A Low-Complexity Divide-and-Conquer Method for Computing Eigenvalues and Eigenvectors of Symmetric Band Matrices

Abstract

A framework for an efficient low-complexity divide-and-conquer algorithm for computing eigenvalues and eigenvectors of an n × n symmetric band matrix with semibandwidth b ≪ n is proposed and its arithmetic complexity analyzed. The distinctive feature of the algorithm—after subdivision of the original problem into p subproblems and their solution—is a separation of the eigenvalue and eigenvector computations in the central synthesis problem. The eigenvalues are computed recursively by representing the corresponding symmetric rank b(p−1) modification of a diagonal matrix as a series of rank-one modifications. Each rank-one modifications problem can be solved using techniques developed for the tridiagonal divide-and-conquer algorithm. Once the eigenvalues are known, the corresponding eigenvectors can be computed efficiently using modified QR factorizations with restricted column pivoting. It is shown that the complexity of the resulting divide-and-conquer algorithm is O (n ² b ²) (in exact arithmetic).