Abstract
If the Jacobian of a differentiable function is singular at a zero of the function, any piecewise linear approximation to it may not have a zero, even when the function has one. In this paper we present a technique for perturbing such functions so that the recent fixed point algorithms that trace zeros of piecewise linear homotopies will succeed in finding a zero of such functions. We also show how to unperturb in case the Jacobian is nonsingular at the solution, and thus not impede the super linear convergence attained by these algorithms.
This work is supported by the grant MCS77-03472 from the National Science Foundation.
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References
L. E. J. Brouwer, "Über Abbildung von Mannigfaltigkeiten," Math. Ann., 71 (1912), pp. 97–115.
B. C. Eaves and R. Saigal, "Homotopies for computing fixed points on unbounded regions," Math. Programming, 3 (1972), pp. 225–237.
J. Katzenelson, "An Algorithm for solving the Nonlinear Resistor Networks," Bell Telephone Tech. J., 44 (1965), pp. 1605–1620.
R. B. Kellogg, T. Y. Li and J. Yorke, "A constructive proof of the Brouwer fixed point theorem and computational results," SIAM J. Numer. Analysis, 13 (1976), pp. 473–483.
H. W. Kuhn and J. G. MacKinnon, "Sandwich Method for Finding Fixed Points," J.O.T.A., 17 (1975), pp. 189–204.
O. H. Merrill, "Applications and Extensions of an Algorithm that Computes Fixed Points of Certain Upper Semi-continuous Point-to-set Mappings," Ph.D. Dissertation, Univ. of Michigan, Ann Arbor, Michigan, 1972.
J. M. Ortega and W. C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
R. Saigal, "On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting," Math. of Operations Res., 2 (1977), pp. 108–124.
R. Saigal, "Fixed Point Computing Methods," Encyclopedia of Computer Science and Technology, Marcel Dekker, New York, 1977.
R. Saigal, "On piecewise linear approximations to smooth mappings," to appear in Math. of Operations Res.
R. Saigal and Y. S. Shin, "Perturbations in fixed point algorithms," in preparation.
H. E. Scarf, "The approximation of fixed points of a continuous mapping," SIAM J. Appl. Math., 15 (1967), pp. 1328–1343.
S. Smale, "A convergent process of price adjustment and global Newton methods," J. Math. Econ., 3 (1976), pp. 107–120.
M. J. Todd, The Computation of Fixed Points and Applications, Springer-Verlag, New York, 1976.
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Saigal, R., Shin, Y.S. (1979). Perturbations in fixed point algorithms. In: Peitgen, HO., Walther, HO. (eds) Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics, vol 730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064328
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DOI: https://doi.org/10.1007/BFb0064328
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