Estimation of the calculation accuracy in the problem of partial identification of a substance

Under consideration is the problem of estimating the accuracy of calculations in the problem of partial identification of the chemical composition of an unknown medium from the results of repeated irradiation of this medium by collimated X-ray fluxes at various energies. The mathematical formulation of the identification problem is presented together with its comparison with a similar problem of finding the chemical composition of an unknown medium. At the first stage of solution, both problems are reduced to studying the singular numbers for a system of algebraic equations that is linear in the products of unknowns.The dimension of this system equals the number of the chemical elements that by assumption can compose the unknown medium. The main role in the identification is played by the intersection of all possible perturbation ellipsoids of the system solution. It is established that the «minimum diameter» of the intersection decreases as the dimension of the problem grows. Thus, in many cases, the solution error can decrease when the problem dimension increases. This result significantly distinguishes the problem of identification of a substance from the problem of finding the chemical composition. The method we propose for solving the identification problem allows us to obtain the set of energy values at which the error of the problem solution is minimal.