Tridiagonal matrix algorithm

Abstract: In this paper, accelerated finite difference method for solving singularly perturbed delay reaction-diffusion equations is presented. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of equations is obtained, which can easily be solved by Thomas algorithm. The consistency, stability and convergence of the method have been established. To increase the accuracy of our established scheme we used Richardson's extrapolation techniques. To validate the applicability of the proposed method, four model examples have been considered and solved for different values of perturbation parameters and mesh sizes. The numerical results have been presented in tables and graphs to illustrate; the present method approximates the exact solution very well. Moreover, the present method gives better accuracy than the existing numerical methods mentioned in the literature.

Abstract: We present a design and implementation of the Thomas algorithm optimized for hardware acceleration on an FPGA, the Thomas Core. The hardware-based algorithm combined with the custom data flow and low level parallelism available in an FPGA reduces the overall complexity from 8N down to 5N serial arithmetic operations, and almost halves the overall latency by parallelizing the two costly divisions. Combining this with a data streaming interface, we reduce memory overheads to 2 N-length vectors per N-tridiagonal system to be solved. The Thomas Core allows for multiple independent tridiagonal systems to be continuously solved in parallel, providing an efficient and scalable accelerator for many numerical computations. Finally we present applications for derivatives pricing problems using implicit finite difference schemes on an FPGA accelerated system and we investigate the use and limitations of fixed-point arithmetic in our algorithm.

Abstract: Quadarant diagonal partitioning (QDP) method is an iterative method for solving linear system proposed by D. J. Evans et al. (see [1],[2],[3]) appropriate for paralell implementation. Here we show that in case of second order elliptic problem, the classical, color and multicolor ordering technique [4] are applicable for further paralellization of QDP iterative methods. Moreover we give a direct algorithm with arithmetical complexity O(n) for solving a special quadrant tridiagonal matrix and using these results we propose a quadrant tridiagonal preconditioned (QTP) conjugate gradient method for solving second order elliptic problems. The numerical experiments presented here shows that this method in several cases are more effective than other PCG algorithm.