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 ? posteriori backward stability at virtually no additional cost, and prove that the algorithm is ? priori forward stable. We then show how we can use the algorithm in order to bound the possible perturbations in the solution components

RELIABLE SOLUTION OF BIDIAGONAL SYSTEMS WITH APPLICATIONS

We show that the stability of Gaussian elimination with partial pivoting relates to the well definition of the reduced triangular systems. We develop refined perturbation bounds that generalize Skeel bounds to the case of ill conditioned systems. We finally develop reliable algorithms for solving general bidiagonal systems of linear equations with applications to the fast and stable solution of tridiagonal systems