Tridiagonal matrix algorithm

Abstract: The recursive doubling algorithm as developed by Stone can be used to solve a tridiagonal linear system of size n on a parallel computer with n processors using O ( log n ) parallel arithmetic steps. Here we describe a limited processor version of the recursive doubling algorithm for the solution of tridiagonal linear systems using O ( n / p + log p ) parallel arithmetic steps on a parallel computer with p

Abstract: The aim of this paper is to show that a kind of boundary value problem for second-order ordinary differential equations which reduces to the problem of solving tridiagonal systems of linear equations can be efficiently solved on modern multicore computer architectures. A new method for solving such tridiagonal systems of linear equations, based on recently developed algorithms for solving linear recurrence systems with constant coefficients, can be easily vectorized and parallelized. Further improvements can be achieved when novel data formats for dense matrices are used.

Abstract: Parallel implementation of algorithm of numerical solution of Navier-Stokes equations for large eddy simulation (LES) of turbulence is presented in this research. The Dynamic Smagorinsky model is applied for sub-grid simulation of turbulence. The numerical algorithm was worked out using a scheme of splitting on physical parameters. At the first stage it is supposed that carrying over movement amount takes place only due to convection and diffusion. Intermediate field of velocity is determined by method of fractional steps by using Thomas algorithm (tridiaginal matrix algorithm). At the second stage found intermediate field of velocity is used for determination of the field of pressure. Three dimensional Poisson equation for the field of pressure is solved using upper relaxation method. Moreover various ways of geometrical decomposition for parallel numerical solution of three dimensional Poisson equations are investigated.

Abstract: The decomposition method which makes the parallel solution of the block-tridiagonal matrix systems possible is presented. The performance of the method is analytically estimated based on the number of elementary multiplicative operations for its parallel and serial parts. The computational speedup with respect to the conventional sequential Thomas algorithm is assessed for various types of the application of the method. It is observed that the maximum of the analytical speedup for a given number of blocks on the diagonal is achieved at some finite number of parallel processors. The values of the parameters required to reach the maximum computational speedup are obtained. The benchmark calculations show a good agreement of analytical estimations of the computational speedup and practically achieved results. The application of the method is illustrated by employing the decomposition method to the matrix system originated from a boundary value problem for the two-dimensional integro-differential Faddeev equations. The block-tridiagonal structure of the matrix arises from the proper discretization scheme including the finite-differences over the first coordinate and spline approximation over the second one. The application of the decomposition method for parallelization of solving the matrix system reduces the overall time of calculation up to 10 times.