Tridiagonal matrix algorithm

Abstract: Many scientific computer codes involve linear systems of equations which are coupled only between nearest neighbors in a single dimension. The most common situation can be formulated as a tridiagonal matrix relating source terms and unknowns. This system of equations is commonly solved using simple forward and back substitution. The usual algorithm is spectacularly ill suited for parallel processing with distributed data, since information must be sequentially communicated across all domains. Two new tridiagonal algorithms have been implemented in FORTRAN 77. The two algorithms differ only in the form of the unknown which is to be found. The first and simplest algorithm solves for a scalar quantity evaluated at each point along the single dimension being considered. The second algorithm solves for a vector quantity evaluated at each point. The solution method is related to other recently published approaches, such as that of Bondeli. An alternative parallel tridiagonal solver, used as part of an Alternating Direction Implicit (ADI) scheme, has recently been developed at LLNL by Lambert. For a discussion of useful parallel tridiagonal solvers, see the work of Mattor, et al. Previous work appears to be concerned only with scalar unknowns. This paper presents a new technique which treatsmore » both scalar and vector unknowns. There is no restriction upon the sizes of the subdomains. Even though the usual tridiagonal formulation may not be theoretically optimal when used iteratively, it is used in so many computer codes that it appears reasonable to write a direct substitute for it. The new tridiagonal code can be used on parallel machines with a minimum of disruption to pre-existing programming. As tested on various parallel computers, the parallel code shows efficiency greater than 50% (that is, more than half of the available computer operations are used to advance the calculation) when each processor is given at least 100 unknowns for which to solve.« less

Abstract: We give a counter example which shows that the non-singularity of the tridiagonal matrix A is not the sufficient condition for the existence of the new quadrant interlocking factorization (Q.I.F.) given by M. M. Chawla and K. Passi [1] for the solution of tridiagonal linear systems. However, we prove the existence of the Q.I.F. when A is diagonally dominant in addition to non-singularity.