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Linear algorithms for testing the sign stability of a matrix and for findingZ-maximum matchings in acyclic graphs

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Summary

Ann×n real matrixA=(a ij ) isstable if each eigenvalue has negative real part, andsign stable (orqualitatively stable) if each matrix B with the same sign-pattern asA is stable, regardless of the magnitudes ofB's entries. Sign stability is of special interest whenA is associated with certain models from ecology or economics in which the actual magnitudes of thea ij may be very difficult to determine. Using a characterization due to Quirk and Ruppert, and to Jeffries, an efficient algorithm is developed for testing the sign stability ofA. Its time-and-space-complexity are both 0(n 2), and whenA is properly presented that is reduced to 0(max{n, number of nonzero entries ofA}). Part of the algorithm involves maximum matchings, and that subject is treated for its own sake in two final sections.

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References

  1. Aho, A. V., Hopcroft, J.E., Ullman, J.D.: The design and analysis of computer algorithms. Reading, Mass.: Addison-Wesley 1975

    Google Scholar 

  2. Anderson, W. N.: Maximum matching and the rank of a matrix. SIAM J. appl. Math.28, 114–123 (1975)

    Article  Google Scholar 

  3. Berge, C.: Two theorems in graph theory. Proc. Nat. Acad. Sci. USA43, 842–844 (1957)

    Google Scholar 

  4. Chartres, B. A.: Numerical methods for the stability analysis of many-dimensional linear systems. Report IBM E-148/1, Division of Applied Mathematics, Brown University, Providence, R.I., 1959

    Google Scholar 

  5. Clarke, B. L.: Theorems on chemical network stability. J. Chem. Phys.62, 773–775 (1975)

    Article  Google Scholar 

  6. Cockayne, E., Goodman, S., Hedetniemi, S.: A linear algorithm for the domination number of a tree. Inform. Processing Letters4, 41–44 (1975)

    Article  Google Scholar 

  7. Edmonds, J.: Paths, trees and flowers. Canadian J. Math.17, 449–467 (1965)

    Google Scholar 

  8. Even, S., Kariv, O.: An 0(N 2.5) algorithm for maximum matching in general graphs. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, pp. 100–112, IEEE Computer Society, 1975

  9. Gabow, H.: An efficient implementation of Edmonds' algorithm for maximum matchings in graphs. J. Assoc. comput. Machin.23, 221–234 (1976)

    Google Scholar 

  10. Hopcroft, J. E., Karp, R. A.: Ann 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Computing2, 225–231 (1973)

    Article  Google Scholar 

  11. Jeffries, C.: Qualitative stability and digraphs in model ecosystems. Ecology55, 1415–1419 (1974)

    Google Scholar 

  12. Jeffries, C., Klee, V., van den Driessche, P.: When is a matrix sign stable? Canadian J. Math.29, 315–326 (1977)

    Google Scholar 

  13. Lawler, E.: Combinatorial optimization: Networks and matroids. New York: Holt, Rinehart and Winston 1976

    Google Scholar 

  14. Levins, R.: The qualitative analysis of partially specified systems. Ann. New York Acad. Sci.231, 123–138 (1974)

    Google Scholar 

  15. Levins, R.: Problems of signed digraphs in ecological theory. In: Ecosystems analysis and prediction (S. Levin, ed.), pp. 264–277. Philadelphia, Pa.: Society for Industrial and Applied Mathematics 1974

    Google Scholar 

  16. Lewin, M.: On maximal circuits in directed graphs. J. combinat. Theory, Ser. B18, 175–179 (1975)

    Google Scholar 

  17. May, R. M.: Qualitative stability in model ecosystems. Ecology54, 638–641 (1973)

    Google Scholar 

  18. May, R. M.: Stability and complexity in model ecosystems. Princeton, N.J.: Princeton Univ. Press 1973

    Google Scholar 

  19. Maybee, J., Quirk, J.: Qualitative problems in matrix theory. SIAM Review11, 30–51 (1969)

    Article  Google Scholar 

  20. Norman, R. Z., Rabin, M. O.: An algorithm for a minimum cover of a graph. Proc. Amer. math. Soc.10, 315–319 (1959)

    Google Scholar 

  21. Quirk, J., Ruppert, R.: Qualitative economics and the stability of equilibrium. Review economic Studies32, 311–326 (1965)

    Google Scholar 

  22. Roberts, F. S., Brown, T. A.: Signed digraphs and the energy crisis. Amer. math. Monthly82, 577–593 (1975)

    Google Scholar 

  23. Schwarz, H. R.: Ein Verfahren zur Stabilitätsfrage bei Matrizen-Eigenwertproblemen. Z. angew. Math. Phys.7, 473–500 (1956)

    Google Scholar 

  24. Tarjan, R. E.: Depth first search and linear graph algorithms. SIAM J. Computing1, 146–160 (1972)

    Article  Google Scholar 

  25. Tyson, J.: Classification of instabilities in chemical reaction systems. J. Chem. Phys.62, 1010–1015 (1975)

    Article  Google Scholar 

  26. Yorke, J. A., Anderson, W. N.: Predator-prey patterns. Proc. nat. Acad. Sci. USA70, 2069–2071 (1973)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    Victor Klee

  2. Department of Mathematics, University of Victoria, V8W 2Y2, Victoria, B.C., Canada

    Pauline van den Driessche

Authors
  1. Victor Klee
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  2. Pauline van den Driessche
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Klee, V., van den Driessche, P. Linear algorithms for testing the sign stability of a matrix and for findingZ-maximum matchings in acyclic graphs. Numer. Math. 28, 273–285 (1977). https://doi.org/10.1007/BF01389968

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  • DOI: https://doi.org/10.1007/BF01389968

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