you can compose the matrix of all eigenvectors of from the -th column of each . Denote the matrix of eigenvectors with . Example The diagonalization of can be written as Polynomials and power series The matrix contains the divided difference scheme for the identity function with respect to the nodes , thus contains the divided differences for the power function with exponent . Consequently, you can obtain the divided differences for a polynomial function by applying to the matrix : If and then This is known as Opitz' formula. Now consider increasing the degree of to infinity, i.e. turn the Taylor polynomial into a Taylor series. Let be a function which corresponds to a power series. You can compute the divided difference scheme for by applying the corresponding matrix series to : If and then Alternative characterizations Expanded form With the help of the polynomial function this can be written as Peano form If and , the divided differences can be expressed as where is the -th derivative of the function and is a certain B-spline of degree for the data points , given by the formula This is a consequence of the Peano kernel theorem; it is called the Peano form of the divided differences and is the Peano kernel for the divided differences, all named after Giuseppe Peano. Forward differences When the data points are equidistantly distributed we get the special case called forward differences. They are easier to calculate than the more general divided differences. Given n+1 data points with the forward differences are defined as The relationship between divided differences and forward differences is See also Difference quotient Neville's algorithm Polynomial interpolation Mean value theorem for divided differences Nörlund–Rice integral Pascal's triangle References External links An implementation in Haskell. Finite differences de:Polynominterpolation#Bestimmung der Koeffizienten: Schema der dividierten Differenzen