Calculating near-singular eigenvalues of the neutron transport operator with arbitrary order anisotropic scattering

The homogeneous one-speed neutron transport equation in plane geometry can be solved using a separation of variables technique. In the case of isotropic scattering and a sub-critical system, it is well known [15,5] that there are exactly two discrete eigenvalues with opposite sign associated to this separation procedure. In the case of arbitrary order anisotropic scattering, there can be many more of those plus–minus pairs of discrete eigenvalues. When in a subsequent step, the corresponding eigenfunctions are used as a basis set for the expansion of a general solution to the neutron transport equation, it is of utmost importance to be able to find all discrete eigenvalues in order to have a complete set. In this paper we briefly describe the three step procedure we have developed to calculate all discrete eigenvalues. During numerical tests with this procedure, we found that there exist cases where there is a discrete eigenvalue located extremely close to the singular point at unity. The main part of this paper is devoted to the description of how we needed to modify our routines in order to calculate these so-called near-singular eigenvalues. Our most important boundary condition was that we did not wish to resort to high-precision fixed point arithmetic but would solely rely on IEEE 754 double precision arithmetic.