Quadrature by differentiation
Macnaughton, Robert Frank
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This paper is divided into five sections. It is concerned with the derivation and application of a formula known as Quadrature by Differentiation. Section One derives the basic formula by applying integration by parts to a suitably chosen 2n^th. degree polynomial. By applying this method to a polynomial of degree m + n, Hummel and Seebeck's Generalized Taylor Expansion is obtained and shown identical with the Quadrature Formula when m is set equal to n. Finally the quadrature approximation is proved convergent if f(x) is analytic in a certain domain of the complex plane. Section Two deals with the representation of certain elementary functions using quadrature methods. These expansions, because they have integer coefficients and appear in a rational form, are far easier to compute than the corresponding Maclaurin Series with the same degree of accuracy. Section Three uses quadrature methods to solve ordinary differential equations whose boundary data are given at a single point. The method that is devoloped is a variation of the predictor corrector type. It is very accurate and is easily extended to solve almost every type of initial value problem. Section Four treats the linear "Two Point" and eigenvalue problem. This is accomplished by transforming the given differential equation into a system of linear algebraic relationships between the known and unknown boundary conditions. This section also deals briefly with the non linear "Two Point Problem" suggesting a iterative method, based on the results of Section Three, to obtain the missing boundary data. Section Five improves on something that Quadrature by Differentiation already is; an accurate integration formula. This is achieved by replacing derivatives with central differences. The final result is three integration formulas based only on the tabular values of the function being integrated. Since these formulas are derived using the basic interval, xg< x < xg + h, integration can be extended into s successive intervals using the same or different values of h.
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