Reliable Parallel Solution of Bidiagonal Systems

This paper presents a new efficient algorithm for solving bidiagonal systems of linear equations on massively parallel machines. We use a divide and conquer approach to compute a representative subset of the solution components after which we solve the complete system in parallel with no communication overhead. We address the numerical properties of the algorithm in two ways: we show how to verify the à posteriori backward stability at virtually no additional cost, and prove that the algorithm is à priori forward stable. We then show how we can use the algorithm in order to bound the possible perturbations in the solution components.