Finding roots of equations involving functions defined by first order algebraic differential equations
Daniel R. Richardson · Birkhäuser Boston eBooks · 1991
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. 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.