Numerical Integration Using Swarm Intelligence Techniques

Numerical Integration Using Swarm Intelligence Techniques

Naceur Khelil (Laboratory of Applied Mathematics, Universite Mohamed Khider de Biskra, Algeria), Leila Djerou (LESIA Laboratory, Universite Mohamed Khider de Biskra, Algeria) and Mohamed Batouche (CCIS-King Saud University, Saudi Arabia)
DOI: 10.4018/978-1-4666-1830-5.ch010

Abstract

This chapter proposes quadrature methods (PSOQF) for approximate calculation of integrals within Particle Swarm Optimization (PSO). PSO is a technique based on the cooperation between particles. The exchange of information between these particles allows to resolve difficult problems. Riemann quadrature formula (RQF) will be discussed first, followed by Trapezoidal quadrature Formula (TQF). Finally, a comparison of these methods presented is given.
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Introduction

The primary purpose of numerical integration (or quadrature) is the evaluation of integrals 978-1-4666-1830-5.ch010.m01which are either impossible or else very difficult to evaluate analytically. Numerical integration is also essential for the evaluation of integrals of functions available only at discrete points. Such functions often arise in the numerical solution of differential equations or from experimental data taken at discrete intervals.

In calculus the integrals are (signed) areas and can be approximated by sums of smaller areas, such as the areas of rectangles. We begin by choosing a partition X that subdivides [a.b]:

The subintervals 978-1-4666-1830-5.ch010.m03 determine the width of each of the approximating rectangles. For the height, we can choose any height of the function 978-1-4666-1830-5.ch010.m04 where978-1-4666-1830-5.ch010.m05

The resulting approximation is:

978-1-4666-1830-5.ch010.m06
(1)

In this chapter we propose a choice (of978-1-4666-1830-5.ch010.m07 using Particle Swarm Optimization.

This chapter is organized as follows. The second section recalls methods of integration the more used. In the third section, a formulation adapted to the strategy of particle swarm optimization and the construction of an algorithm to generate the different agents in a swarm. The fourth section exposes an illustrated example to show how the PSO algorithm can lead to a satisfactory result for numerical integration. In the fifth section, we discuss the future research directions, finally the comments and conclusion are made in sixth section.

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Background

To use (1) in order to approximate integrals in with actual numbers, we need to have a specific 978-1-4666-1830-5.ch010.m08 in each interval 978-1-4666-1830-5.ch010.m09 . Several ways to choose 978-1-4666-1830-5.ch010.m10 are (for example):

  • 1.

    Approximation with Riemann Quadrature Formula (RQF): if we choose 978-1-4666-1830-5.ch010.m11the right-hand point, then the resulting approximation is:978-1-4666-1830-5.ch010.m12

Figure 1.

The left sums, RQF

978-1-4666-1830-5.ch010.f01
  • 2.

    Approximation with Riemann Quadrature Formula (RQF): if we choose 978-1-4666-1830-5.ch010.m13 the left-hand point, then the resulting approximation is:978-1-4666-1830-5.ch010.m14

Figure 2.

The right sums, RQF

978-1-4666-1830-5.ch010.f02

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