A simplicial mapping method for locating the zeros of a function
Authors:
C. S. Hsu and W. H. Zhu
Journal:
Quart. Appl. Math. 42 (1984), 41-59
MSC:
Primary 65H05
DOI:
https://doi.org/10.1090/qam/736504
MathSciNet review:
736504
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Abstract: The currently available methods of finding the zeros of a vector function are quite effective but they do require a prior knowledge of the approximate locations of the zeros. Such a knowledge is, however, often not available, hence making the task of finding the zeros very difficult. Developed in this paper is a method which enables us to search the whole domain of interest in a systematic way to locate all the simple zeros approximately. This method complements naturally the various conventional iteration procedures. The method involves (i) partitioning the domain into a system of simplexes and (ii) at each simplex examining a local simplicial mapping induced by the given function in order to test whether a zero is present. Special properties of the barycentric coordinates are used in devising a simple testing procedure which makes the method viable and effective.
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P. S. Aleksandrov, Combinatorial topology, Vol. 1, Graylock Press, Rochester, N.Y., 1956
K. M. Brown, A quadratically convergent Newton-like method based upon Gaussian elimination, SIAM J. Numer. Anal. 6, 560–569 (1969)
S. S. Cairns, Introductory topology, Ronald Press, New York, 1968
B. Carnahanm H. A. Luther and H. O. Wilkes, Applied numerical methods, Wiley, New York, 1969
E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955
C. S. Hsu, A theory of index for dynamical systems of order higher than two, J. Appl. Mech., 47, 421–427 (1980)
C. S. Hsu and W. H. Leung, Singular entities and an index theory for cell functions, J. Math. Anal. Appl. (to appear)
S. Lefschetz, Introduction to topology, Princeton University Press, Princeton, N.J., 1949
K. Nomizu, Fundamentals of linear algebra, McGraw-Hill, New York, 1966
S. Perlis, Theory of matrices, Addison-Wesley, Reading, Mass., 1952
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Article copyright:
© Copyright 1984
American Mathematical Society
