The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Computational methods of linear algebra

We consider the inversion of block tridiagonal, block Toeplitz matrices and comment on the behaviour of these inverses as one moves away from the diagonal. Using matrix Mobius transformations, we first present an representation (with respect to the number of block rows and block columns) for the inverse matrix and subsequently use this representation to characterize the inverse matrix. There are four symmetry-distinct cases where the blocks of the inverse matrix (i) decay to zero on both sides of the diagonal, (ii) oscillate on both sides, (iii) decay on one side and oscillate on the other and (iv) decay on one side and grow on the other. This characterization exposes the necessary conditions for the inverse matrix to be numerically banded and may also aid in the design of preconditioners and fast algorithms. Finally, we present numerical examples of these matrix types.

https://iopscience.iop.org/article/10.1088/1749-4680/8/1/015001/pdf

Characterizing the inverses of block tridiagonal, block Toeplitz matrices

A new iterative method for the solution of linear systems, based upon a new splitting of the coefficient matrix A , is presented. The method is obtained by considering splittings of the type A = ( A − M ) + M , where M −1 is a symmetric tridiagonal matrix, and by minimizing the Frobenius norm of the iteration matrix so derived. Numerical examples are provided, showing that our algorithm improves the rate of convergence of Jacobi method, without increasing the order of magnitude of the computational efforts required.

New techniques for the solution of linear systems by iterative methods

Parallel solution of block tridiagonal linear-systems

Algorithm 15 inversion of a blockwise tridiagonal matrix

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Algorithm 15 inversion of a blockwise tridiagonal matrix