ON A CLASS OF GAUSS-LIKE QUADRATURE-RULES

We consider a problem that arises in the evaluation of computer graphi cs illumination models. In particular, there is a need to find a finit e set of wavelengths at which the illumination model should be evaluat ed. The result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emana ting from a point in the scene. These values are then used to determin e the RGB coordinates of the light by evaluating three definite integr als, each with a common integrand (the SPD) and interval of integratio n but with distinct weight functions. We develop a method for selectin g the sample wavelengths in an optimal manner. More abstractly, we exa mine the problem of numerically evaluating a set of m definite integra ls taken with respect to distinct weight functions but related by a co mmon integrand and interval of integration. It is shown that when m gr eater-than-or-equal-to 3 it is not efficient to use a set of m Gauss r ules because valuable information is wasted. We go on to extend the no tions used in Gaussian quadrature to find an optimal set of shared abc issas that maximize precision in a well-defined sense. The classical G auss rules come out as the special case m = 1 and some analysis is giv en concerning the existence of these rules when m > 1. In particular, we give conditions on the weight functions that are sufficient to guar antee that the shared abcissas are real, distinct, and lie in the inte rval of integration. Finally. we examine some computational strategies for constructing these rules.