Tridiagonal matrix algorithm

Abstract: An adaptive Richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right-hand side vectors. This scheme ``learns'' about the linear system to be solved by computing inner products of residual matrices during the iterations. These inner products are interpreted as block modified moments. A block version of the modified Chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials. The eigenvalues of this block tridiagonal matrix define an interval, which determines the choice of relaxation parameters for Richardson iteration. Only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right-hand side vectors. We outline the changes required.

Abstract: While numerically stable techniques have been available for deflating a fulln byn matrix, no satisfactory finite technique has been known which preserves Hessenberg form. We describe a new algorithm which explicitly deflates a Hessenberg matrix in floating point arithmetic by means of a sequence of plane rotations. When applied to a symmetric tridiagonal matrix, the deflated matrix is again symmetric tridiagonal. Repeated deflation can be used to find an orthogonal set of eigenvectors associated with any selection of eigenvalues of a symmetric tridiagonal matrix.

Abstract: Many algorithms for solving the Navier-Stokes equations require the solution of periodic block tridiagonal systems of equations. By applying a splitting to the matrix representing this system of equations, it may first be reduced to a block tridiagonal matrix plus an outer product of two block vectors. The Sherman-Morrison-Woodbury formula is then applied. The algorithm thus reduces a periodic banded system to a non-periodic banded system with additional right-hand sides and is of higher efficiency than standard Thomas algorithm/LU decompositions.

Abstract: The first step in the development of a chip set to support eigenvalue-eigenvector-based estimation algorithms is presented. It is based on the assumption that an averaging technique will produce a symmetric covariance matrix. Such a matrix can be reduced to a symmetric tridiagonal matrix, and hence the eigenvalues and eigenvectors can be found by successive iterations involving QR decomposition. The architecture is unique in that other architectures either solve only for the eigenvalues or use methods other than QR iteration. It has potential for use in a systolic computer for computer intensive digital signal processing based on modern spectral-analysis techniques. >