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: An algorithm for reducing a nonsymmetric matrix to tridiagonal form as a first step toward finding its eigenvalues is described. The algorithm uses a variation of threshold pivoting, where at each step, the pivot is chosen to minimize the maximum entry in the transformation matrix that reduces the next column and row of the matrix. Situations are given where the tridiagonalization process breaks down, and two recovery methods are presented for these situations. Although no existing tridiagonalization algorithm is guaranteed to succeed, this algorithm is found to be very robust and fast in practice. A gradual loss of similarity is also observed as the order of the matrix increases.

Abstract: Basic linear algebra vectors, matrices and linear transformations introduction to linear systems Gaussian elimination, theory and practice partial pivoting as a valuable technique within Gaussian elimination elementary error analysis applied to floating point arithmetic and Gaussian elimination in particular matrix norms introduction to iterative schemes bisection, secant and Newton's method guarantees of convergence and rates of convergence extensions of methods secant method extensions of methods Newton's method and the Jacobian matrix.

Abstract: Two methods are presented which efficiently solve tridiagonal systems on vector supercomputers and parallel computers with a moderate degree of parallelism. The first algorithm for diagonally dominant systems uses incomplete Gaussian elimination, the other for more general systems applies Gaussian elimination with partial pivoting. The methods are based on wrap-around partitioning, which is closely related to the partitioning used in Wang's algorithm. The first algorithm delivers an asymptotic speedup by a factor ofp on ap-processor computer if compared to the scalar algorithm, whereas the second algorithm delivers a speedup by a factor of roughlyp/2, which is also typical for cyclic reduction. For the incomplete factorization, existence and approximation properties are proved. Timing experiments were run on a Cray X-MP.

Abstract: A transient three-dimensional finite difference mode (FDM) of the heat flow in the circumferential GTA welding of pipes was developed and applied to calculate the temperature distribution in the workpiece. In order to minimize the computing time required for solving the FDM equations as much as possible, the alternating direction implicit (ADI) scheme which makes use of the tridiagonal matrix algorithm efficiently was adopted.Based on the characteristics of the pipe welding process, the periodic boundary condition was applied to calculate the temperature distribution in the 8 direction. For treating the moving heat source effectively, the grid meshes with variable spacings were regenerated at each time step. In order to decrease the interpolating error by grid remeshing, the temperature values at new meshes were interpolated from those at old meshes by using the periodic spline function. The temperature-dependent thermal properties, the latent heat and the convective and radiative boundary conditions were...

Abstract: A new proof is presented of the existence of commuting tridiagonal matrices for a particular family of Toeplitz matrices.

Abstract: Systems of tridiagonal equations frequently arise in practical applications related to solving ordinary or partial differential equations by discrete numerical methods. In this paper a new direct method called the recursive tri-reduction method is developed for the tridiagonal system. The method is simple and has the advantage over the Gaussian Elimination procedure when we use the parallel computer.

Abstract: The work presented in this thesis mainly concerns the analysis of parallel algorithms for the solution of tridiagonal linear systems and the design of a new tridiagonal equation solver, which can be run on a MIMD (Multiple Instruction Multiple Data stream) type parallel computer, in particular the Balance 8000 Sequent system at Loughborough University of Technology. In the first chapter, an introduction to the existing computer models is given, together with a brief description of the process that has led from the uniprocessor machine to the development of different parallel architectures. Enhancement is given to MIMD shared memory systems. In this respect, the main characteristics of the Sequent system are presented, as well as the main programming features supported by the Balance Operating System, the Dynix....cont'd

Abstract: A new parallel algorithm is presented for computing the determinant of a tridiagonal matrix The algorithm is based on a divide-and-conquer strategy and has a parallel time complexity of O(log/sub 2/n) The algorithm is adaptive and the effect of the available number of processors on the computation time is studied on a simulated MIMD static dataflow machine >

Abstract: Periodic tridiagonal linear systems of equations typi- cally arise from discretizing second order differential equations with periodic boundary conditions. In this paper a parallel-vector algorithm is introduced to solve such systems. Implementation of the new algorithm is carried out on an Intel iPSC/2 hypercube with vector processor boards attached to each node processor. It is to be noted that t his algorithm can be extended to solve other periodic banded linear systems.

Abstract: - Parallel iterative algorithms for solving tridiagonal systems of equations are derived from the symplectic factorization of the odd-even permuted matrix of coefficients. These algorithms have halved parallel computational costs with respect to Accelerated Parallel Gauss, under weaker conditions for converg- ence. 1. Introduction The solution of tridiagonal linear systems of equations is a central problem in numerical linear algebra. The developement of parallel architectures has addressed the major interest towards algorithms suitable to be implemented on parallel computers [5], [8], [10], [11], [12], [13]. In 1973 Traub devised a parallel version of Gauss method for tridiagonal systems of equations (Parallel Gauss) [ 14], consisting essentially of vector itera- tions, fastly convergent under simple assumptions resembling diagonal domi- nance. This result was improved in [7], where the convergence of Parallel Gauss was fastened by updating alternately the odd and the even components in the vector iterations. The new algorithm (Accelerated Parallel Gauss or APG), hav- ing a rate of convergence not depending on the size of the problem, turned out to be superior to the other known tridiagonal iterative solvers. m Received 21 march 1991. (l) Dipartimento di Informatica, Universit~ di Pisa, Corso Italia 40, 56100 Pisa, Italy. (2) Dipartimento di Matematica, Universit~ di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy.

Abstract: A new algorithm, the double-bordering algorithm, for the solution of linear systems of equations is derived, and its complexity is shown to be compatible with the complexity of Gaussian elimination, The application of the algorithm to the solution of block-tridiagonal systems is also described

Abstract: It has been conjectured that when Gaussian elimination with complete pivoting is applied to a real n-by-n matrix, the maximum possible growth is n. In this note, a 13-by-13 matrix is given, for which the growth is 13.0205. The matrix was constructed by solving a large nonlinear programming problem. Growth larger than n has also been observed for matrices of orders 14, 15, and 16.

Abstract: This paper presents a method for computing the inverse of a complex n-block tridiagonal quasi-Hermitian matrix using an adequate number of partitions of the complete matrix. This type of matrix is very usual in quantum mechanics and, more specifically, in solid state physics (e.g. interfaces and super-lattices), when the tight-binding approximation is used. The efficiency of the method is analysed by comparing the required CPU time and work-area with other techniques.