On computing the eigenvectors of a class of structured matrices

A real symmetric matrix of order n   has a full set of orthogonal eigenvectors. The most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one, i.e., tridiagonal matrices or semiseparable matrices. This step is accomplished in O(n3)O(n3) operations. Once the latter symmetric structured matrix is available, its spectrum is computed in an iterative fashion by means of the QR   method in O(n2)O(n2) operations.In principle, the whole set of eigenvectors of the latter structured matrix can be computed by means of inverse iteration in O(n2)O(n2) operations.The blemish in this approach is that the computed eigenvectors may not be numerically orthogonal if clusters are present in the spectrum. To enforce orthogonality the Gram–Schmidt procedure is used, requiring O(n3)O(n3) operations in the worst case.In this paper a fast and stable method to compute the eigenvectors of tridiagonal and semiseparable matrices which does not suffer from the loss of orthogonality due to the presence of clusters in the spectrum is presented. The algorithm requires O(n2)O(n2) floating point operations if clusters of small size are present in the spectrum.