Tridiagonal matrix algorithm

Abstract: The popular sequential Thomas algorithm for the numerical solution of tridiagonal linear algebraic equation systems is extended on a class of quasi-block-tridiagonal equation systems arising from finite-difference discretisations of boundary value and initial boundary value problems of reaction-migration-advection-diffusion type in one space dimension, occurring in electrochemistry. The extension allows for a simultaneous consideration of: (a) multiple space intervals with common boundaries; (b) additional algebraic or differential-algebraic equations coupled with mixed boundary conditions, that may express e.g. adsorption at the boundaries; (c) three-point finite-difference approximations to the gradients of the solutions of the initial/boundary value problems at the boundaries; (d) periodic or non-periodic boundary conditions at the external boundaries. The resulting equation matrix may include nonzero off-diagonal corner blocks associated with periodic boundary conditions, may be locally block-pentadiagonal at a number of isolated rows corresponding to internal spatial boundaries, and its blocks may have variable dimensions. Testing calculations are performed.

Abstract: In this paper, we consider whose invereses are tridiagonal Z-matrices Based on a characterization of symmetric tridiagonal matirices by Gantmacher and Kein, we show that a matrix is the inverse of a tridiagonal Z-matrix if and only if up to a positive scaling of the rows,it is the Hadamard product of a so called weak type D matrix and a flipped weak type D matrix whose parameters satisfy certiain quadratic conditions we predict from these parameters to which class of Z-matices the inverse belings to In particular, we give a characterization of inverse tridiagonal M-matrices Moreover ,we charactetrize inverese of diagonal M-matrices that saftisfy certain row sum ceriteria. This leads to the cyclopses that are matrices constructed from type D and flipped type D matrices .we establish some properties of the cyclopses and provide explicit formulae for the entries of the inverse of a nonsingular cyclopses. we also shoe that the cyclopses are the only generalized ultrametric matrices whose inverses are tridiagonal

Abstract: In this paper, the point A.G.E. method is extended to obtain the solution of block tridiagonal linear systems derived from the discretisation of multidimensional elliptic boundary value problems. The numerical results obtained agree with the theoretical results presented earlier.