Numerical Evaluation of Integrals and Derivatives

Abstract

The numerical evaluation of integrals referred to as a quadrature is an important aspect of a large number of applied problems in science and engineering. In Chap. 2, we derived several different methods for the numerical evaluation of integrals. These include the trapezoidal and Simpson’s rules, the higher order Newton-Cotes algorithms, the Clenshaw-Curtis scheme and the Gauss quadrature methods based on classical and nonclassical polynomials. In this chapter, general principles for the accurate and efficient numerical evaluation of integrals that occur in the modeling of physical systems are provided. This is the basis for an efficient numerical method of solution of integral equations discussed in Chap. 5. The physical systems considered vary considerably from section to section and a brief introduction is provided in each case with numerous references to textbooks and current research publications. We consider radial integrals that occur in density functional theory, integrals for chemical and nuclear fusion rate coefficients and also for the solution of the Boltzmann equation. The numerical evaluation of matrix elements in kinetic theory and quantum mechanics is also presented with important implications for pseudospectral methods. The latter section of the chapter is devoted to the pseudospectral method for numerical differentiation based on the Lagrange and Sinc interpolants. The numerical solution of Sturm-Liouville differential eigenvalue problems for the classical polynomials is also presented.