Study Of Tridiagonal Matrix About Theory And Algorithm

We say A being irreducibly diagonally dominant if |?|?|?|+|?|,sub-diagonally dominant if |?|?|?|+|?| and super-diagonally dominant if |?|?|?|+|?|.Let A be a tridi-agonal Toeplitz matrix denoted by A=Tritoep(?,?,?).Firstly,we consider the solution of a tridiagonal Toeplitz system with the coefficient matrix being sub-diagonally dominan-t,super-diagonally dominant or irreducibly diagonally dominant,respectively.Compared with the LU factorization method with pivoting,our algorithm takes less floating-point operations(flops),needs less memory storage and data transmission.In particular,our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency.Numerical experiments are given to illustrate the effectiveness of our algorithms.we applied the above fast algorithm to the calculation process of practical examples and found that calculation effectiveness of most examples are remarkable,but the calculation precision of some examples could not reach the accuracy of computing precision.So the next chapter of this paper mainly discusses how to numerically solve tridiagonal Toeplitz linear systems Ax=b precisely.Aiming at example of this kind of short of computing precision,in the fourth chapter,based on the direct algorithm of solving tridiagonal Toeplitz linear equations of Ax=b,the iterative refinement is carried out to improve the computational accuracy of such example.Numerical experiments show that the computational accuracy of our algorithm can reach computing precision by iterative refinementThis thesis consists of four chapters:In Chapter 1,we mainly introduce the research background,the related solution and the research status of matrix equations,as well as the innovation point in this paperIn Chapter 2,we review some basic definitions and theorems frequently used in the sequel.In Chapter 3,fast algorithms for solving tridiagonal Toeplitz linear systems are studiedIn Chapter 4,the algorithm of solving tridiagonal Toeplitz linear systems precisely is studied.