An efficient algorithm for calculating taylor polynomials of implicit functions

Abstract

An iterative procedure is introduced which generates the Taylor expansion of an implicitly defined function , that satisfies an equation F(x, f(x)) = 0. In particular the mth iterate is precisely the mth degree Taylor polynomial of f(x). The algorithm is implemented by using a symbolic manipulation language. Examples are presented which illustrate the pertinent features of the method and the procedure is employed to solve some transcendental equations that arise in neutron transport calculations.

Keywords:

• Implicit function
• fixed point
• contractive map
• Taylor polynomials
• symbolic processor
• neutron transport