Abstract
An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples.
Highlights
The Lagrange interpolation operator associates to each function its Lagrange interpolating polynomial of degree less than or equal to at distinct nodes 0 on the interval
In formula (2) of [5], a conditioning associated to a representation of the form is introduced, cond where 0 are functionals belonging to the space generated by the evaluation functionals 0 and 0 is a basis of
By formula (4) and Theorem 4 of [5], we have cond cond that is, the conditioning of the Lagrange representation coincides with the Lebesgue function and it is lower than the conditioning of any other representation, and in particular, than the Newton representation
Summary
. Using the Lagrange interpolation formula, the Lagrange interpolation operator can be written in the form where. In formula (2) of [5], a conditioning associated to a representation of the form is introduced, cond (1). Where 0 are functionals belonging to the space generated by the evaluation functionals 0 and 0 is a basis of. The space of polynomials of degree not greater than. That is, the conditioning of the Lagrange representation coincides with the Lebesgue function and it is lower than the conditioning of any other representation, and in particular, than the Newton representation.
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