https://engcourses-uofa.ca/books/numericalanalysis/polynomial-interpolation/newton-interpolating-polynomials/

As stated in the introduction, the matrix formed in Equation 1 can be ill-conditioned and difficult to find an inverse for. A simpler method can be used to find the interpolating polynomial using Newton’s Interpolating Polynomials formula for fitting a polynomial of degree n through n+1 data points (x_i,y_i) with 1 < = I < = n+1:

where the coefficients b_i are defined recursively using the divided differences as follows:

This recursive divided differences are illustrated in the following table:

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