Diagonally Dominant Matrix Sexual Problems Related To Research

Diagonally dominant matrix is a classification of matrices which is widely used. Itsexcellent properties attract a large number of scholars to explore. In scientific andengineering computing, such as computational electromagnetics, computational fluiddynamics, optimization, the basic principles in relevant disciplines are expressed aspartial differential equations or integral equations. After discretization by finite differre-nce method, finite element method, domain decomposition algorithms, these partialdifferential equations are often treated into largescale sparse linear equations, of whosesolution, the existence, uniqueness, and convergence properties and stability of relatedmethods are all related with certain diagonal dominance of the coefficient matrix. Thisthesis gives new theorems which judge the diagonally dominant matrices are singular ornot and gives the excellent applications of diagonally dominant matrices. This thesisconsists of five chapters, giving the following contents:The first chapter introduces the history of diagonally dominant matrices, also givesthe theoretical importance and the practical importance that tell us that the research fordiagonally dominant matrix is necessary. In the last of this chapter there are someconceptions for matrix and some notations for this article.The second chapter gives common iteration methods which are convergence fordiagonally dominant matrices. In the last gives a lot of fine properties of diagonallydominant matrices, for example, diagonally dominant matrices can keep diagonaldomination in Gauss elimination method.The third chapter first gives the theoretical foundation of diagonally dominantmatrices. Establish new theorems according the elements’ properties of the diagonallydominant matrix. The new theorems are established to judge a diagonally dominantmatrix is singular or not. These theorems have developed the old ones. Finally gives theconclusions that one row can be represented by the remaining rows in the diagonallydominant matrix, the row keep equivalent diagonally domination.The forth chapter gives the application of diagonally dominant matrix when it isused in the input-output model and how it is used to solute an integral equation. Described how can solve the second type of Fredholm linear integral equation by thediagonally dominant matrix in detail. This is one of the most important methods thatwe solve the equations of this type. Then give an example of solving the equation. Bythis way iteration algorithm has smaller amount of calculation and the computer needssmaller memory requirements.The fifth Chapter gives the conclusion and the prospect.