New Formulas and Methods for Interpolation, Numerical Differentiation and Numerical Integration
For researchers in numerical analysis, the paper presents new theoretical formulas but lacks empirical validation, making it an incremental contribution.
The paper introduces new formulas for divided differences and interpolation, numerical differentiation, and numerical integration for both evenly and unevenly spaced data, claiming advantages over existing methods. No concrete numerical results are provided.
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.