  Function approximation is a generic process in a variety of computational
problems, from data interpolation to the solution of differential equations and
inverse problems. In this work, a unified approach for such techniques is
demonstrated, by utilizing partial sums of Taylor series in high arithmetic
precision. In particular, the proposed method is capable of interpolation,
extrapolation, numerical differentiation, numerical integration, solution of
ordinary and partial differential equations, and system identification. The
method is based on the utilization of Taylor polynomials, by exploiting some
hundreds of computer digits, resulting in highly accurate calculations.
Interestingly, some well-known problems were found to reason by calculations
accuracy, and not methodological inefficiencies, as supposed. In particular,
the approximation errors are precisely predictable, the Runge phenomenon is
eliminated and the extrapolation extent may a-priory be anticipated. The
attained polynomials offer a precise representation of the unknown system as
well as its radius of convergence, which provide a rigor estimation of the
prediction ability. The approximation errors have comprehensively been
analyzed, for a variety of calculation digits and test problems.
