The Complexity of the AlgebraicEigenproblem

The eigenproblem for an n-by-n matrix A is the problem of the approximation (within a relative error bound 2 ?b) of all the eigenvalues of the matrix A and computing the associated eigenspaces of all these eigenvalues. We show that the arithmetic complexity of this problem is bounded by O(n 3 + (n log 2 n) log b). If the characteristic and minimum polynomials of the matrix A coincide with each other (which is the case for generic matrices of all classes of general and special matrices that we consider), then the latter de-terministic cost bound can be replaced by the randomized bound O(K A (2n) + n 2 + (n log 2 n) log b) where K A (2n) denotes the cost of the computation of the 2n ?1 vectors A i v, i = 1; : : : ; 2n ?1, maximized over all n-dimensional vectors v; K A (2n) = O(M(n) log n), for M(n) = o(n 2:376) denoting the arithmetic complexity of n n matrix multiplication. This bound on the complexity of the eigenproblem is optimal up to a logarithmic factor and implies much faster solution of the eigenproblem for the important special classes of matrices. In particular, we prove the bound O(n 2 log n + (n log 2 n) log b) on the randomized arithmetic complexity of the eigenproblem for generic matrices of the classes of nn Toeplitz, Hankel, Toeplitz-like, Hankel-like and Toeplitz-like-plus-Hankel-like matrices. Then again, this bound is optimal (up to a logarithmic factor) for each of the latter classes of input matrices. We also prove similar nearly optimal upper bounds for the generic Cauchy-like, Vandermonde-like and sparse matrices. All our complexity estimates improve the known ones by order of magnitude.