Numerical Integration Using Swarm Intelligence Techniques

Introduction

The primary purpose of numerical integration (or quadrature) is the evaluation of integrals which 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 determine the width of each of the approximating rectangles. For the height, we can choose any height of the function where

The resulting approximation is:

In this chapter we propose a choice (of 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.

Background

To use (1) in order to approximate integrals in with actual numbers, we need to have a specific in each interval . Several ways to choose are (for example):

• 1.

  Approximation with Riemann Quadrature Formula (RQF): if we choose the right-hand point, then the resulting approximation is:

Figure 1.

The left sums, RQF

• 2.

  Approximation with Riemann Quadrature Formula (RQF): if we choose the left-hand point, then the resulting approximation is:

Figure 2.

The right sums, RQF