Orthogonal polynomials, Gaussian quadratures, and PDEs

them particularly important in solvinG physical problems. Also, Gaussian integration provides a highly accurate and efficient algorithm for integrating functions. The value of the methods I describe in this installment of “Computing Prescriptions” depends on the basic assumption that a finite-order polynomial can effectively approximate a function. Therefore, a finite sum of orthogonal polynomials can accurately represent this function. By using the ideas of Gaussian integration, a function can be integrated or expanded in terms of orthogonal polynomials. I discovered that many of my colleagues thought that Gaussian quadrature required finding the necessary table and entering it into the computer. The inclusion of these tables in the Handbook of Mathematical Functions1 tends to obscure the fact that they are essentially irrelevant, owing to the modern workstation’s computing power and to a method that Herbert Wilf devised.2 (Although Wilf’s method is wellknown to numerical-analysis professionals, it is not wellknown to physicists and engineers. Despite having taught courses in numerical analysis for the last 30 years, I learned of this method only a few years ago. I also discovered that I was not alone in my ignorance. For example, only in the second edition of Numerical Recipes3 is it even mentioned.) Using Wilf’s method, you can generate the nodes and weights for Gaussian quadrature quickly and simply from the known recurrence relations. This article presents a formulation of this method in terms of eigenvalues and eigenvectors of an N × N matrix. I attempt to clearly demonstrate why the Gaussian quadrature works and how to use these quantities to produce polynomial expansions, showing their application to PDEs. I also explain how to use the symmetry of some polynomials to reduce computer time by a factor of four.