Tridiagonal matrix algorithm

Abstract: In this paper, a new class of parallel Gaussian elimination algorithms is presented for the solution of tridiagonal matrix systems. The new algorithms, called ACER (alternating cyclic elimination and reduction), combine the advantages of the well known cyclic elimination algorithm (which is fast) and the cyclic reduction algorithms (which requires fewer operations). The ACER algorithms are developed with the unifying graph model.

Abstract: : There has been no stable direct method for solving symmetric systems of linear equations which takes advantage of the symmetry. If the system is also positive definite, then fast, stable direct methods (e.g., Cholesky and symmetric Gaussian elimination) exist which preserve the symmetry. These methods are unstable for symmetric indefinite systems. Such systems often occur in the calculation of eigenvectors. Gaussian elimination with partial or complete pivoting is currently recommended for solving symmetric indefinite systems, and here symmetry is lost. A generalization of symmetric Gaussian elimination is presented here, called the diagonal pivoting method, in which pivots of order two as well as one are allowed in the decomposition. It is shown that the diagonal pivoting method for symmetric indefinite matrices takes advantage of symmetry so that only 1/6 n cubed multiplications, at most 1/3 n cubed additions, and 1/2 n squared storage locations are required to solve A x = b, where A is a non-singular symmetric matrix of order n. Furthermore, it is shown that the method is nearly as stable as Gaussian elimination with complete pivoting, while requiring only half the number of operations and half the storage.

Abstract: This article deals with the problem of estimating the error in the computed solution to a system of equations when that solution is obtained by using Gaussian elimination without pivoting. The corresponding problem, where either partial or complete pivoting is used, has received considerable attention, and efficient and reliable methods have been developed. However, in the context of solving large sparse systems, it is often very attractive to apply Gaussian elimination without pivoting, even though it cannot be guaranteed a-priori that the computation is numerically stable. When this is done, it is important to be able to determine when serious numerical errors have occurred, and to be able to estimate the error in the computed solution. In this paper a method for achieving this goal is described. Results of a large number of numerical experiments suggest that the method is both inexpensive and reliable.