对称非负可约矩阵的最大特征值算法及应用

发布时间：2018-04-16 17:37

本文选题：对称非负可约矩阵 + 最大特征值　；参考：《曲阜师范大学》2017年硕士论文

【摘要】：矩阵的特征值理论是计算数学中最重要的研究问题之一,广泛应用于经济、工程和军事等领域,并且大多数实际问题最后常常归结为矩阵的最大特征值问题.因此,矩阵的最大特征值计算就变得尤为重要.许多学者对非负不可约矩阵设计了高效的求解算法.实际问题的计算中,对于高维矩阵,要判断其可约性,是及其花费时间的.所以我们的目的就是给出一种求解非负可约矩阵最大特征值的算法.本文基于非负不可约矩阵的最大特征值的研究,我们把已有结论和算法推广到对称非负可约矩阵上,给出计算对称可约矩阵最大特征值的算法,进一步,把算法应用到H-矩阵以及Z-矩阵正定性的判定上.第一章介绍了可约与不可约矩阵的一些基础知识以及求解非负不可约矩阵最大特征值的方法.第二章基于非负不可约矩阵的最大特征值的对角变换算法的研究,提出了求解对称非负可约矩阵的最大特征值的算法.该算法既不需要判断矩阵的可约性,也不需要分解矩阵.我们给出算法收敛性的证明,并给出数值例子说明了算法的可行性.最后,把算法应用到H-矩阵的判定上.第三章结合非负不可约矩阵最大特征值的幂算法的研究,给出了一个求解对称非负可约矩阵的最大特征值的新算法.新算法在选取初始向量时,要保证各个分量是严格大于零的,并且在每次迭代后,要对向量进行归一化处理.该算法对于任意的对称非负可约矩阵是收敛的,并给出数值实例说明了算法的优越性.进一步,我们给出算法的一个实际应用,即把算法应用到Z-矩阵正定性的判定上.最后,我们对论文进行总结并给出今后研究的方向.
[Abstract]:The eigenvalue theory of matrix is one of the most important research problems in computational mathematics. It is widely used in the fields of economy, engineering and military, and most practical problems are usually reduced to the maximum eigenvalue problem of matrix.Therefore, the calculation of the maximum eigenvalue of a matrix becomes particularly important.Many scholars have designed efficient algorithms for solving nonnegative irreducible matrices.In the calculation of practical problems, it takes time to judge the reducibility of high dimensional matrices.So our aim is to give an algorithm to solve the maximum eigenvalue of nonnegative reducible matrix.In this paper, based on the study of the maximum eigenvalues of nonnegative irreducible matrices, we extend the existing conclusions and algorithms to symmetric nonnegative reducible matrices, and give an algorithm to calculate the maximum eigenvalues of symmetric irreducible matrices.The algorithm is applied to determine the positive definiteness of H-matrix and Z-matrix.The first chapter introduces some basic knowledge of reducible and irreducible matrices and the method of solving the maximum eigenvalues of nonnegative irreducible matrices.In chapter 2, based on the study of diagonal transformation algorithm for the maximum eigenvalues of nonnegative irreducible matrices, an algorithm for solving the maximum eigenvalues of symmetric nonnegative reducible matrices is proposed.The algorithm needs neither the reducibility of judgment matrices nor the decomposition of matrices.We prove the convergence of the algorithm and give a numerical example to illustrate the feasibility of the algorithm.Finally, the algorithm is applied to the judgment of H-matrix.In chapter 3, a new algorithm for solving the maximum eigenvalue of symmetric nonnegative reducible matrix is presented, which combines the power algorithm of the maximum eigenvalue of nonnegative irreducible matrix.When selecting initial vectors, the new algorithm should ensure that each component is strictly greater than zero, and normalize the vectors after each iteration.The algorithm is convergent for any symmetric nonnegative reducible matrix. A numerical example is given to illustrate the superiority of the algorithm.Furthermore, we give a practical application of the algorithm, that is, the algorithm is applied to the determination of the positive definiteness of the Z-matrix.Finally, we summarize the paper and give the direction of future research.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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