Abstract: If numerical analysts understand anything, surely it must be Gaussian elimination. This is the oldest and truest of numerical algorithms. To be precise, I am speaking of Gaussian elimination with partial pivoting, the universal method for solving a dense, unstructured n X n linear system of equations Ax = b on a serial computer. This algorithm has been so successful that to many of us, Gaussian elimination and Ax = b are more or less synonymous. The chapter headings in the book by Golub and Van Loan [3] are typical -- along with "Orthogonalization and Least Squares Methods," "The Symetric Eigenvalue Problem," and the rest, one finds "Gaussian Elimination," not "Linear Systems of Equations."

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.

Abstract: An algorithm symbolically calculating the trace of the power of a tridiagonal matrix is proposed. The setting is based on techniques developed from structure analysis and combinatorics. The complexity analysis, the extension and the possible applications of this algorithm are also discussed.

Abstract: In many programs solving difference equations, problem size is restricted by the number of available memory cells. A strategy has been developed to permit trade-offs between the number of floating point operations required and storage requirements for the solution of certain problems such as block tridiagonal systems of equations. This is done by recomputing some intermediate results instead of storing them. Reducing the storage to the square root of the current requirement will roughly double the number of computations. In theory, if m is the order of each sub-matrix in the block tridiagonal matrix, one can solve any linear system with only 5 sq m + 1 temporary storage cells. This method lends itself to efficient use on computers with parallel processing or vector processing architectures. On these computers the larger number of floating point operations is more than offset by the decrease in I/O and the increased percentage of vector operations made possible by this algorithm.

Abstract: : We propose a few implementations of the Alternating Direction Method for solving parabolic partial differential equations on multiprocessors. A careful complexity analysis of these implementations shows that, contrary to what is generally believed, the method can be made highly efficient on parallel architectures by using pipelining and variations of the classical Gaussian elimination algorithm for solving tridiagonal systems.

Abstract: A bounded cyclic self-adjoint operator C defined on a separable Hilbert space H can be represented as a tridiagonal matrix with respect to the basis generated by the cyclic vector. An operator / can then be defined so that CJ - JC = -2iK where K also has tridiagonal form. If the subdiagonal elements of C converge to a non-zero limit and if K is of trace class then C must have an absolutely continuous part.

Abstract: In this paper, we consider the problem of solving a sparse nonsingular system of linear equations. We show that the structures of the triangular matrices obtained in the $LU$-decomposition of a sparse nonsingular matrix A using Gaussian elimination with partial pivoting are contained in those of the Cholesky factors of $A^T A$, provided that the diagonal elements of A are nonzero. Based on this result, a method for solving sparse linear systems is then described. The main advantage of this method is that the numerical computation can be carried out using a static data structure. Numerical experiments comparing this method with other implementations of Gaussian elimination for solving sparse linear systems are presented and the results indicate that the method proposed in this paper is quite competitive with other approaches.