Some thought on Numerical Integration Based on Interpolation

Carpathian Journal of Mathematics

In this paper we have considered the asymptotic expressions for remainder term of quadrature formulas of the interpolator type. We derive some corrected versions of the quadrature formulas of interpolatory type, which provide a better approximation accuracy than the original rules. A method to improve the degree of exactness of the quadrature formulas is also considered. A numerical example of the proposed method is given.

International Journal for Research in Applied Science and Engineering Technology -IJRASET, 2020

In this paper we will come across introduction to interpolation and calculus of finite differences. It further includes various polynomial interpolation methods like that of Lagrange's, Newton's forward, and backward & central difference method. These help us to calculate any number of numerical integrations with minimal error. The main idea lies in increasing the coefficients rather than an interval .In order to reduce the numerical computations a formula has been derived from Newton's interpolation method. Application of this formula can be seen and is formulated below.

In this chapter we are going to explore various ways for approximating the integral of a function over a given domain. There are various reasons as of why such approximations can be useful. First, not every function can be analytically integrated. Second, even if a closed integration formula exists, it might still not be the most efficient way of calculating the integral. In addition, it can happen that we need to integrate an unknown function, in which only some samples of the function are known. In order to gain some insight on numerical integration, it is natural to review Rie-mann integration, a framework that can be viewed as an approach for approximating integrals. We assume that f (x) is a bounded function defined on [a, b] and that {x 0 ,. .. , x n } is a partition (P) of [a, b]. For each i we let

Advanced Journal of Graduate Research, 2019

This paper presents a numerical technique for solving fractional integrals of functions by employing the trapezoidal rule in conjunction with the finite difference scheme. The proposed scheme is only a simple modification of the trapezoidal rule, in which it is treated as an algorithm in a sequence of small intervals for finding accurate approximate solutions to the corresponding problems. This method was applied to solve fractional integral of arbitrary order α > 0 for various values of alpha. The fractional integrals are described in the Riemann-Liouville sense. Figurative comparisons and error analysis between the exact value, two-point and three-point central difference formulae reveal that this modified method is active and convenient.

Scientia et Technica

Using the Intermediate Value Theorem we demonstrate the rules of Trapeze and Simpson's. Demonstrations with this approach and its generalization to new formulas are less laborious than those resulting from methods such as polynomial interpolation or Gaussian quadrature. In addition, we extend the theory of approximate integration by finding new approximate integration formulas. The methodology we used to obtain this generalization was to use the definition of the integral defined by Riemann sums. Each Riemann sum provides an approximation of the result of an integral. With the help of the Intermediate Value Theorem and a detailed analysis of the Middle Point, Trapezoidal and Simpson Rules we note that these rules of numerical integration are Riemann sums. The results we obtain with this analysis allowed us to generalize each of the rules mentioned above and obtain new rules of approximation of integrals. Since each of the rules we obtained uses a point in the interval we have ca...

Numerical Algorithms, 1993

In this paper product quadratures based on quasi-interpolating splines are proposed for the numerical evaluation of integrals with anL 1-kernel and of Cauchy Principal Value integrals.

Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1972

Applied Mathematics and Sciences: An International Journal, 2019

The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson's 1/3, and Simpson's 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson's 1/3 is more effective and accurate subdivision is only even the when the condition of for solving a definite integral.

Archives of Computational Methods in Engineering, 2021

This work develops formulas for numerical integration with spline interpolation. The new formulas are shown to be alternatives to the Newton-Cotes integration formulas. These methods have important application in integration of tables or for discrete functions with constant steps. An error analysis of the technique was conducted. A new type of spline interpolation is proposed in which a polynomial passes through more than two tabulated points. The results show that the proposed formulas for numerical integration methods have high precision and absolute stability. The obtained methods can be used for the integration of stiff equations. This paper opens a new field of research on numerical integration formulas using splines.