Accurate ordering of eigenvectors and singular vectors without eigenvalues and singular values

The rank of an eigenvector of an unreduced real symmetric tridiagonal matrix can be determined by just knowing the signs of the elements of the eigenvector and the signs of the off-diagonal entries of the tridiagonal matrix. Surprisingly, no arithmetic operations involving real numbers are required to determine this ordinal count. The absence of real arithmetic operations guarantees an error free algorithm. Thus, it is possible to rank and order eigenvectors without knowing the corresponding eigenvalues. It is known that the singular value decomposition (SVD) of bidiagonal matrices are closely related to three tridiagonal eigenvalue problems. Using this connection, it is possible to order singular vectors of bidiagonal matrices without knowing the singular values. Again, no real arithmetic is needed. The ordinal count is a theoretical result, which is valid in exact arithmetic. If eigenvectors and singular vectors are poorly determined in floating-point arithmetic then the ordering procedure can detect faulty eigenpairs. Three standard symmetric eigenproblem routines and two SVD routines from LAPACK are investigated to verify whether the ordinal counts give valid results for computed eigenvectors. For some difficult problems such as the Wilkinson Wn+ matrix, the ordinal counts disagree with that given by the routines for n 25. In general, standard LAPACK routines appear to be reasonably resilient even though the routines are not particularly designed to be robust in this context. For some classes of tridiagonals, no failures were detected for n exceeding 3000. However, one of the new LAPACK routines cannot be considered to be robust since it destroys information by thresholding small eigenvector elements to zero. An SVD routine also gave poor results for graded matrices. This could be due to a bug. Many eigenvalue algorithms do a global sort of the eigenvalues. However, if the ordinal count is reliable then ranking of eigenvectors across processors can be achieved without global communication.