Tridiagonal matrix algorithm

Abstract: In this note, we present a numerical scheme for finding an approximate solution of an equation which can be viewed as a model for spatial diffusion of age-dependent biological populations. Discretization of the model yields a linear system with a block tridiagonal matrix. Our main concern will be discussion of stability for this scheme by examining the eigenvalues of the block tridiagonal matrix. Numerical results are presented.

Abstract: A rank-two divide and conquer algorithm is developed for calculating the eigensystem of a symmetric tridiagonal matrix. This algorithm is compared to the LAPACK recommended path for this problem and the rank-one divide and conquer algorithm. The timing results on a Sequent Symmetry S81b show that this algorithm has potential as a parallel alternative to the QR algorithm. >

Abstract: A method of solving the uniform bicubic B-spline surface fitting algorithm is proposed which introduces parallelism in a way that may be effectively exploited by a suitable parallel architecture. This method is based on the observation that a tensor product spline surface fitting problem can be split into two spline curve fitting problems and each of these problems can be realized by a macropipeline of fixed size VLSI arrays. In fact, the heart of curve fitting problem consists of a block tridiagonal linear system. Based on the state-of-art electronic and packaging technologies, the size of VLSI arithmetic devices is limited due to the bounded chip area and I/O packaging constraints. A modular approach to achieve VLSI matrix arithmetic solution for the block tridiagonal linear system is amenable from the viewpoints of feasibility and applicability. A matrix partitioning approach is presented to overcome those technological constraints imposed by the number of I/O pins. A block tridiagonal linear system of size mn is then divided into m simple tridiagonal systems of size n and n simple tridiagonal systems of size m by the Dc Boor partitioning theorem. Each of the simple tridiagonal linear systems could be partitioned and mappied into a series of two fixed size primitive VLSI matrix arithmetic arrays including L-U decomposer and triangular system solver. The L-U decomposer and triangular system solver could be realized by a hex-connected processor array and an inverse perfect shuffle machine respectively. It would be shown that a B-spline surface fitting problem for a grid of mn points can be solved by m hex-connected processor arrays having 4 processors, m inverse perfect shuffle machines having n processors and n inverse perfect shuffle machines having m processors in (3(m+n)+2({logzn1 +flog2n)+4J units of time.

Abstract: Givens well known bisection method for the calculation of the eigenvalues af a symmetric tridiagonal matrix is extended to directly decomposable matrices. It is given an easy, algebraic correct algorithm, which is free of divisions.