Tridiagonal matrix algorithm

Abstract: When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations.

Abstract: We present a new algorithm for solving banded diagonal matrix problems efficiently on distributed-memory parallel computers, designed originally for use in dynamic alternating-direction implicit partial differential equation solvers The algorithm optimizes efficiency with respect to the number of numerical operations and to the amount of interprocessor communication This is called the ``delayed coupling method`` because the communication is deferred until needed We focus here on tridiagonal and periodic tridiagonal systems

Abstract: The definite integral is generally interpreted geometrically as an “area”. An alternate interpretation as a steady-state “flux” through a unit slab is derived, which leads to a new method of numerical integration. The usual sum of a large number of approximate areas is replaced by the flux through a “single” increment. The method involves the solution of a system of linear finite difference equations. The coefficient matrix is tri-diagonal and is solved efficiently by the Thomas algorithm. During the iterative process the coefficients are determined by simple quadrature schemes applied to each increment. Error analysis revealed that an expression could be derived for the roundoff error associated with the final Thomas iteration. It is shown that the roundoff error is smallest when the matrix coefficients a k \S>1. The method is shown to be superior to the classical methods due to its simplicity and tolerance for variable increment size. In addition, a new function is determined which is useful in diffusion studies. Numerical data are presented confirming these results.

Abstract: The inverse of a class of block tridiagonal matrices is investigated With the LU decomposition of the block tridiagonal matrix,an explicit expression of the block inverse elements is obtained A relation between the inverse elements is found,and a new algorithm for inverting a block tridiagonal matrix is established The computing complexity and computing time of this algorithm is lower than that of existed algorithms