. VAN DER WOUDE, The generic dimension of a minimal realization of an AR-system, Math. Control Signals Systems, 8 (1995), pp. 50-64.