Tridiagonal matrix algorithm

Abstract: Although it is known that Gaussian elimination method for solving simultaneous linear equations is not asymptotically optimal, it is still one of the most useful methods for solving systems of moderate size. This paper proposes some ideas how to speed-up the standard method. First, the trick which takes the advantage of the eventual symmetry of the system is presented, which speeds up the calculation by the factor slightly less than 2. Second, it is shown that by using some rearrangement of the calculation, it is possible to get additional speed-up, no matter whether the system is symmetric or not, although the eventual symmetry additionally doubles the execution speed. This rearrangement is performed using similar approach as in LU factorization, but retaining basic features of the Gaussian elimination method, like producing the triangular form of the system. As the required modifications in the original method are quite simple, the improved method may be used in all engineering applications where the original Gaussian elimination is used.

Abstract: Iterative methods such as generalized minimal residual (GMRES) method are used to solve large sparse linear systems. This paper is considered the GMRES method for solving tridiagonal block Toeplitz linear systems with diagonal blocks, and establishes upper bounds for GMRES residuals. The coefficient matrix becomes an m-tridiagonal Toeplitz matrix, and tridiagonal toeplitz systems are subcases of these systems. Also, we show that the GMRES method on linear system computes the exact solution in at most N steps.

Abstract: The problem of finding solution of a tridiagonal operator equation through its finite dimensional truncations is discussed. Effectively verifiable sufficient conditions are given. An algorithm is presented to compute the numerical approximation to the solution of Tx = y for a given tridiagonal operator. This is illustrated with a numerical example.