Tridiagonal matrix algorithm

Abstract: : A one-dimensional fluid model of a surface-type dielectric barrier discharge is created using He as the background gas. This simple model, which only considers ionizing collisions and recombination in the electropositive gas, creates an important framework for future studies into the origin of experimentally observed flow control effects of the DBD. The two methods employed in this study include the semi-implicit sequential algorithm and the fully implicit simultaneous algorithm. The first involves consecutive solutions to Poisson's, the electron continuity, ion continuity and electron energy equations. This method combines a successive over relaxation algorithm as a Poisson solver with the Thomas algorithm tridiagonal routine to solve each of the continuity equations. The second algorithm solves an Ax=b system of linearized equations simultaneously and implicitly. The coefficient matrix for the simultaneous method is constructed using a Crank-Nicholson scheme for additional stability combined with the Newton-Raphson approach to address the non-linearity and to solve the system of equations. Various boundary conditions, flux representations and voltage schemes are modeled. Test cases include modeling a transient sheath, ambipolar decay and a radio-frequency discharge. Results are compared to validated computational solutions and/or analytic results when obtainable. Finally, the semi-implicit method is used to model a DBD streamer.

Abstract: In parallel solving weak diagonal dominant tridiagonal systems,the approximate error of the Parallel Diagonal Dominant(PDD) algorithm cannot be ignored.An iterated PDD algorithm was presented.In the algorithm,the solution of the correction value was calculated by iterative method,and the computational accuracy was obviously improved.Through error analysis on the algorithm,an estimation formula of iteration number was derived for a given error tolerance.And the numerical experiment shows the validity.Based on the complexity analysis of the iterative and non-iterative PDD algorithm,the increase of iterative algorithm computational complexity is very small,but the communication complexity increases exponentially with the iteration number.

Abstract: Matrix inverse computation is one of the fundamental mathematical problems of linear algebra and has been widely used in many fields of science and engineering. In this paper, we consider the inverse computation of an opposite-bordered tridiagonal matrix which has attracted much attention in recent years. By exploiting the low-rank structure of the matrix, first we show that an explicit formula for the inverse of the opposite-bordered tridiagonal matrix can be obtained based on the combination of a specific matrix splitting and the generalized Sherman–Morrison–Woodbury formula. Accordingly, a numerical algorithm is outlined. Second, we present a breakdown-free symbolic algorithm of $$O(n^2)$$ for computing the inverse of an n-by-n opposite-bordered tridiagonal matrix, which is based on the use of GTINV algorithm and the generalized symbolic Thomas algorithm. Finally, we have provided the results of some numerical experiments for the sake of illustration.

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