Abstract: The Parallel Diagonal Dominant (PDD) algorithm is an efficient tridiagonal solver. In this paper, a detailed study of the PDD algorithm is given. First the PDD algorithm is extended to solve periodic tridiagonal systems and its scalability is studied. Then the reduced PDD algorithm, which has a smaller operation count than that of the conventional sequential algorithm for many applications, is proposed. Accuracy analysis is provided for a class of tridiagonal systems, the symmetric and skew-symmetric Toeplitz tridiagonal systems. Implementation results show that the analysis gives a good bound on the relative error, and the PDD and reduced PDD algorithms are good candidates for emerging massively parallel machines.

Abstract: Gaussian elimination is the algorithm of choice for the solution of dense linear systems of equations. However, Gauss himself originally introduced his elimination procedure as a way of determining the precision of least squares estimates and only later described the computational algorithm. This article tells the story of Gauss, his algorithm, and its relation to his probabilistic development of least squares.

Abstract: We study nonsymmetric tridiagonal operators acting in the Hilbert space ?2 and describe the spectrum and the resolvent set of such operators in terms of a continued fraction related to the resolvent. In this way we establish a connection between Pade approximants and spectral properties of nonsymmetric tridiagonal operators.

Abstract: Tridiagonal, pentadiagonal, and cyclic triagonal matrix algorithms are well-eestablished elements of line-by-line iterative procedures for the solution of algebraic decretized equations yielded by finite-difference, finite-volumes, finate-element, and controt-volume finite-element methods for fluid flow and heat transfer In this article, five extensions of these algorithms are presented: a coupled tridiagonal matrix algorithm, a coupled cyclic tridiagonal matrix algorithm, the cyclic pentadiagonal matrix algorithm, a coupled pentadiagonal matrix algorithm, and a coupled cyclic pentadiagonal matrix algorithm

Abstract: The present paper describes a numerical method, aimed to simulate the flow field of vertical-axis wind turbines, based on the solution of the steady, incompressible, laminar Navier-Stokes equations in cylindrical coordinates. The flow equations, written in conservation law form, are discretized using a control volume approach on a staggered grid. The effect of the spinning blades is simulated by distributing a time-averaged source terms in the ring of control volumes that lie in the path of turbine blades. The numerical procedure used here, based on the control volume approach, is the widely known “SIMPLER” algorithm. The resulting algebraic equations are solved by the TriDiagonal Matrix Algorithm (TDMA) in the r- and z-directions and the Cyclic TDMA in the 0-direction. The indicial model is used to simulate the effect of dynamic stall at low tip-speed ratio values. The viscous model, developed here, is used to predict aerodynamic loads and performance for the Sandia 17-m wind turbine. Predictions of the viscous model are compared with both experimental data and results from the CARDAAV aerodynamic code based on the Double-Multiple Streamtube Model. According to the experimental results, the analysis of local and global performance predictions by the 3D viscous model including dynamic stall effects shows a good improvement with respect to previous 2D models.

Abstract: Olver has given an elegant construction of recessive solutions of nonhomogeneous scalar three term recurrence relations. We investigate the feasibility of applying his methods to block tridiagonal nonhomogeneous systems where the coefficient matrices A B C are n × n matrices and Y and D are n × m. Such systems with m = 1 arise in numerical methods for solving partial differential equations [7,11]. Of course, the results obtained are also applicable to the symmetric case with and symmetric Bk . The homogeneous case with m = n arises in matrix continued fractions [4] and the well-studied symmetric homogeneous case is closely realted to discrete Hamiltonian systems and, for m = n, to discrete Riccati equations [2,1,3,5].

Abstract: This paper establishes a new entrywise relative perturbation result for the inverse of a nonsingularM-matrixA. It is shown that a version of Gaussian elimination with one step of iterative refinement solves the systemAx =b, whereb is nonnegative, with small entrywise relative error. IfA is tridiagonal, the Gaussian elimination alone suffices.

Abstract: A divide-and-conquer algorithm for the resolution of distributed linear tridiagonal systems of equations is implemented within a binary tree connection architecture. A new scheme for the distribution of the data among the computing nodes allows a dilation-one implementation of a recursive substitution scheme for the solution of the global system. In this way, computation time decreases linearly with the number of nodes, and the data communication required becomes proportional to the logarithm of the number of nodes. This takes place within a network with a fixed connectivity degree of three.

Abstract: Generalizing Muller and Scheerer's method which is used to parallelize the tridiagonal solvers, this paper presents a parallel method for solving the circulant block-tridiagonal systems. The applications of our result to solve the block-tridiagonal systems, the banded systems, and the circulant tridiagonal systems (for example, solving the closed B-spline curve fitting) are also addressed.

Abstract: A method is described to solve the systems of tridiagonal linear equations that result from discrete approximations of the Poisson or Helmholtz equation with either periodic, Dirichlet, Neumann, or shear-periodic boundary conditions. The problem is partitioned into a set of smaller Dirichlet problems which can be solved simultaneously on parallel or vector computers leaving a smaller tridiagonal system to be solved on one of the processors.

Abstract: A compact scheme is a discretization scheme that is advantageous in obtaining highly accurate solutions. However, the resulting systems from compact schemes are tridiagonal systems that are difficult to solve efficiently on parallel computers. Considering the almost symmetric Toeplitz structure, a parallel algorithm, simple parallel prefix (SPP), is proposed. The SPP algorithm requires less memory than the conventional LU decomposition and is efficient on parallel machines. It consists of a prefix communication pattern and AXPY operations. Both the computation and the communication can be truncated without degrading the accuracy when the system is diagonally dominant. A formal accuracy study has been conducted to provide a simple truncation formula. Experimental results have been measured on a MasPar MP-1 SIMD machine and on a Cray 2 vector machine. Experimental results show that the simple parallel prefix algorithm is a good algorithm for symmetric, almost symmetric Toeplitz tridiagonal systems and for the compact scheme on high-performance computers.

Abstract: In this paper a finite element formulation is proposed for the calculation of advection and diffusion in a thin cavity. For these kinds of systems, very high aspect ratio elements are necessary for cost-effective simulation. Locally, element dimensions, say in x and y, are comparable, whereas the dimension in the transverse direction z is orders of magnitude smaller than those for x and y. In our formulation, the three-dimensional basis functions for interpolation are constructed as a tensor product of the basis functions that span the lateral (x, y) plane of an element and those that span the transverse direction. Unknowns along the transverse direction are solved implicitly, in a line-by-line fashion, using the tridiagonal matrix algorithm, while the “out-of-line” unknowns are treated either explicitly or semi-implicitly. Several applications to material processing are discussed, as are the computationally-intensive components of three variations of our basic procedure.

Abstract: Based on the parallel minimal norm method an algorithm is derived to solve tridiagonal linear systems with a high degree of parallelism. No conditions need to be posed with respect to the system. Experiments indicate that the numerical stability of the algorithm is similar to Gaussian elimination with partial pivoting. >

Abstract: In this paper, we give the fast algorithm for inverting a tridiagonal matrix of n order. Its calculating quantity of arithmetic operation is only n2+7n-8. At the same time, we give the expression on the elements of the inverse tridiagonal matrix, on which we get exact estimate. It greatly expands and improves the results of [2], [3].

Abstract: The partition method for the parallel solution of tridiagonal linear systems is discussed and the coefficients of the reduced global system derived. It is shown that if the full system is diagonally dominant then the reduced system retains this property. This has important implications for the stability of calculations in this reduced system and eliminates the need for global pivoting with its expensive communication overhead.

Abstract: We present and analyze a parallel method for the solution of partial differential equation models of the nervous system. These models mathematically are one-dimensional nonlinear parabolic equations defined on branching domains. Implicit methods for these equations leads to numerical solution of diagonally dominant almost tridiagonal linear systems at each time step. We first review some exact methods for the solution of these linear systems that includes an Exact Domain Decomposition. This EDD leads to the solution of many tridiagonal linear systems one for each branch. The sizes of these systems is equal to the number of grid points on the branch. Since the branches of realistic neurons vary widely in size, the decomposition leads to a very poor a priori load balance. This problem may be solved with the Overlapped Partition Method, a method for decomposing large diagonally dominant tridiagonal systems. We describe and analyze an algorithm based on EDD and OPM that can be load balanced.

Abstract: A scalable algorithm for the reduction to tridiagonal form of symmetric matrices is developed. It uses one-sided rotations instead of similarity transforms. This allows a data distribution independent implementation with low communication volume. Timings on the Fujitsu AP 1000 and VPP 500 show good performance. >

Abstract: The use of threshold pivoting with the purpose to reduce fill-in during sparse Gaussian elimination has been generally acknowledged. Here we describe the application of threshold pivoting in dense Gaussian elimination for improving the performance of a parallel implementation. We discuss the effect on the numerical stability and conclude that the consequences are only of minor importance as long as the threshold is not chosen

Abstract: : An algorithm for the parallel solution of tridiagonal and pentadiagonal linear systems having nonzero elements at the top right and bottom left corners. Tridiagonal systems of this kind arise from the solution of two point boundary value problems with periodic boundary conditions. Penta- diagonal systems of this kind arise from e.g the approximation of the shallow water equations by the two-stage Galerkin method combined with a high accuracy compact approximation to the first derivative (Navon, 1983).

Abstract: We show classical elimination procedure can be simply extended to uncouple partitioned tridiagonal systems for parallel processing of their solution. In each block of equations, we now need two simultaneous eliminations; one usual forward elimination and one backward from across the succeeding block. Significantly, unlike Wang's method [6], our is a one-stage elimination procedure, at the end of which the core system is reached. Once the core system is solved, the uncoupled subsystems are solved in parallel by back substitution.

Abstract: Publisher Summary In this chapter, two codes that employ a similar methodology for the solution of the incompressible Navier–Stokes equations are developed for parallel architectures. The first code (REACT) solves the Reynolds averaged Navier–Stokes equations to model chemically reacting flow in an axisymmetric pipe. The geometry can be modified by the addition of obstacles and baffles (where a baffle is considered to be an infinitely thin obstacle). The second code (FLAME) solves the Favre-averaged Navier–Stokes equations to model turbulent combustion in an axisymmetric geometry. A standard grid partitioning strategy is employed for both FLAME and REACT. However, the combustion code uses a staggered grid approach, while the chemically reacting model has a collocated grid arrangement. For both codes, the coupling between the pressure and velocity fields is handled by the SIMPLE algorithm. The discretization process results in a set of sparse algebraic equations, coupling the field values at neighboring points and a source term representing the contribution from other variables at the same point. The method chosen to solve this large system of linear equations is the tridiagonal matrix algorithm, and the pressure correction algorithm is solved by a preconditioned conjugate gradient algorithm.

Abstract: We consider linear systems of algebraic equations Su = f with tridiagonal interval matrix S and interval vector f An interval version of the sweep method allows us to find an interval vector u = ( u t , u 2 , . . . , u n ) T that contains the united set of solutions of the system. In the paper we present estimates of the absolute value and the width of the intervals ui, i = 1, 2 , . . . , ~z tinder certain assumptions on the elements of the matrix S that do not include the traditional condition of diagonal dominance. The width estimates are three orders of magnitude narrnwer, and the assumptions on the system's coefficients are weaker than those in works published so far.

Abstract: The solution of tridiagonal and block tridiagonal systems of equations represents one of the basic problems of numerical linear algebra. In the present paper, we describe a parallel method which is based on the use of Schur complements. When compared to previously introduced parallel methods, this method preserves all necessary mathematical properties in all its parts while a high numerical efficiency can be guaranteed. The paper includes a short comparison with other relevant methods, illustrated by the results of numerical tests carried out on a CRAY Y-MP. The applicability of our method is, however, not restricted to a specific architecture.

Abstract: A parallel variant of the block Gauss–Seidel iteration is presented for the solution of block tridiagonal linear systems. In this method parallel computations derive from a block reordering of the ...

Abstract: Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given.

Abstract: A new algorithm is presented, designed to solve tridiagonal matrix problems efficiently with parallel computers (multiple instruction stream, multiple data stream (MIMD) machines with distributed memory). The algorithm is designed to be extendable to higher order banded diagonal systems.