On Incorporating Linear Constraints into H-Matrix Calculations

Some years ago Kriese and Siewert [1] used an idea of Pahor [2], and Pahor and Larson [3], which can be seen to be closely related to the computational scheme suggested by Shure and Natelson [4], to incorporate the linear constraint into the calculation of the H matrix applicable to the scattering of polarized light. It was found [ 1 ] that for weakly absorbing media, co ~ 1, the developed equation for the A matrix converged by iteration much more quickly than the usual H equation, which as discussed by Lenoble [5] appeared, from a numerical point-of-view, not to converge at all. This same approach of developing an equation for a function simply related to the H matrix was also used in neutron-transport studies by Kriese et al. [6] whose L equation is very similar to the mentioned A equation. In a work devoted to the scalar version of the L equation [7], it was shown that the non-linear L equation had a unique solution that yielded the correct H function. The question of how to improve the calculation of the H matrix has recently been discussed by Mullikin [8]. In addition to the fact that the L equation has led to a more expedient method for computing H matrices, we note that Bowden and Zweifel [9] were able to use the L equation for one-speed theory to prove, from the functional analysis approach, the existence and uniqueness of the scalar H function, as defined by the non-linear H equation and one linear constraint, for multiplying media, c > 1. Thus, since to date only the singular integral equation approach [10] has been able to deal with factorizations of dispersion matrices for multiplying media, the utility of the L equation appears to be greater than one only of computational merit. Here we wish to discuss the general case for H matrices when there may be more than one linear constraint and to show that the developed L equation has a unique solution that yields the desired H matrix. We consider first that the dispersion matrix can be written as