Inverse Polynomial Interpolation Method

Muller s method has to select the new point between the two solutions of the interpolating parabola. This difficulty is enhanced by interpolating the function with a higher degree polynomial. 

However, it is possible to overcome it by adopting an inverse interpolation. In other words, a polynomial in the dependent variable y can be developed. 

Given n + 1 support points, it is possible to calculate the polynomial parameters, approximating t with respect to y  

Since the polynomial is useful to predict the point t = tj+i where y = 0, the Neville method, which does not require any parameter estimation, can be adopted (see Buzzi-Ferraris and Manenti, 2010b). 

The inverse polynomial interpolation method is never used in the BzzMalh library classes dedicated to root-finding. 

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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. 

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