本文選題：對(duì)稱非負(fù)可約矩陣 + 最大特征值�。� 參考：《曲阜師范大學(xué)》2017年碩士論文

【摘要】：矩陣的特征值理論是計(jì)算數(shù)學(xué)中最重要的研究問(wèn)題之一,廣泛應(yīng)用于經(jīng)濟(jì)、工程和軍事等領(lǐng)域,并且大多數(shù)實(shí)際問(wèn)題最后常常歸結(jié)為矩陣的最大特征值問(wèn)題.因此,矩陣的最大特征值計(jì)算就變得尤為重要.許多學(xué)者對(duì)非負(fù)不可約矩陣設(shè)計(jì)了高效的求解算法.實(shí)際問(wèn)題的計(jì)算中,對(duì)于高維矩陣,要判斷其可約性,是及其花費(fèi)時(shí)間的.所以我們的目的就是給出一種求解非負(fù)可約矩陣最大特征值的算法.本文基于非負(fù)不可約矩陣的最大特征值的研究,我們把已有結(jié)論和算法推廣到對(duì)稱非負(fù)可約矩陣上,給出計(jì)算對(duì)稱可約矩陣最大特征值的算法,進(jìn)一步,把算法應(yīng)用到H-矩陣以及Z-矩陣正定性的判定上.第一章介紹了可約與不可約矩陣的一些基礎(chǔ)知識(shí)以及求解非負(fù)不可約矩陣最大特征值的方法.第二章基于非負(fù)不可約矩陣的最大特征值的對(duì)角變換算法的研究,提出了求解對(duì)稱非負(fù)可約矩陣的最大特征值的算法.該算法既不需要判斷矩陣的可約性,也不需要分解矩陣.我們給出算法收斂性的證明,并給出數(shù)值例子說(shuō)明了算法的可行性.最后,把算法應(yīng)用到H-矩陣的判定上.第三章結(jié)合非負(fù)不可約矩陣最大特征值的冪算法的研究,給出了一個(gè)求解對(duì)稱非負(fù)可約矩陣的最大特征值的新算法.新算法在選取初始向量時(shí),要保證各個(gè)分量是嚴(yán)格大于零的,并且在每次迭代后,要對(duì)向量進(jìn)行歸一化處理.該算法對(duì)于任意的對(duì)稱非負(fù)可約矩陣是收斂的,并給出數(shù)值實(shí)例說(shuō)明了算法的優(yōu)越性.進(jìn)一步,我們給出算法的一個(gè)實(shí)際應(yīng)用,即把算法應(yīng)用到Z-矩陣正定性的判定上.最后,我們對(duì)論文進(jìn)行總結(jié)并給出今后研究的方向.
[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.
【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.6

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 李良;黃廷祝;;非負(fù)不可約矩陣譜半徑的估計(jì)[J];應(yīng)用數(shù)學(xué)學(xué)報(bào);2008年02期

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本文編號(hào)：1759929