Tridiagonal matrix algorithm

Abstract: Let T be a tridiagonal operator on which has strict row and column dominant property except for some finite number of rows and columns. This matrix is shown to be invertible under certain conditions. This result is also extended to double infinite tridiagonal matrices. Further, a general theorem is proved for solving an operator equation using its finite-dimensional truncations, where T is a double infinite tridiagonal operator. Finally, it is also shown that these results can be applied in order to obtain a stable set of sampling for a shift-invariant space.

Abstract: We present and analyze a parallel method for the solution of partial differential equation models of the nervous system. These models mathematically are one-dimensional nonlinear parabolic equations defined on branching domains. Implicit methods for these equations leads to numerical solution of diagonally dominant almost tridiagonal linear systems at each time step. We first review some exact methods for the solution of these linear systems that includes an Exact Domain Decomposition. This EDD leads to the solution of many tridiagonal linear systems one for each branch. The sizes of these systems is equal to the number of grid points on the branch. Since the branches of realistic neurons vary widely in size, the decomposition leads to a very poor a priori load balance. This problem may be solved with the Overlapped Partition Method, a method for decomposing large diagonally dominant tridiagonal systems. We describe and analyze an algorithm based on EDD and OPM that can be load balanced.

Abstract: Motivated by the folding algorithm of Evans and Hatzopoulos [6] (see also [1]) for the solution of certain banded systems of linear equations, we describe a “new” folding Gaussian elimination algorithm for linear systems Ax = d with a full general coefficient matrix . We introduce a series of transformations Wm which simultaneously eliminate the elements and . For n even, the transformed system has a coefficient matrix with two half-size triangular subsystems uncoupled, obviating the need to solve 2×2 core subsystems as in [6]. The new algorithm has an arithmetical operations count of which is consistent with of the unidirectional algorithm; thus, it could possibly attain a speed up of 1.6 if implemented on a dual processor machine. The present algorithm can be adapted for banded linear systems; we consider its adaptation for tridiagonal linear systems where it exhibits a near perfect speed up over Thomas' algorithm.

Abstract: We show that certain integral positive definite symmetric tridiagonal matrices of determinant $n$ are in one to one correspondence with elements of $(\mathbb Z/n\mathbb Z)^*$ We study some properties of this correspondence In a somewhat unrelated second part we discuss a construction which associates a sequence of integral polytopes to every integral symmetric matrix