Tridiagonal matrix algorithm

Abstract: A scalable parallel algorithm of block tridiagonal systems for solving the initial boundary value problem of 3D-parabolic equation with the Dirichlet boundary condition is discussed A parallel degree of the difference scheme is proposed for showing its intrinsic parallelism of the difference scheme The relation between the parallel degree and the performance of the parallel algorithm is investigated The method proposed in this paper has been implemented on the super computer "ZiQiang 3000" of Shanghai University, and the numerical results match closely with theoretical analysis With the given accuracy, the line speedup is obtained, and the parallel implementation efficiency over 90% is reached

Abstract: A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented.

Abstract: The principal minors of a tridiagonal matrix satisfy two-term and three-term recurrences [1, 2]. Based on these facts, the current article presents a new efficient and reliable hybrid numerical algorithm for evaluating general n-th order tridiagonal determinants in linear time. The hybrid numerical algorithm avoid all symbolic computations. The algorithm is suited for implementation using computer languages such as FORTRAN, PASCAL, ALGOL, MAPLE, MACSYMA and MATHEMATICA. Some illustrative examples are given. Test results indicate the superiority of the hybrid numerical algorithm.

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.