Метод сингулярного разложения матрицы в задаче нахождения химического состава среды

The problem of finding the chemical composition of a homogeneous medium by the method of multi-energy radiography is considered. The main attention is paid to the choice of optimal energy values of X-ray radiation and to derivation of accurate estimates of the perturbation of the solution of the system of equations for given perturbations of the right-hand side. For this purpose the method of Singular Value Decomposition of the matrix was used. Singular numbers and singular vectors corresponding to a number of groups of chemical elements, often encountered in practice, are calculated. Optimum values of X-ray radiation energy for these groups of elements are found. The results obtained has been compared with the case of overdetermined system solution. Several examples of the solution of the problem for media of different chemical composition have been constructed.