Function inverse polynomial interpolation

Calculate the prediction of function zeroing when an inverse polynomial interpolation and an inverse rational interpolation are used. To perform an inverse interpolation is sufficient to invert the independent and dependent variables t and y. In other words, the objective is to find the value of t such that y = 0. 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

An inverse interpolation can be effectively exploited using a rational function rather than a polynomial and the Bulirsch-Stoer algorithm rather than the Neville method. 

---