The Research Of Sign Regular Matrix Decomposition And Its Preservativeness Of Solving Tridiagonal Linear System

An n×n matrix A is sign regular, for each order k(k≤n), if all k×k sub-matrices of A have determinant with the same sign or be zero. These matricesare widely applied in computer aided geometric design, approximation theory, eco-nomics, statistics and numerical algebra and etc.. And many large-scale science en-gineering calculation and numerical simulation, for example, nuclear physics, fluidmechanics, reservoir engineering, earthquake prediction and weather forecasting andetc., depend on the tridiagonal linear system. So the research of the algorithm forthis linear system has important theoretical and practical signifcance.In this paper, we considered the stability of the sign regular matrices in theorthogonal decomposition, and analysed the question of the sign structure preservesand the stability of the algorithms for solving the tridiagonal sign regular linearsystem. Research in this study was mainly concentrated on the following threeaspects:1. Considering the sign structure preserves of sign regular matrices in the orthogo-nal decomposition, according to Neville elimination methods and Givens rota-tion theory, the sign regular matrices will be simplifed to m-bounded matricesor tridiagonal sign regular matrices(when m=1)with the same signature.2. The partitioning algorithm was frstly put forward by H.H.WAN G[44]. We willuse some properties of the sign regular matrices to illustrate the sign structurepreserves of the schur complement in this algorithm and then give the symbolregularity of the block matrices.3. R.W. Hockney[24] put forward cycle reduction algorithm frstly. We will use thisalgorithm to solve the tridiagonal and totally nonnegative linear systems, andresearch the sign structure preserves of the schur complement in this algorithm.Then the backward error bounds of the algorithm was given.