Making any radiosity matrix symmetric positive definite

The present work demonstrates how a device known in numerical analysis as a ''pre-conditioner'' can transform any radiosity matrix into a symmetric positive definite matrix, regardless of the possibly different sizes of the elements that discretize the surfaces of the environment. This device also allows for the use of other methods to accelerate the convergence, for instance, ''conjugate grandients'' or ''Krylov's subspaces.'' The theorem that every radiosity matrix may be transformed into a symmetric positive defininte matrix regardless of the different sizes of the elements has been proved. Some of the acceleration methods considered here, for example, successive over-relaxation (SOR), require either the exact value or a lower bound for the smallest eigenvalue of the resulting symmetric radiosity matrix. Additional theorems, already proved, give the exact value of the smallest eigenvalue for special environments, and lower bounds for all other environments.