Tridiagonal Splittings in the Conditioning and Parallel Solution of Banded Linear Systems

We study the conditioning and the parallel solution of banded linear systems of algebraic equations. We propose an iterative method for solving the linear system Au = b based on a tridiagonal splitting of the real coefficient matrix A which permits the study of the conditioning and the parallel solution of banded linear systems using the theoretical results known for tridiagonal systems. Sufficient conditions for the convergence of this method are studied, and the definition of tridiagonal dominant matrices is introduced, observing that for this class of matrices the iterative method converges. When the iterative method converges, the conditioning of A nay be studied using that of its tridiagonal part. Finally, we consider a parallel version of this iterative method and show some parallel numerical tests.