Newton’s Method Estimates from Data at One Point

Conference paper


Newton’s method and its modifications have long played a central role in finding solutions of non-linear equations and systems. The work of Kantorovich has been seminal in extending and codifying Newton’s method. Kantorovich’s approach, which dominates the literature in this area, has these features: (a) weak differentiability hypotheses are made on the system, e.g., the map is C 2 on some domain in a Banach space; (b) derivative bounds are supposed to exist over the whole of this domain. In contrast, here strong hypotheses on differentiability are made; analyticity is assumed. On the other hand, we deduce consequences from data at a single point. This point of view has valuable features for computation and its theory. Theorems similar to ours could probably be deduced with the Kantorovich theory as a starting point; however, we have found it useful to start afresh.


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  1. [1]
    J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960.zbMATHGoogle Scholar
  2. [2]
    W. Gregg and R. Tapia, “Optimal error bounds for the Newton-Kantorovich theorem,” SIAM J. Numer. Anal., 11 (1974), 10–13.MathSciNetCrossRefGoogle Scholar
  3. [3]
    L. Kantorovich and G. Akilov, Functional Analysis in Normed Spaces, MacMillan, New York, 1964.zbMATHGoogle Scholar
  4. [4]
    M. Kim, Ph.D. Thesis, City University of New York, to appear, 1985.Google Scholar
  5. [5]
    S. Lang, Real Analysis, Addison-Wesley, Reading, Massachusetts, 1983.zbMATHGoogle Scholar
  6. [6]
    Ostrowski, Solutions of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1973.Google Scholar
  7. [7]
    L. Rail, “A note on the convergence of Newton’s method,” SIAM J. Numer. Anal, 11 (1974), 34–36.MathSciNetCrossRefGoogle Scholar
  8. [8]
    J. Renegar, “On the efficiency of Newton’s method in approximating all zeroes of a system of complex polynomials,” Mathematics of Operations Research, to appear.Google Scholar
  9. [9]
    M. Shub and S. Smale, “Computational complexity: on the geometry of polynomials and a theory of cost, part I,” Ann. Sci. Ecole Norm. Sup. 4 serie t, 18 (1985), 107–142.MathSciNetzbMATHGoogle Scholar
  10. [10]
    M. Shub and S. Smale, “Computational complexity: on the geometry of polynomials and a theory of cost, part II,” SIAM J. Computing, 15 (1986), 145–161.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    S. Smale, “The fundamental theorem of algebra and complexity theory,” Bull. Amer. Math. Soc. (N.S.), 4 (1981), 1–36.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    S. Smale, “On the efficiency of algorithms of analysis,” Bull. Amer. Math. Soc. (N.S.), 13 (1985), 87–121.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    J. Traub and H. Wozniakowski, “Convergence and complexity of Newton iteration for operator equations,” J. Assoc. Comp. Mach., 29 (1979), 250–258.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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