Tridiagonal matrix algorithm

Abstract: Based on the idea of chasing method,a new algorithm is developed to solving the circular and quasi-circular tridiagonal systems in this paper.The computational costs of multiplication and division are 8N and 3N,respectively.Compared with the traditional method,the new chasing method saves the computational cost.The numerical experiments indicate that,the exact solutions can be obtained in several seconds by using this method.

Abstract: In this paper we propose a method for the numerical solution of singularly perturbed two-point boundary value problems (BVPs) with the boundary layer at left or right point. A fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the system. Error estimates are derived and numerical examples are solved to illustrate the present method.

Abstract: In this article we discuss a small modification of the bisection routines in EISPACK and LAPACK for finding a few of the eigenvalues of a symmetric tridiagonal matrix A. When the principal minors of the matrix A yield good approximations to the desired eigenvalues, these modifications can yield about 30% reduction in the computation times.

Abstract: Let M be a square matrix of order Nn. Partition M into N 2 square matrices M(A, B), 1 < A, B < N, of order n; and let M(A, B; i , j ) , 1 < i,j < n, denote the element in the ith row and j th column o f M ( A , B). In general, for large values of Nn the problem of determining the eigenvalues and eigenvectors can only be solved by extensive calculation. For special classes of matrices M, of the type we shall consider, the eigenvalues and eigenvectors can be found explicitly• To motivate our specialization we remark that some problems involving the approximation of linear second order partial differential equations in two independent variables by difference equations lead to block tridiagonal coefficient matrices M, i.e. M(A,B) = 0 when [A -B] > 1. In particular, if the partial differential equation being approximated has constant coefficients, one may have