Parallel Tridiagonal Equation Solvers

Harold S. Stone
1975 ACM Transactions on Mathematical Software  
This paper compares three parallel algorithms for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC IV and CDC STAR. For array computers similar to ILLIAC IV, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not
more » ... l, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases. When the tridiagonal system satisfies a strong diagonal dominance condition, the intermediate values computed by cyclic odd-even reduction form a rapidly convergent sequence, and thus the algorithm can be terminated early when values are correct to within machine accuracy. The convergence is linear until off diagonal terms fall below 1/3 the magnitude of the diagonal element, at which point the convergence becomes quadratic. The quadratic convergence is superior to the linear convergence reported by Traub for several parallel iterative tridiagonal solvers. Parallel Tridiagonal Equation Solvers by Harold S. Stone
doi:10.1145/355656.355657 fatcat:dcegxoxuajhllhqsig6yz75ota