Tridiagonal matrix algorithm

Abstract: In this paper we consider the application of polynomial root-finding methods to the solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed results at a modest increase in computational cost. Numerical experiments illustrate that under some restriction on the conditioning the novel iterations can approximate and/or refine the eigenvalues of a tridiagonal matrix with high relative accuracy.

Abstract: A cyclic reduction method is described for the fast numerical solution of constant tridiagonal Toeplitz linear systems which occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. In this paper, we show that the linear systems can be solved efficiently by the Stride of 3 reduction algorithm.

Abstract: Solving special tridiagonal systems often arise in the fields of engineering and science. This special tridiagonal system is diagonally dominant and circulant near-Toeplitz. This paper presents two fast vectorized algorithms for solving special tridiagonal systems. Both algorithms employ the matrix perturbation technique and have many computational advantages on vector supercomputer. The related error analysis are also given. Some experimental results are illustrated on vector uniprocessor of the CRAY X-MP EA/116se.

Abstract: Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal matrix with $-1$'s on the sub-diagonal, $1$'s on the super-diagonal, $-1$ in the $(1,1)$ entry, $1$ in the $(n,n)$ entry and zeros elsewhere. We find the eigen-pairs of the matrices $R_n$.