Spectral analysis and optimization of the condition number problem.

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. [ABSTRACT FROM AUTHOR]