Updating a numerical code for the solution of the Kompaneets equation in cosmological context,

In this report we describe the fundamental approach and, in par- ticular, the recent update to recent NAG versions of a numerical code, KYPRIX, specifically written for the solution of the Kompaneets equation in cosmologi- cal context, first implemented in the years 1989-1991, aimed to the very accu- rate computation of the cosmic microwave background spectral distortions under quite general assumptions. Specifically, we discuss the main subdivisions of the code and the most relevant aspects about technical specifications and code imple- mentation. After a presentation of the equation formalism and of the boundary conditions added to the set of ordinary differential equations derived from the original parabolic partial differential equation, we provide details on the adopted space (i.e. dimensionless frequency) grid, on the output results, on the accuracy parameters, and on the used integration routines. The problem of introducing the time dependence of the ratio between electron and photon temperatures and of the radiative Compton scattering term, both of them introducing integral terms in the Kompaneets equation, is addressed in the specific context of the recent NAG versions. Finally, we describe the introduction of the cosmological constant in the terms controlling the general expansion of the universe, in agreement with the fundamental discoveries of the last years.