Tridiagonal matrix algorithm

Abstract: It is known that certain tridiagonal matrices have exact eigenvalues and eigenvectors There are sixteen documented tridiagonal matrix families, from the discretization of the one-dimensional Helmholtz equation that possess such properties Extended members of these matrices share a same set of eigenvectors making them commutative with respective to matrix multiplication We may therefore construct, in a fairly straightforward way, exact closed-form solutions of certain tridiagonal generalized matrix eigenvalue problems

Abstract: In this note we examine the algebraic variety of complex tridiagonal matrices , such that , where is a fixed real diagonal matrix. If then is , the set of tridiagonal normal matrices. For , we identify the structure of the matrices in and analyze the suitability for eigenvalue estimation using normal matrices for elements of . We also compute the Frobenius norm of elements of , describe the algebraic subvariety consisting of elements of with minimal Frobenius norm, and calculate the distance from a given complex tridiagonal matrix to .

Abstract: For linear systems with coefficient matrices of classical structure, WZ factorizations for matrices are basic mathematical theories to design a class of parallel algorithms. Based on WZ factorization, a parallel algorithm is provided for symmetric tridiagonal linear systems. The method estimates the computation task carefully so that it assigns the system skillfully to get even load balance. In addition, the algorithm makes full use of the overlap between computation and communication to reduce waiting time in each processor. Both the subsystem assigned in each processor and the reduced subsystem have the same computing logic, as a result, a two-level method forms. By theory analysis and experiment results, it can be concluded that our method is effective in load balance and efficiency.

Abstract: The computation of the eigenvalues and orthogonal eigenvectors of an N × N real symmetric tridiagonal matrix is a well known problem in numerical analysis. The problem frequently arises in the determination of eigenvalues and eigenvectors of dense and banded symmetric matrices and in connection with various families of orthogonal polynomials and special functions satisfying three term recurrence relations. Numerous algorithms exist for the solution of this problem, which typically require O(N2) operations for the determination of eigenvalues and O(N3) operations for the determination of orthogonal eigenvectors. In this thesis we propose a new class of fast algorithms for the computation of the eigenvalues of a symmetric tridiagonal matrix in O( N ln N) operations. Such an algorithm may be combined with any one of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues. The underlying technique is a divide-and-conquer approach which determines eigenvalues of a larger tridiagonal matrix from those of constituent matrices by the use of relations of their characteristic polynomials. The evaluation of characteristic polynomials is accelerated by the use of a technique known as the Fast Multipole Method. We provide a detailed presentation of a prototype for this class of algorithms and discuss several generalizations. An implementation of a prototype for this class of algorithms has been developed in FORTRAN, which serves to provide a comparison with existing techniques in terms of running time and accuracy. We present numerical results which demonstrate the effectiveness of the method.