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Finding roots of equations involving functions defined by first order algebraic differential equations

  • Daniel Richardson
Part of the Progress in Mathematics book series (PM, volume 94)

Abstract

A method is given for approximating solutions of f (x) = 0, for f in a certain class F of real valued analytic functions of one real variable. The method depends on being able to decide the sign of f (r) for given f in F and rational r. That is, approximation of roots is Turing reduced to the constant problem. The last root problem is evaded: essentially solutions are only found in given finite intervals.

The class of functions F contains x, exp(x), log(x) for x > 0, sin(x) for x in (—π/2, π/2), and is closed under field operations, differentiation and integration. F is built up by successively adding solutions g of first order differential equations q(x, g, g′) = 0, where q is a polynomial whose coefficients may involve previously defined functions.

The main technique used is construction of a false derivative. A false derivative of f (x) is a function f *(x) which is continuous and has the same sign as f′(x) whenever f(x) = 0.

Keywords

Normal Form Order Differential Equation Finite Interval Algebraic Differential Equation Quotient Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Arnon, D. S., and B. Buchberger, (Editors), “Algorithms in real algebraic geometry,” Academic Press, 1988.Google Scholar
  2. [2]
    Davenport, J. H., Computer Algebra for Cylindrical Decomposition, University of Bath computer science technical report 88-110.Google Scholar
  3. [3]
    van den Dries, L., On the elementary theory of restricted elementary functions, The Journal of Symbolic Logic 53 (1988), 769–808.Google Scholar
  4. [4]
    van den Dries, L. and J. Denef, P-adic and real subanalytic sets, Annals of Mathematics 128 (1988), 79–138.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Khovanskii, A. G., Fewnomials and Pfaff manifolds, in “Proc. Int. Congress of Mathematicians,” Warsaw 1983, pp. 549–564.Google Scholar
  6. [6]
    Khovanskii, A.G., On a class of systems of transcendental equations, Soviet Math. Dokl. 22 (1980), 762–765.Google Scholar
  7. [7]
    Lombardi, H., Algebre Elementaire en Temps Polynomial, Thesis, University of Nice, 1989.Google Scholar
  8. [8]
    Richardson, D., Wu’s method and the Khovanskii finiteness theorem, Bath Computer Science, technical report 89-29.Google Scholar
  9. [9]
    Shackell, J., Zero-Equivalence in function fields defined by algebraic differential equations.Google Scholar
  10. [10]
    Shackell, J., A Differential-equations approach to functional equivalence.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Daniel Richardson
    • 1
  1. 1.Department of MathematicsUniversity of BathUK

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