Tridiagonal matrix algorithm

Abstract: We propose a stable algorithm for the parallel solution of banded and periodically banded linear systems. While most of the known parallel algorithms are stable only for symmetric positive de nite or diagonally dominant systems, the new algorithm incorporates pivoting without sacri cing e ciency. The principle ingredient of the algorithm is a bidiagonal cyclic reduction that admits pivoting. We report on numerical experiments conducted on various multiprocessor computers.

Abstract: In this paper, we analyze fluid queues driven by truncated birth-death processes with general birth and death rates. We compute the equilibrium distribution of the content of the fluid buffer by providing efficient numerical procedures to compute the eigenvalues and the eigenvectors of the associated real sign-asymmetric tridiagonal matrix. We illustrate the effectiveness of the procedures through tables and graphs.

Abstract: SUMMARY A method for efficient implementation of a combined spectral finite difference algorithm for computation of incompressible stratified turbulent flows on distributed memory computers is presented. The solution technique is the fractional step method with a semi-implicit time advancement scheme. A single-programme multiple-data abstraction is used in conjunction with a static data-partitioning scheme. The distributed FFTs required in the explicit step are based on the transpose method and the large sets of independent tridiagonal systems of equations arising in the implicit steps are solved using the pipelined Thomas algorithm. A speed-up analysis of a model problem is presented for three partitioning schemes, namely unipartition, multipartition and transpose partition. It is shown that the unipartitioning scheme is best suited for this algorithm. Performance measurements of the overall as well as individual stages of the algorithm are presented for several different grids and are discussed in the context of associated dependency and communication overheads. An unscaled speed-up efficiency of up to 91% on doubling the number of processors and up to 60% on an eightfold increase in the number of processors was obtained on the Intel Paragon and iPSC 860 Hypercube. Absolute performance of the code was evaluated by comparisons with performance on the Cray-YMP. On 128 Paragon processors, performance up to five times that of a single-processor Cray-YMP was obtained. The validation of the method and results of grid refinement studies in stably stratified turbulent channel flows are presented. 1997 by John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids24: 1129-1158, 1997.

Abstract: The QR iteration for the eigenvalues of a symmetric tridiagonal matrix can be accelerated by incorporating a sequence of origin shifts. The origin shift may be either subtracted directly from the diagonal elements of the matrix or incorporated by means of an implicit algorithm. Both methods have drawbacks: the direct method can unnecessarily degrade small eigenvalues, while the implicit method can effectively loose the shift and thereby retard the convergence. This paper presents a new method which has neither drawback.