New Formulas and Methods for Interpolation, Numerical Differentiation and Numerical Integration

TL;DR

This paper introduces new formulas and methods for interpolation, numerical differentiation, and integration that work with both evenly and unevenly spaced data, offering higher accuracy and computational advantages.

Contribution

It develops generalized interpolation formulas and new differentiation and integration schemes that improve upon existing methods for unevenly spaced data.

Findings

New interpolation formulas generalizing Newton's and Lagrange's methods.

Two novel numerical differentiation methods without polynomial differentiation.

New numerical integration formulas with improved accuracy.

Abstract

We present a new formula for divided difference and few new schemes of divided difference tables in this paper. Through this, we derive new interpolation, numerical differentiation and numerical integration formulas with arbitrary order of accuracy for evanly and unevanly spaced data. First, we study the new interpolation formula which generalizes both Newton's and Lagrange's interpolation formula with the new divided difference table for unevanly spaced points and using this; we derive other interpolation formulas, in terms of differences for evanly spaced data. Second, we study two new different methods of numerical differentiation for both evanly and unevanly spaced points without differentiating the interpolating polynomials or the use of operators. Third, we derive new numerical integration formulas using new differentiation formulas and Taylor formula for both evanly and unevanly spaced data. Basic computer algorithms for few new formulas are given. In Comparison to former polynomial interpolation, numerical differentiation and numerical integration formuals, these new formulas have some featured advantages for approximating functional values, numerical derivatives of higher order and approximate integral values for evanly and unevanly spaced data.