Appl. Math. Inf. Sci. 10, No. 1, 1-9 (2016) 1 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/100101 On the Application of Multicomplex Algebras in Numerical Integration S. Shirinkam 1,∗ , A. Alaeddini 2 and H. R. Millwater 2 1 Department of Mathematics, University of Texas at San Antonio, TX 78249, USA 2 Department of Mechanical Engineering, University of Texas at San Antonio, TX 78249, USA Received: 4 Jun. 2015, Revised: 2 Aug. 2015, Accepted: 3 Aug. 2015 Published online: 1 Jan. 2016 Abstract: In this paper, we propose a methodology to numerically integrate functions using multicomplex algebras and their corresponding matrix representations. The methodology employs multicomplex Taylor series expansion (MCTSE) to adaptively approximate and integrate a function using sufficiently small number of points. We investigate this methodology by presenting three different algorithms for various approximation strategies. We also use numerical studies to demonstrate the performance of the proposed methodology. Keywords: Numerical Integration, Multicomplex Algebra, Taylor Series Expansion. 1 Introduction Numerical integration, also referred to as numerical quadrature, constitutes a broad family of algorithms for calculating the numerical value of a definite integral [13]. There are several reasons for carrying out numerical integration. The integrand function may not be known at some points, it may be difficult or impossible to find an antiderivative, or it may be easier to compute a numerical approximation than to compute the antiderivative. Numerical integration methods can be generally described as combining evaluations of the integrand to get an approximation of the integral [1]. The integrand is evaluated at a finite set of integration points and a weighted sum of these values is used to approximate the integral. The integration points and weights depend on the specific method used and the accuracy required from the approximation. There has been a large body of literature around numerical integration [9]. A large class of methods uses Newton-Cotes formulas, also known as quadrature formulas, which approximate the function with various degrees of polynomials evaluated at equally spaced points, of which the trapezoidal rule and Simpson’s rule are among common examples [13]. Some of these methods have been integrated with Taylor series approximation as proposed in [3]. In addition, a generalization of the trapezoidal rule is Romberg integration, which can yield more accurate results for many fewer function evaluations [1]. Another group of quadrature formulas allow intervals between interpolation points to vary, which includes Gaussian quadrature formulas [5]. When the integrand is smooth, a Gaussian quadrature rule is typically more accurate than a NewtonCotes rule. Other quadrature methods with varying intervals include GaussKronrod and ClenshawCurtis quadrature methods [2] and [4]. The other group of quadrature, known as adaptive quadrature, approximates the function using static quadrature rules on adaptively refined subintervals of the integration domain [7]. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for ”well behaved” integrands, but are also effective for ”badly behaved” integrands for which traditional algorithms tend to fail. There are also other numerical integrations methods based on information theory, which have been developed to simulate information systems such as computer controlled systems, communication systems, and control systems [11]. An important part of the analysis of any numerical integration method is to study the behavior of the approximation as a function of the number of integrand evaluations. Generally, a method that yields a small error ∗ Corresponding author e-mail: sara.shirinkam@utsa.edu c  2016 NSP Natural Sciences Publishing Cor.