Inverse Rational Interpolation Method

An exact polynomial interpolation can deteriorate as the polynomial degree increases (Buzzi-Ferraris and Manenti, 2010b), whereas this problem does not arise with rational functions. 

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. 

It is worth remarking that this method cannot be implemented on its own in a program, rather certain additional (and less efficient) methods must be integrated with it to ensure convergence. These additional algorithms should be used whenever the inverse rational function interpolation encounters any difficulty. 

Inverse interpolation with rational functions is very efficient and is the ideal basis for the development of a general function root-finding program, even though it is slightly more complex than the other algorithms. 

In the Bzz Interpolation class, the Neville function allows an interpolation with polynomials to be performed, whereas BulirschStoer performs an interpolation with rational functions. 

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The inverse rational interpolation method is used as the basic method when more than three iterations of the BzzMath classes for function root-finding have been performed. 

If an inverse interpolation method is used and the Bulirsch-Stoer algorithm in particular is applied to rational functions, the differences between the predictions of different rational functions can be calculated in order to check whether they tend to zero as well as to estimate the error arising in using the differences between the various previsions. 

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