A maximally pipelined tridiagonal linear equation solver

Abstract

A new method of solving tridiagonal systems of linear equations is proposed which introduces parallelism in a way that may be effectively exploited by a suitable parallel computer architecture. The scheme is based on the program transformation techniques which can produce machine code to be executed in a maximally pipelined fashion. Compared to the existing parallel solution techniques such as the cyclic reduction algorithms, the new method has the following advantages: (1) It eliminates the substantial data rearrangement overhead incurred by many existing parallel algorithms; (2) it sustains a relatively constant parallelism during various phases of program execution; (3) the code generation is independent of the length of the vectors to be computed, hence is more flexible; (4) in general the size of machine code is much smaller and hence is more efficient in terms of memory usage. The new method is also numerically stable. Using the new method, we outline the code structure of a maximally pipelined tridiagonal linear equation solver for a static data flow super-computer.