The Lanczos' tau method is a spectral method originally proposed to solve linear differential problems with polynomial coefficients. Generalizations to extending its domain of application to integro-differential, nonlinear and functional-differential problems, among others, have been made. Despite all these important contributions, the method have never been widespread used, mostly as a consequence of the lack of automatic procedures as well as of the numerical instabilities present in their implementations.

The purpose of this undergoing work is three-fold: (i) generalizing the method to a wider class of applications, (ii) overcoming some numerical instabilities and (iii) building a toolbox to efficiently solve problems with the tau method and to allow for the implementation of new procedures based on building blocks. After showing how to use properties of orthogonal polynomials to improve numerical stability of the method, we will illustrate the use of these improvements to compute approximate solution for nonlinear differential and integro-differential problems in the tau sense.