An Accelerated Recursive Doubling Algorithm for Block Tridiagonal Systems

TL;DR: This work presents a novel algorithm, called the accelerated recursive doubling algorithm, that delivers O(R) improvement when solving block tridiagonal systems with R distinct right hand sides and this improvement translates to very significant speedups in practice.

Abstract: Block tridiagonal systems of linear equations arise in a wide variety of scientific and engineering applications. Recursive doubling algorithm is a well-known prefix computation-based numerical algorithm that requires O(M3(N/P + logP)) work to compute the solution of a block tridiagonal system with N block rows and block size M on P processors. In real-world applications, solutions of tridiagonal systems are most often sought with multiple, often hundreds and thousands, of different right hand sides but with the same tridiagonal matrix. Here, we show that a recursive doubling algorithm is sub-optimal when computing solutions of block tridiagonal systems with multiple right hand sides and present a novel algorithm, called the accelerated recursive doubling algorithm, that delivers O(R) improvement when solving block tridiagonal systems with R distinct right hand sides. Since R is typically ~ 102--104, this improvement translates to very significant speedups in practice. Detailed complexity analyses of the new algorithm with empirical confirmation of runtime improvements are presented. To the best of our knowledge, this algorithm has not been reported before in the literature.