Acknowledgments References

17 where we take the finite sum as an infinite one. This does not affect significatively the bound but simplifies the eventual handling of the formulae. Thus, the value of the mth component of the error is bounded by and the explicit value of m leads to the proposition. A generalized vision of some parallel bidiagonal systems solvers. [Ston73] H. S. Stone. An efficient parallel algorithm for the solutionof a tridiagonal linear system of equations. e m [ ] δ m − 1 δ 1 − − s ∞ ≤ 16 Lemma 2. Let be an N x N linear system with the following characteristics: i). ii). Then, it is possible to verify that the necessary amount of overlapping m for OPM in order to have an absolute error less than ε. Proof. We take a general partition of the system , as explained in section 4. The indices for this general partition range from 1 to. The error of the elements of with respect to the equivalent elements of the actual solution is determined by. We know that e satisfies and. Only the first component of is different from 0. From lemma 1 we know that the value of the mth component of the error vector is bounded by for strictly diagonal dominant systems of equations. We also know that thus, it is necessary to bound. We bound by finding a bound for the solution of. Since is strictly diagonal dominant we can bound. We can write , where. Therefore is determined by a finite sum. Due to the structure of the matrix, it is easy to see that. At s = Ax 1 0 a 2 [ ] 1 … … 0 a N [ ] 1 t 1 [ ] … t N [ ] s 1 [ ] … s N [ ] = = min i 1 a i [ ] ⁄ () δ 1 > = m ε 1 δ 1 − − () log s ∞ δ 1 − log ≥ At s = D' p q' p s' p = R m + q' p x p e t p q' p − = D' p e c = c c 1 [ ] 0 … 0 T = = l' p 1 [ ] t p 0 [ ] 0 … 0 T = l' p e m [ ] δ …