The Applications Of α-polynomial

By reviewing the conventional polynomial, a summary is proposed which contains theconcepts, properties and applications of the conventional polynomial. It includes mainly asfollow: the concept of the conventional polynomial interpolation; various kinds of theconventional polynomial interpolation, the formulas of the numerical integration.Then, by giving the concept of α-polynomial and taking the knowledge of theconventional polynomial as reference, some corresponding properties and applications ofα-polynomial are obtained. They are mainly as follow: Lagrange interpolation based onα-polynomial and its optimization, Newton’s interpolatory divided difference formulabased on α-polynomial and its optimization, α-Hermite interpolation and itsoptimization, piecewise-polynomial interpolation based on α-polynomial and itsoptimization; α-polynomial interpolatory differentiation formulas; Trapezoidal rule andSimpson rule based on α-polynomial interpolation and their optimization, compositeTrapezoidal rule based on α-polynomial interpolation and its optimization.Finally, the numerical examples for these applications are presented. There are thecompassions between α-polynomial and the conventional polynomial in the numericalexamples. The advantages of α-polynomial are seen from the examples easily.