非hermitian正定线性方程组的分裂迭代解法的分析研究.pdf

太原理工大学硕士研究生学位论文非Herm tan正定线性方程组的分裂迭代解法研究摘要大型线性代数方程的求解在现代的各种科学工程研究中发挥着越来越重要的作用,如在流体力学,最优化问题,电磁学,油藏模拟等领域都有着广泛的应用。因此对于大型稀疏线性方程组解法的研究已经成为如今科学工程研究的重要课题之一,有着深远而广泛的理论意义和应用价值。本文在研究非Hermitian正定线性方程组中系数矩阵的HSS分裂的基础上,提出了一种新的分裂格式,并利用这种新的分裂方法构造出了非 Hermitian正定线性方程组的单分裂收敛理论,在一定精度要求下,数值算例结果令人满意。此外,本文把此方法应用到预处理算法中,进一步得到一种比较好的分裂形式,并得出了可行的收敛理论。文中研究的分裂格式放宽了HSS分裂的限制条件,使算法更简单,易处理。这就说明了本文算法的可行性和有效性,实验结果表明,此算法迭代次数减少,是对非Hermitian正定线性方程组的单分裂迭代算法的进一步完善与发展。关键词:非Hermitian正定矩阵,矩阵分裂,分裂迭代格式,收敛性,线性方程组,奇异矩阵太原理工大学硕士研究生学位论文太原理工大学硕士研究生学位论文 Il-ESEARCH ON SPLITTING ITERATIVE METHODS OF POSITⅣE DEFnⅢTE LINEAR SYSTEMS ABSTRACT Solving Large linearalgebraic equations plays amore andmore important roleinvariousmodem scientificproject filed,such asin thefiledoffluid mechanics, optimization problem, ics and reservoir ,thestudy of how to solve large spare linear equations es one oftheimpotent taskinmodem engineering researchstudy,and has major theoreticvalueandwide-ranging application. Inthispaper,we give anew splitting scheme based on thestudy of the coefficient matrix ofnon-Hermitianpositive the new method we constructed a single splitting convergence theory of non-hermitianpositive definiteequations,the resultissatisfactory withsome precision. Besides,theapplication ofthenew method inpreprocessing calculate provide abettersplitting form,and obtainfeasibleconvergence theory. The new method we studied inthispaper relaxed theconditionsofHSS splitting,making thecalculationeasierandthetreatmentquicker,hence thenew method resultsindicatethat,the algorithm method cutdown iterationtimes,and furtherimproved anddeveloped thesinglesplitting iterationmethod oflarge linearequations. KEY WORDS:non-Hermitian positive definite matrix,matrix spliaing, splitting iterative format,convergence,linear equation, singular matrix IV 太堕里三奎堂堡主堕窒皇堂垡笙奎主要符号说明么叫 H ㈣: A≥0 A>0 么>-O p(A) diag(dI,?,以) 圆实数集复n维列向量空间实n维列向量空间 m×拧复矩阵集肌×,z实矩阵集单位矩阵矩阵彳的转置矩阵么