Tridiagonal matrix algorithm

Abstract: We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the 'level' process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Abstract: The parallel homotopy algorithm for finding few or all eigenvalues of a symmetric tridiagonal matrix is presented. The computations were executed on an NCUBE, a distributed memory multiprocessor. The numerical results show that the performance of our algorithm is strongly competitive with “divide and conquer” and bisection/multisection algorithms. The almost 100 percent efficiency seems to suggest that the natural parallelism of the homotopy method makes the algorithm an excellent candidate for a variety of architectures.

Abstract: In this paper, a fast algorithm for solving the special tridiagonal system is presented. This special tridiagonal system is a symmetric diagonally dominant and Toeplitz system of linear equations. The error analysis is also given. Our algorithm is quite competitive with the Gaussian elimination, cyclic reduction, specialLU factorization, reversed triangular factorization, and Toeplitz factorization methods. In addition, our result can be applied to solve the near-Toeplitz tridiagonal system. Some examples demonstrate the good efficiency and stability of our algorithm.