Tridiagonal matrix algorithm

Abstract: The approximability of the resolvent of an operator induced by a band matrix by the resolvents of its finite-dimensional sections is studied. For bounded perturbations of self-adjoint matrices a positive result is obtained. The convergence domain of the sequence of resolvents can be described in this case in terms of matrices involved in the representation. This result is applied to tridiagonal complex matrices to establish conditions for the convergence of Chebyshev continued fractions on sets in the complex domain. In the particular case of compact perturbations this result is improved and a connection between the poles of the limit function and the eigenvalues of the tridiagonal matrix is established.

Abstract: This paper describes programs to reduce a nonsymmetric matrix to tridiagonal form, to compute the eigenvalues of the tridiagonal matrix, to improve the accuracy of an eigenvalue, and to compute the corresponding eigenvector. The intended purpose of the software is to find a few eigenpairs of a dense nonsymmetric matrix faster and more accurately than previous methods. The performance and accuracy of the new routines are compared to two EISPACK paths: RG and HQR-INVIT. The results show that the new routines are more accurate and also faster if less than 20 percent of the eigenpairs are needed.

Abstract: An algorithm combining Gaussian elimination with the modified Gram-Schmidt (MGS) procedure is given for solving the linear least squares problem. The method is based on the operational efficiency of Gaussian elimination for LU decompositions and the numerical stability of MGS for unitary decompositions and is designed for slightly overdetermined linear systems.

Abstract: Two methods are presented which efficiently solve tridiagonal systems on vector supercomputers and parallel computers with a moderate degree of parallelism. The first algorithm for diagonally dominant systems uses incomplete Gaussian elimination, the other for more general systems applies Gaussian elimination with partial pivoting. The methods are based on wrap-around partitioning, which is closely related to the partitioning used in Wang's algorithm. The first algorithm delivers an asymptotic speedup by a factor ofp on ap-processor computer if compared to the scalar algorithm, whereas the second algorithm delivers a speedup by a factor of roughlyp/2, which is also typical for cyclic reduction. For the incomplete factorization, existence and approximation properties are proved. Timing experiments were run on a Cray X-MP.