Parallel Solvers for Almost-Tridiagonal Linear Systems

The problem of solving tridiagonal systems on parallel machines has been studied extensively. This paper examines an existing parallel solvers for tridiagonal systems and extends this divide-and-conquer algorithm to solving almost-tridiagonal systems, systems consisting of a tridiagonal matrix with non-zeros elements in the upper right and lower left corners. In addition to a sketch of a solver already used in global climate modeling code, where almost-tridiagonal systems arise because the problem domain wraps around a sphere, two new algorithms are proposed, one using a pipeline to send many small messages, and another sending a few large messages. In-depth descriptions are provided with pseudocode where considered helpful. The algorithms are generalized to the case where there are multiple matrices and multiple right hand side vectors to be solved for. The proposed algorithm which minimizes messages has been implemented in CMMD and a discussion of preliminary timing results is given.