Abstract

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.

中文翻译：

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物质部分鉴定问题中计算精度的估计

摘要

正在考虑的问题是，根据在各种能量下通过准直X射线通量对该介质进行反复辐照的结果来部分识别未知介质的化学组成的问题中，估计计算准确性的问题。提出了识别问题的数学公式，并与发现未知介质化学成分的类似问题进行了比较。在解决方案的第一阶段，这两个问题都简化为研究在未知数乘积中线性的代数方程组的奇异数。该系统的尺寸等于假设可以构成未知介质的化学元素的数量。识别的主要作用是系统解决方案所有可能的扰动椭圆​​的交集。可以确定的是，交叉口的“最小直径”随着问题尺寸的增大而减小。因此，在许多情况下，当问题尺寸增大时，求解误差会减小。该结果显着地将物质的识别问题与寻找化学组成的问题区分开。我们提出的解决识别问题的方法使我们能够获得问题解决方案的误差最小的一组能量值。当问题维度增加时，解决方案错误会减少。该结果显着地将物质的识别问题与寻找化学组成的问题区分开。我们提出的解决识别问题的方法使我们能够获得问题解决方案的误差最小的一组能量值。当问题维度增加时，解决方案错误会减少。该结果显着地将物质的识别问题与寻找化学组成的问题区分开。我们提出的解决识别问题的方法使我们能够获得问题解决方案的误差最小的一组能量值。