ALGORITHM 652: HOMPACK: A Suite of Codes for Globally Convergent Homotopy Algorithms

ACM Transactions on Mathematical Software (TOMS)

Volumn 13, Issue 3, 1987, Pages 281-310

  Watson, Layne T ^a   Billups, Stephen C ^b   Morgan, Alexander P ^c  

^a UNIVERSITY OF MICHIGAN   (United States)

^b SANDIA NATIONAL LABORATORIES   (United States)

^c GENERAL MOTORS CORPORATION   (United States)

Author keywords

Chow Yorke algorithm; continuation method; curve tracking; fixed point; globally convergent; homotopy methods; nonlinear systems; polynomial systems; zero

Indexed keywords

COMPUTER PROGRAMMING - ALGORITHMS;

GLOBALLY CONVERGENT HOMOTOPY ALGORITHMS; HOMPACK; MATHEMATICAL SOFTWARE; ROOTS OF NONLINEAR EQUATIONS;

MATHEMATICAL TECHNIQUES;

EID: 0023415939     PISSN: 00983500     EISSN: 15577295     Source Type: Journal    
DOI: 10.1145/29380.214343     Document Type: Article

References (39)

1. Billups, S. C. An augmented Jacobian matrix algorithm for tracking homotopy zero curves. Master's thesis, Dept. of Computer Science, Virginia Polytechnic Institute and State University, Blacksburg, Va., Sept. 1985.
2. Brunovskö, P., and Meravö, P. Solving systems of polynomial equations by bounded and real homotopy. Numer. Math. 43 (1984), 397–418.
3. Businger, P., AND Golub, G. H. Linear least squares solutions by Householder transformations. Numer. Math. 7 (1965), 269–276.
4. Chan, T. F., AND Saad, Y. Iterative methods for solving bordered systems with applications to continuation methods. SIAM J. Sci. Stat. Comput. 6 (1985), 438–451.
5. Chow, S. N., Mallet-Paret, J., and Yorke, J. A. Finding zeros of maps: Homotopy methods that are constructive with probability one. Math. Comput. 32 (1978), 887–899. ACM Transactions on Mathematical Software, Vol. 13, No. 3, September 1987.
6. Chow, S. N., Mallet-Paret, J. and Yorke, J. A. A homotopy method for locating all zeros of a system of polynomials. In Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics Vol. 730. H. O. Peitgen and H. O. Walther, Eds. Springer-Verlag, New York, 1979, 228–237.
7. Craig, E. J. Iteration procedures for simultaneous equations. Ph.D. dissertation, MIT, Cambridge, 1954.
8. Dennis, J. E., and Schnabel, R. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, N.J., 1983.
9. Fadeev, D. K., and Fadeeva, V. N. Computational Methods of Linear Algebra. Freeman, London, 1963.
10. Garcia, C. B., and Zangwill, W. I. Finding all solutions to polynomial systems and other systems of equations. Math. Program. 16 (1979), 159–176.
11. Gill, P. E., and Murray, W. Newton type methods for unconstrained and linearly constrained optimization. Math. Program. 7 (1974), 311–350.
12. Haselgrove, C. B. A solution of nonlinear equations and of differential equations with two-point boundary conditions. Comput. J. 4 (1961), 255–259.
13. Hestenes, M. R. The conjugate gradient method for solving linear systems. In Proceedings of the Symposium on Applied Mathematics, 6, Numer. Anal., AMS, New York, 1956, 83–102.
14. Kubicek, M. Dependence of solutions of nonlinear systems on a parameter. ACM Trans. Math. Softw. 2 (1976), 98–107.
15. Matthies, H., and Strang, G. The solution of nonlinear finite element equations. Int. J. Numer. Math. Eng., 14 (1979), 1613–1626.
16. Meintjes, K., and Morgan, A. P. A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22 (1987), 333–361.
17. Mejia, R. CONKUB: A conversational path-follower for systems of nonlinear equations, J. Comput. Phys. 63 (1986), 67–84.
18. Menzel, R., and Schwetlick, H. Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen. Numer. Math. 30 (1978), 65–79.
19. Morgan, A. P. A transformation to avoid solutions at infinity for polynomial systems. Appl. Math. Comput. 18 (1986), 77–86.
20. Morgan, A. P. Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems. Prentice-Hall, Englewood Cliffs, N.J., 1987.
21. Morgan, A. P., and Sommese, A. A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput. 23 (1987).
22. Rheinboldt, W. C. Numerical analysis of continuation methods for nonlinear structural problems. Comput. Struct. 13 (1981), 103–113.
23. Rheinboldt, W. C. Numerical Analysis of Parameterized Nonlinear Equations. Wiley-Inter-science, New York, 1986.
24. Rheinboldt, W. C., and Burkardt, J. V. Algorithm 596: A program for a locally parameterized continuation process. ACM Trans. Math. Softw. 9 (1983), 236–241.
25. Shampine, L. F., and Gordon, M. K. Computer Solution of Ordinary Differential Equations: The Initial Value Problem. W. H. Freeman, San Francisco, 1975.
26. Tsai, L.-W., and Morgan, A. P. Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods. ASME J. Mech. Transmissions, Aut. Des. 107 (1985), 48–57.
27. van der Waerden, B. L. Modern Algebra, Vols. 1 and 2. Frederick Ungar, New York, 1953.
28. Watson, L. T. A globally convergent algorithm for computing fixed points of C2 maps. Appl. Math. Comput. 5 (1979), 297–311.
29. Watson, L. T. Fixed points of C2 maps. J. Comput. Appl. Math. 5 (1979), 131–140.
30. Watson, L. T. Solving the nonlinear complementarity problem by a homotopy method. SIAM J. Contr. Optim. 17 (1979), 36–46.
31. Watson, L. T. An algorithm that is globally convergent with probability one for a class of nonlinear two-point boundary value problems. SIAM J. Numer. Anal. 16 (1979), 394–401.
32. Watson, L. T. Computational experience with the Chow-Yorke algorithm. Math. Program. 19 (1980), 92–101.
33. Watson, L. T. Solving finite difference approximations to nonlinear two-point boundary value problems by a homotopy method. SIAM J. Sci. Stat. Comput. 1 (1980), 467–480.
34. Watson, L. T. Engineering applications of the Chow-Yorke algorithm. Appl. Math. Comput. 9 (1981), 111–133.
35. Watson, L. T. Numerical linear algebra aspects of globally convergent homotopy methods. Tech. Rep. TR-85–14, Dept. of Computer Science, Virginia Polytechnic Institute and State University, Blacksburg, Va., 1985.
36. Watson, L. T., and Fenner, D. Chow-Yorke algorithm for fixed points or zeros of C2 maps. ACM Trans. Math. Softw. 6 (1980), 252–260.
37. Watson, L. T., Kamat, M. P., and Reaser, M. H. A robust hybrid algorithm for computing multiple equilibrium solutions. Eng. Comput. 2 (1985), 30–34.
38. Watson, L. T., and Scott, M. R. Solving spline collocation approximations to nonlinear two-point boundary value problems by a homotopy method. Tech. Rep. TR-84–15, Dept. of Computer Science, Virginia Polytechnic Institute and State University, Blacksburg, Va., 1984.
39. Wright, A. H. Finding all solutions to a system of polynomial equations. Math. Comput. 44 (1985), 125–133.