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

Manifold Mora Vica 

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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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