Nonlinear Recurrence Relations for Classical Orthogonal Polynomials

1958 ◽  
Vol 65 (3) ◽  
pp. 195 ◽  
Author(s):  
T. S. Chihara
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relations ◽  
Classical Orthogonal Polynomials

Numerical quadrature

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Recurrence Relations ◽  
Classical Mechanics ◽  
Numerical Quadrature ◽  
Classical Orthogonal Polynomials ◽  
Gaussian Integration ◽  
Physical Problems ◽  
Important Means ◽  
Improper Integrals

This chapter discusses the numerous applications of numerical quadrature (integration) in classical mechanics, in semiclassical approaches to quantum mechanics, and in statistical mechanics; and then describes several ways of implementing integration in C++, for both proper and improper integrals. Various algorithms are described and analyzed, including simple classical quadrature algorithms as well as those enhanced with speedups and convergence tests. Classical orthogonal polynomials, whose properties are reviewed, are the basis of a sophisticated technique known as Gaussian integration. Practical implementations require the roots of these polynomials, so an algorithm for finding them from three-term recurrence relations is presented. On the computational side, the concept of polymorphism is introduced and exploited (prior to the detailed treatment later in the text). The nondimensionalization of physical problems, which is a common and important means of simplifying a problem, is discussed using Compton scattering and the Schrödinger equation as an example.

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