Splitting of Expanded Tridiagonal Matrices

The article addresses a regular splitting of tridiagonal matrices. The given tridiagonal matrix A is rst expanded to an equivalent matrix e A and then split as e A = B R for which B is block-diagonal and every eigenvalue of B R is zero, i.e., (M N) = 0. The optimal splitting technique is applicable to various algorithms that incorporate one-dimensional solves or their approximations. Examples can be found in the parallelization of alternating direction iterative (ADI) methods and e cient parameter choices for domain decomposition (DD) methods for elliptic and parabolic problems. Numerical results solving the Helmholtz wave equation in two dimensions are presented to demonstrate usefulness and e ciency of the splitting technique.