An Accelerated Recursive Doubling Algorithm for Block Tridiagonal Systems

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 computationbased numerical algorithm that requires O(M 3 ( N P + log P )) 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 ∼ 10 2 − 10 4 , 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.

Optimizing the Accelerated Recursive Doubling Algorithm for Block Tridiagonal Systems of Equations

“…The original formulation of E i as described in [2] given in the theorem below, which also states the equivalence of the formulations described in this work and [2].…”

“…The Accelerated Recursive Doubling Algorithm presented in [2] accelerates the computation of solving for x for multiple right-hand sides b by separating the computation dependent on b (right-hand side dependent) and the computation independent of the right-hand side. In order to do this, B i is separated into two matrices, one right-hand side independent (C i ), and one right-hand side dependent (F i ).…”

“…C i and F i matrices have the following properties that help simplify expressions involving their products, as presented in [2].…”

“…Using this formulation, we can start from E 1 and determine all subsequent E i using the recursive formulation. This formulation of E i is simpler but equivalent to the original formulation in [2].…”

“…Note: A method suggested in [2] to make ARDA numerically stable for the same classes of matrices for which Cyclic Reduction is stable is to take the LU-decomposition of A, and then solve Ax = LU x = b in two steps: First solving Ly = b, and then solve U x = y.…”