Abstract Spectral analysis systems using gamma radiation to determine the percentage fraction of several compounds in a sample need several stages. The conditioning number of covariance matrices associated with the least-squares system needs to be very close to one so that the determined values are as close as possible to the actual values. Scientific evidence gives us reason to believe that these matrices make it possible to obtain fractions of compounds in the sample close to the true values. This work focuses on the use of the metaheuristic Greedy Randomized Adaptation Search Procedure (GRASP) to estimate percent counts of constituents of a compound represented by prompt gamma ray spectra, based on the minimization of the condition number of the covariance matrix derived from the underlying linear system. For this purpose, steps of GRASP were modified: Algorithms are used for building a solution and for searching in the vicinity of that built solution. Following the stages of building a solution and searching around the built solution, carried out by GRASP, there are attempts to achieve a better quality solution than determined by local search by executing the linking algorithm between two solutions. The results presented in our work improve the condition numbers of the covariance matrices found in articles that are currently published, observing the characteristics of the application data under study. The proposed algorithm obtained the spectral count matrices as input, obtaining average improvements of 31.30% in the condition number of the covariance matrices in the execution of GRASP in the partitioned data. The objective was reached in 7 of the 9 instances. The improvement of the values in condition number was on average of 84.62% in cases of orthogonalized covariance matrices. The average error of the spectral count percentages for each element was 7 . 2 × 1 0 − 3 % . Results lead us to the conclusion that it is possible to obtain covariance matrices that minimize numerical instability in the least squares solution.

中文翻译：

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条件数问题的谱分析与优化

摘要 使用伽马辐射来确定样品中几种化合物的百分比分数的光谱分析系统需要几个阶段。与最小二乘系统相关联的协方差矩阵的条件数需要非常接近于 1，以便确定的值尽可能接近实际值。科学证据使我们有理由相信，这些矩阵可以使样品中获得接近真实值的化合物分数成为可能。这项工作的重点是使用元启发式贪婪随机适应搜索程序 (GRASP) 来估计由即时伽马射线光谱表示的化合物成分的百分比计数，基于从底层线性推导出的协方差矩阵的条件数的最小化。系统。以此目的，修改了 GRASP 的步骤：算法用于构建解决方案并在构建的解决方案附近进行搜索。在构建解决方案和围绕构建解决方案进行搜索的阶段之后，由 GRASP 执行，尝试通过执行两个解决方案之间的链接算法来获得比本地搜索确定的更好质量的解决方案。我们工作中提出的结果改进了当前发表的文章中发现的协方差矩阵的条件数，观察了所研究的应用程序数据的特征。该算法以频谱计数矩阵为输入，在分区数据中执行GRASP时协方差矩阵的条件数平均提高了31.30%。9 个实例中有 7 个达到了目标。在正交协方差矩阵的情况下，条件数值的改进平均为 84.62%。每个元素的光谱计数百分比的平均误差为 7 . 2 × 1 0 − 3 % 。结果使我们得出结论，可以在最小二乘解中获得最小化数值不稳定性的协方差矩阵。