Tridiagonal matrix algorithm

Abstract: The ScaLAPACK library contains a pair of routines for solving banded linear systems which are strictly diagonally dominant by rows. Mathematically, the algorithm is complete block cyclic reduction corresponding to a particular block partitioning of the system. In this paper we extend Heller's analysis of incomplete cyclic reduction for block tridiagonal systems to the ScaLAPACK case. We obtain a tight estimate on the significance of the off diagonal blocks of the tridiagonal linear systems generated by the cyclic reduction algorithm. Numerical experiments illustrate the advantage of omitting all but the first reduction step for a class of matrices related to high order approximations of the Laplace operator.

Abstract: In this paper, a fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at both end (left and right) points. We have introduced a fitting factor in Dahlquist tridiagonal finite difference scheme and obtained its value from the theory of singular perturbations. This fitting factor takes care of the rapid changes that occur in the boundary layer. The efficient Thomas algorithm is used to solve the tridiagonal system. Maximum absolute errors are presented in tables to show the efficiency of the method.

Abstract: The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU. Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.

Abstract: The efficiency of the second algorithm described in [1] for determining the zeros of a polynomial has been investigated in connection with the computation of the eigenvalues of a tridiagonal matrix, all the eigenvalues being computed at the same time together with their range of approximation. Let A be the given matrix of order N whose coefficients (real or complex) are arranged in the following way: