The authors consider iterative methods for the solution of tridiagonal systems and present a new iteration whose rate of convergence is comparable to that of the optimal two-cyclic Chebyshev iteration but which does not require the calculation of optimal parameters. The theory has a natural extension to block tridiagonal systems. Numerical experiments suggest that on a parallel computer this new algorithm is the best of the iterative algorithms considered.

Mathematics of Computing -- General.

Mathematics of Computing -- Parallelism.

This volume deals with problems of modern effective algorithms for the numerical solution of the most frequently occurring elliptic partial differential equations. From the point of view of implementation, attention is paid to algorithms for both classical sequential and parallel computer systems. The first two chapters are devoted to fast algorithms for solving the Poisson and biharmonic equation. In the third chapter, parallel algorithms for model parallel computer systems of the SIMD and MIMD types are described. The implementation aspects of parallel algorithms for solving model elliptic boundary value problems are outlined for systems with matrix, pipeline and multiprocessor parallel computer architectures. A modern and popular multigrid computational principle which offers a good opportunity for a parallel realization is described in the next chapter. More parallel variants based in this idea are presented, whereby methods and assignments strategies for hypercube systems are treated in more detail. The last chapter presents VLSI designs for solving special tridiagonal linear systems of equations arising from finite-difference approximations of elliptic problems. For researchers interested in the development and application of fast algorithms for solving elliptic partial differential equations using advanced computer systems.