本文選題：嚴(yán)格對角占優(yōu)矩陣 + 非奇異H-矩陣　； 參考：《吉首大學(xué)》2017年碩士論文

【摘要】：非奇異H-矩陣是計(jì)算數(shù)學(xué)、數(shù)學(xué)物理、控制論和矩陣?yán)碚撝休^為活躍的研究領(lǐng)域,它在經(jīng)濟(jì)學(xué)、生物學(xué)、動力系統(tǒng)理論及智能科學(xué)等許多學(xué)科中都有著廣泛的應(yīng)用.但是判定一個(gè)矩陣是否為H-矩陣是較為困難的問題,因此,研究H-矩陣的簡捷實(shí)用的判定條件,構(gòu)造高效快速的判別算法,具有十分重要的理論價(jià)值和實(shí)際應(yīng)用價(jià)值.本文主要研究了非奇異H-矩陣的直接判定法,遞進(jìn)判定法和交叉迭代判定算法,改進(jìn)了近期的一些結(jié)果,主要內(nèi)容如下:首先,介紹了非奇異H-矩陣判定問題的研究背景和現(xiàn)狀,給出了有關(guān)的符號和定義,以及本文所做的主要工作.其次,研究了非奇異H-矩陣的直接判定法.根據(jù)非奇異H-矩陣的定義和性質(zhì),通過添加新的系數(shù)來構(gòu)造正對角矩陣變換因子,利用不等式放縮技巧,得到了一種新的判定條件,改進(jìn)了近期的一些結(jié)果,并用數(shù)值例子驗(yàn)證了這個(gè)方法的優(yōu)越性.再次,研究了非奇異H-矩陣的遞進(jìn)判定.遞進(jìn)判定法主要通過對正對角矩陣因子進(jìn)行遞進(jìn)選取,給出了幾個(gè)新的遞進(jìn)判定法.數(shù)值實(shí)例驗(yàn)證了新的遞進(jìn)判定法的判定范圍更加廣泛,改進(jìn)了近期的結(jié)果.最后,研究了非奇異H-矩陣的算法判定,給出了非奇異H-矩陣的一種新的無參數(shù)交叉迭代判定算法.對于一個(gè)給定的不可約矩陣A,總能通過有限步迭代來判斷這個(gè)矩陣是否為非奇異H-矩陣.并用數(shù)值實(shí)例驗(yàn)證了新算法的有效性和優(yōu)越性,新算法比舊算法迭代次數(shù)更少、應(yīng)用范圍更廣,并且對于所給矩陣為非H-矩陣的情況,同樣能夠更快地進(jìn)行判斷,改進(jìn)了近期的結(jié)果.
[Abstract]:Non-singular H-matrix is an active research field in computational mathematics, mathematical physics, cybernetics and matrix theory. It has been widely used in many disciplines such as economics, biology, dynamic system theory and intelligent science. However, it is difficult to determine whether a matrix is a H-matrix or not. Therefore, it is very important to study the simple and practical judgment conditions of H- matrix and to construct an efficient and fast discriminant algorithm, which has very important theoretical value and practical application value. In this paper, the direct decision method, the progressive decision method and the cross iterative decision algorithm of the nonsingular H-matrix are studied, and some recent results are improved. The main contents are as follows: first of all, In this paper, the research background and present situation of nonsingular H-matrix decision problem are introduced, and the related symbols and definitions are given, as well as the main work done in this paper. Secondly, the direct decision method of nonsingular H-matrix is studied. According to the definition and properties of nonsingular H-matrices, the transformation factors of diagonal matrices are constructed by adding new coefficients. By using the technique of inequality scaling, a new judgment condition is obtained, and some recent results are improved. A numerical example is given to verify the superiority of this method. Thirdly, the progressive decision of nonsingular H-matrix is studied. In this paper, several new progressive decision methods are given by selecting the positive diagonal matrix factors. Numerical examples show that the new progressive decision method is more extensive and improves the recent results. Finally, the algorithm of nonsingular H-matrix is studied, and a new algorithm of nonsingular H-matrix is presented. For a given irreducible matrix A, it is always possible to determine whether the matrix is a nonsingular H-matrix by finite step iteration. The effectiveness and superiority of the new algorithm are verified by numerical examples. The new algorithm has fewer iterations than the old one and has a wider range of applications. The new algorithm can also be used to judge the non-H-matrix more quickly. The recent results have been improved.
【學(xué)位授予單位】：吉首大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O151.21

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