Publications of

Tien-Yien Li

1. Existence of solutions for ordinary differential equation in Banach spaces,
   J. Differential Equations, Vol. 18, (1975), No. 1, pp. 29-40.
    

2. Bounds for the periods of periodic solutions of differential delay equations,
   J.  Math. Anal. Appl., Vol. 49, (1975), pp. 124-129.
    

3. Period three implies chaos  (with J.A. Yorke),
   Amer. Math. Monthly, Vol. 82, (1975), No. 10, pp. 985-992.
    

4. The simplest dynamical system  (with J.A. Yorke),
   Dynamical Systems, Academic Press, New York (1976), pp. 203-206.
    

5. Finite approximation for the Frobenius-Perron operator—A solution to Ulam’s conjecture
   J. Approximation Theory, 17 (1976), pp. 177-186.
    

6. A constructive proof of the Brouwer fixed point theorem and computational results (with R. B. Kellogg and J.A. Yorke),
   SIAM J. Numer. Anal., 13 (1976), pp. 473-483.
    

7. The numerical solution of Hilbert problem  (with Y. Ikebe and F. Stenger)
   Theory of Approximation with Applications, Academic Press, New York (1976), pp. 338-358.
    

8. Computing the Brouwer fixed point by following the continuation curve, 
   Fixed Point Theory and Its Applications, Academic Press, New York (1976), pp. 131-135.
    

9. A method of continuation for calculating a Brouwer fixed point  (with R. B. Kellogg and J.A. Yorke),
   Fixed points, algorithms and applications, S. Karamardien ed., Academic Press, New York (1977), pp. 133-147.
    

10. Ergodic transformations from an interval into itself, 
    Trans. Amer. Math. Soc., Vol. 235 (1978), pp. 183-192.
     

11. Ergodic maps on [0, 1] and nonlinear pseudo-random number generators  (with J.A. Yorke),
    J. of Nonlinear Analysis, Vol. 2, No. 4 (1978), pp. 473-481.
     

12. The generalized Boole’s transformation is ergodic  (with F. Schweiger),
    Manuscripta Mathematica, 25 (1978), pp. 161-167.
     

13. The elliptic porous slider — A homotopy method  (with L. Watson and C.Y. Wang),
    J. of Applied Mechanics, Vol. 45, No. 2 (1978), pp. 435-436.
     

14. Path following approaches for solving nonlinear equations: homotopy, continuous Newton and projection  (with J.A. Yorke),
    Functional differential equations and approximation of fixed points, (1978) pp. 257-261.
     

15. On the number of solutions to polynomial systems of equations  (with C.B. Garcia),
    SIAM J. Numer. Anal., Vol. 17 (1980), pp. 540-546.
     

16. A simple reliable numerical algorithm for following homotopy paths  (with J.A. Yorke),
    Analysis and Computation of Fixed Points, Academic Press (1980), pp. 73-91.
     

17. On a path following method for systems of equations  (with C.B. Garcia),
    Bull. Inst. Math. Acad. Sinica, Vol. 9, No. 2 (1981), pp. 249-259.
     

18. Odd chaos  (with M. Misiurewicz, G. Pianigiani and J.A. Yorke),
    Physics Letter A, vol. 87A, No. 6 (1982), pp. 271-273.
     

19. No division implies chaos  (with M. Misiurewicz, G. Pianigiani and J.A. Yorke),
    Trans. Amer. Math. Soc., Vol. 273 (1982), pp. 191-199.
     

20. Piecewise smooth continuation  (with J.C. Alexander and J.A. Yorke),
    Homotopy methods and global convergence, Plenum Pub. Co. (1983), pp. 1-14.
     

21. Iterating piecewise expanding maps: Asymptotic dynamics of probability densities  (with J.A. Yorke).
    On locating all the zeros of an analytic function within a bounded domain by revised Relves/Lyness method,  SIAM J. Numer. Anal., Vol. 20, No. 4 (1983), pp. 865-871.
     

22. On Chow, Mallet-Paret and Yorke homotopy for solving system of polynomials,
    Bull. Inst. Math. Acad. Sinica, Vol. 11, No. 3, (1983), pp. 433-437.
     

23. Asymptotic periodicity of the iterates of Markov operators, 
    Trans. Amer. Math. Soc., Vol. 186, No. 2 (1984), pp. 751-764.
     

24. Regularity results for real analytic homotopies  (with J. Mallet-Paret and J.A. Yorke),
    Numer. Math., 46, (1985), pp. 43-50.
     

25. Regularity results for solving systems of polynomials by homotopy method  (with T. Sauer), Numer. Math., 50, (1987), pp. 283-289.
     

26. Numerical solution of a class of deficient polynomial systems  (with T. Sauer and J.A. Yorke), SIAM J. Numer. Anal., Vol. 24, No. 2 (1987), pp. 435-451.
     

27. Homotopy method for generalized eigenvalue problems Ax = _Bx  (with T. Sauer),
    Linear Alg. Appl., Vol. 91 (1987), pp. 65-74.
     

28. Solving polynomial systems, 
    Mathematical Intelligencer, Vol. 9, No. 3 (1987), pp. 33-39.
     

29. The random product homotopy and deficient polynomial systems  (with T. Sauer and J.A. Yorke), Numer. Math., 51, (1987), pp. 481-500.
     

30. Homotopy algorithm for symmetric eigenvalue problems (with N. Rhee), Northeastern
    Math. J., 3(4), (1987), pp. 379-383.
     

31. Homotopy method for general _-matrix problems  (with M. Chu and T. Sauer),
    SIAM J. Matrix Anal. and Appl., Vol. 9, No. 4 (1988), pp. 528-536.
     

32. Numerically determining solutions of systems of polynomial equations  (with T. Sauer and J.A. Yorke),
    A.M.S. Bull., Vol. 18, No. 2 (1988), pp. 173-177.
     

33. Consequences of the Li-Yorke theorem on chaos,  (in Japanese)
    Sugaku Seminar, Vol. 27, No. 10 (1988), pp. 44-47.
     

34. Solving all the isolated zeros of polynomial systems,  (in Chinese)
    Advances in Math., Vol. 17, No. 3 (1988), pp. 260-266.
     

35. The cheater’s homotopy: An efficient procedure for solving systems of polynomial equations  (with T. Sauer and J.A. Yorke),
    SIAM J. Numer. Anal., Vol. 26, No. 5 (1989), pp. 1241-1251.
     

36. Homotopy algorithm for symmetric eigenvalue problems  (with N. Rhee),
    Numer. Math., 55 (1989), pp. 265-280.
     

37. A simple homotopy for solving deficient polynomial systems  (with T. Sauer)
    Japan J. Appl. Math., 6 (1989), pp. 409-419.
     

38. Entropy,  (in Chinese)
    Advances in Math., Vol. 19, No. 3 (1990), pp. 301-320.
     

39. A homotopy for solving the kinematics of the most general six-and-five-degree of freedom manipulators  (with X.Wang),
    Proc. of ASME Conference on Mechanisms, D1 - Vol.25 (1990), pp. 249-252.
     

40. An algorithm based on weighted logarithmic barrier functions for linear complementarity problems  (with Jiu Ding),
    Arabian J. Sci. Eng., Vol. 15, No. 4B (1990), pp. 679-685.
     

41. A polynomial-time predictor-corrector algorithm for linear complementarity problem  (with Jiu Ding),
    SIAM J. Optimization, Vol.1, No.1 (1991), pp. 83-92.
     

42. Parallel homotopy algorithm for symmetric tridiagonal eigenvalue problem  (with H. Zhang and X.H. Sun),
    SIAM J. Sci. Stat. Comput., Vol. 12, No. 3 (1991), pp. 469-487.
     

43. Solving deficient polynomial systems with homotopies which keep the subschemes at infinity invariant  (with X. Wang), Math. Comp., Vol. 56, No. 194 (1991), pp. 693-710.
     

44. Markov finite approximation of Frobenius-Perron operator  (with J. Ding),
    J. Nonlinear Anal., Theory, Methods & Applications, Vol. 17, No. 8 (1991), pp. 759-772.
     

45. Solving eigenvalue problems of real nonsymmetric matrices with real homotopies  (with Z. Zeng and L. Cong),
    SIAM J. Numer. Anal., Vol. 29, No. 1 (1992), pp. 229-248.
     

46. Nonlinear homotopies for solving deficient polynomial system with parameters  (with X. Wang), SIAM J. Numer. Anal., Vol. 29, No. 4 (1992), pp. 1104-1118.
     

47. Error estimates of the Markov finite approximation of the Frobenius-Perron operator  (with C. Chiu and Q. Du),
    J. Nonlinear Anal., Theory, Methods & Applications, Vol. 19, No. 4 (1992), pp. 291-308.
     

48. Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems  (with Z. Zeng),
    Math. Comp., Vol. 59, No. 200 (1992), pp. 483-502.
     

49. Projection solutions of Frobenius-Perron Operator Equations  (with Jiu Ding),
    International J. of Math. & Math. Sci., Vol. 16, No. 3 (1993), pp. 465-484.
     

50. An algorithm for symmetric tridiagonal eigen-problems - divide and conquer with homotopy continuation  (with K. Li),
    SIAM J. Sci. Comput., Vol. 14, No. 3 (1993), pp. 735-751.
     

51. A homotopy algorithm for a symmetric generalized eigenproblem  (with K. Li),
    Numerical Algorithm, 4 (1993), pp. 167-195.
     

52. Solving real polynomial systems with real homotopies  (with X.Wang),
    Math Comp., Vol. 60, No. 202 (1993), pp. 669-680.
     

53. High order approximation of the Frobenius-Perron operator  (with J. Ding and Q. Du),
    Appl. Math. Comput., 53 (1993), pp. 157-171.
     

54. Entropy - An introduction  (with J. Ding),
    Nankai Ser. Pure, Appl. Math. Theoret. Phys., 4, World Sci. Publishing, River Edge, NJ, (1993), pp. 26-54.
     

55. Solving polynomial systems by homotopy continuation methods ,
    Computer Mathematics, Nankai Ser. Pure, Appl. Math, Theoret. Phys., 5, World Sci. Publishing, River Edge, NJ, (1993), pp. 18-35.
     

56. Laguerre’s iteration in solving the symmetric tridiagonal eigenproblem – Revisited  (with Z. Zeng),
    SIAM J. Sci. Comput., Vol. 15 , No. 5 (1994), pp. 1445-1473.
     

57. Homotopy method for the singular symmetric tridiagonal eigenproblem  (with K. Li),
    Missouri J. Math. Sci., Vol. 6, No. 1, (1994) pp. 34-46.
     

58. The spectral analysis of Frobenius-Perron operators (with J. Ding and Q. Du)
    J. Math. Anal. Appl., Vol. 184, No. 2, (1994) pp. 285-301.
     

59. Higher order turning points  (with X. Wang),
    Appl. Math. & Comput., Vol. 64, No. 2, (1994) pp. 155-166.
     

60. An algorithm for the generalized symmetric tridiagonal eigenproblem  (with K. Li and Z. Zeng), Numerical Algorithm, Vol. 8, No. 3, (1994) pp. 269-291.
     

61. An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices  (with N. Rhee and Z. Zeng),
    Numer. Math., Vol. 69, (1995) pp. 283-301.
     

62. A scalable eigenvalue solver for symmetric tridiagonal matrices (with C. Trefftz, P. McKinley and Z. Zeng),
    Parallel Computing, 21 (1995) pp. 1213-1240.
     

63. Parallel homotopy algorithm for symmetric large sparse eigenproblems  (with L. Huang)
     J. Comput. & Appl. Math., 60 (1995) pp. 77-100.
     

64. Lyapunov iterations for solving coupled algebraic Riccati equations for Nash differential games and algebraic Riccati equation of zero-sum games  (with Z. Gajic),
    G. J. Olsder ed., New Trends in Dynamic Games and Applications, Annals of the International Society of Dynamic Games, Vol. 3, Birkhauser, (1995) pp. 333-351.
     

65. The BKK root count in Cn  (with X. Wang),
    Math. Comp. Vol. 65, No. 216, (1996) pp. 1477-1484.
     

66. Quasi-Laguerre iteration in solving symmetric tridiagonal eigenvalue problems  (with Q. Du, M. Jin and Z. Zeng),
    SIAM J. Sci. Comput. Vol. 17, No. 6, (1996) pp. 1347-1368.
     

67. Random product homotopy with minimal BKK bound (with T. Wang and X. Wang),
    J. Renegar, M. Shub and S. Smale ed., The Mathematics of Numerical Analysis, A. M. S., (1996) pp. 503-512.
     

68. Determining the structure of the Jordan normal form of a matrix by symbolic computation (with Z. Zhang and T.Wang),
    Linear Alg. Appl., Vol. 252, (1997) pp. 221-259.
     

69. The Quasi-Laguerre iteration (with Q. Du, M. Jin and Z. Zeng),
    Math. Comp., Vol. 66, No. 217, (1997) pp. 345-361.
     

70. Numerical solution of multivariate polynomial systems by homotopy continuation methods ,
    Acta Numerica (1997), pp. 399-436.
     

71. A parallel method for the banded eigenvalue problem (with K. Li),
    AJSE, Vol. 22, No. 2c, (1997), pp. 31 - 40.
     

72. A convergence rate analysis for Markov finite approximations to a class of Frobenius - Perron operators (with J. Ding),
    Nonlinear Analysis, Vol. 31, No. 5/6, (1998), pp. 765-776.
     

73. Counterexamples to the connectivity conjecture of the mixed cells  (with X. Wang),
    Disc. & Comput. Geom., 20 (1998), pp. 515-521.
     

74. On multivariate Descartes’ rule - a counterexample (with X. Wang),
    Beitr¨age zur Algebra und Geometrie, Vol. 39, No. 1 (1998), pp. 1-5.
     

75. Homotopy algorithm for nonsymmetric eigenvalue problem  (with Z.Zeng)
    SIAM J. Sci. Comput., Vol. 20, No. 5 (1999), pp. 1627-1651.
     

76. Implementing the parallel qusi-Laguerre’s algorithm for symmetric tridiagonal
    eigenproblems (with X. Zou),
    SIAM J. Sci. Comput., Vol. 20, No. 6 (1999), pp. 1954-1963.
     

77. Finding all isolated zeros of polynomial systems in Cn via stable mixed volumes, (with T. Gao and X. Wang)
    J. of Symbolic Comput., Vol. 28 (1999), pp. 187-211.
     

78. Solving polynomial systems by polyhedral homotopies ,
    Taiwanese J. of Math., Vol. 3, No. 3 (1999), pp. 251-279.
     

79. A root-finding method for eigenproblem  (with K. Li),
    AJSE, Vol. 25, No. 2A (2000), pp. 121-130.
     

80. Balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation Methods (with T. Gao, J. Verschelde and M.Wu),
    App. Math. & Comput. 114 (2000), pp. 233-247.
     

81. Mixed volume computation via linear programming (with T. Gao),
    Taiwanese J. of Math., Vol. 4, No. 4 (2000), pp. 599-619.
     

82. Exploring chaos on an interval (with J. Yorke),
    The Chaos Avant-Garde, R. Abraham & Y. Ueda edited, World Scientific Series on Nonlinear Science, Series A, Vol.39 (2000), pp. 201-208.
     

83. Finding mixed cells in the mixed volume computation  (with X. Li),
    Found. Comput. Math., Vol. 1 (2001), pp. 161-181.
     

84. Numerical Schubert calculus by the Pieri homotopy algorithm  (with X. Wang & M. Wu),
    SIAM J. Numer. Anal., Vol. 40, No. 2 (2002), pp. 578-600.
     

85. SLYRB measures: natural invariant measures for chaotic systems (with B. Hunt, J. Kennedy and H. Nusse),
    Physica D, Vol. 170, No. 1 (2002), pp. 50-71.
     

86. Finite approximations of Markov operators (with J. Ding and A. Zhou),
    J. of Comput. & Appl. Math., Vol. 147 (2002), pp. 137-152.
     

87. Mixed volume computation for semi-mixed polynomial systems  (with T. Gao)
    Disc. & Comput. Geom., 29 (2003), pp. 257-277.
     

88. Solving polynomial systems by the homotopy continuation method ,
    Handbook of numerical analysis, Vol. XI (2003), pp. 209-304, Edited by P.G. Ciarlet, North-Holland, Amsterdam.
     

89. Counting real connected components of trinomial curve intersections and mnomial hypersurfaces (with J. M. Rojas and X. Wang),
    Disc. & Comput. Geom. 30 (2003), pp. 379-414.
     

90. Book “The theory of chaotic attractors  (co-edited with B. Hunt, J. Kennedy and H. Nusse), (2004)
    Springer-Verlag, New York.
     

91. Determine whether a numerical solution of a polynomial system is isolated
    (Submitted).
     

92. A rank-revealing method with updating, downdating, and applications (with Z. Zeng
    SIAM J. Matrix Appl., to appear.
     

93. MixedVol: A software package for mixed volume computation  (with T. Gao & M. Wu)
    (Submitted).