Electronic Edition on a Class of Gauss-like Quadrature Rules

We consider a problem that arises in the evaluation of computer graphics illumination models. In particular, there is a need to nd a nite set of wavelengths at which the illumination model should be evaluated. The result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emanating from a point in the scene. These values are then used to determine the RGB coordinates of the light by evaluating three deenite integrals, each with a common integrand (the SPD) and interval of integration but with distinct weight functions. We develop a method for selecting the sample wavelengths in an optimal manner. More abstractly, we examine the problem of numerically evaluating a set of m deenite integrals taken with respect to distinct weight functions but related by a common integrand and interval of integration. It is shown that when m 3 it is not eecient to use a set of m Gauss rules because valuable information is wasted. We go on to extend the notions used in Gaussian quadrature to nd an optimal set of shared abcissas that maximize precision in a well-deened sense. The classical Gauss rules come out as the special case m = 1 and some analysis is given concerning the existence of these rules when m > 1. In particular, we give conditions on the weight functions that are suucient to guarantee that the shared abcissas are real, distinct, and lie in the interval of integration. Finally, we examine some computational strategies for constructing these rules.