Numerical Methods with MS Excel

's 3 / 1 rule, Simpson's 8 / 3 rule and rule.

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.

International Journal of Innovative Research in Science, Engineering and Technology, 2013

It is observed that most of the numerical problems have been solved by developing algorithm using high level languages ,such as FORTRAN ,C,C++, Visual Basic ect ,which are not so easy to handle by all readers . In this paper we have developed formulae in excel worksheet for solving numerical problems without using high level language which is very familiar and easy accessible to all .

The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.

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.

INDIAN JOURNAL OF PURE AND …, 2000

Pure and Applied Mathematics Journal, 2020

In this paper, we mainly propose the approximate solutions to solve the integration problems numerically using the quadrature method including the Trapezoidal method, Simpson's 1/3 method, and Simpson's 3/8 method. The three proposed methods are quite workable and practically well suitable for solving integration problems. Through the MATLAB program, our numerical solutions are determined as well as compared with the exact values to verify the higher accuracy of the proposed methods. Some numerical examples have been utilized to give the accuracy rate and simple implementation of our methods. In this study, we have compared the performance of our solutions and the computational attempt of our proposed methods. Moreover, we explore and calculate the errors of the three proposed methods for the sake of showing our approximate solution's superiority. Then, among these three methods, we analyzed the approximate errors to prove which method shows more appropriate results. We also demonstrated the approximate results and observed errors to give clear idea graphically. Therefore, from the analysis, we can point out that only the minimum error is in Simpson's 1/3 method which will beneficial for the readers to understand the effectiveness in solving the several numerical integration problems.

The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0) calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.

Abstract: Numerical analysis concerns the development of algorithms for solving various types of problems of mathematics; it is a vast-ranging field having deep interaction with computer science, mathematics, engineering, and the sciences. Numerical analysis mainly consists of Numerical Integration, Numerical Differentiation and finding Roots numerically.

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

In this paper, we will demonstrate how nonadaptive integration quadratures such as Simp- son's Rule, and Trapezoidal's Rule with equally spaced divisions can be easily modifled into adapative quadratures with non-equally spaced divisions by using \matrices or ar- rays". The necessicity of adaptive quadratures depends upon the behavior of a function. These quadratures can be used in treating functions which are monotone, or highly oscil- lating with singularities (see (2)). All these quadratures could not have been experimented without the presence of computer algebra system such as Maple. We assume readers are familiar with basic Maple commands. The following example will show why the adapted quadratures are necessary and how we use matrices to device such quadratures: Example :L et f ( x )= 1 = p xfor x2(0;1), and f(0) = 0. We use Maple to demonstrate