The complexity of the matrix eigenproblem

The eigenproblem for an n-by-n matrix A is the problem of the approximation (within a relative error bound 2-‘) 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(n3 + (nlog’n)log b). If the characteristic and minimum polynomials of the matrix A coincide with each other (which is the ca8e for generic matrices of all classes of general and special matrices that we consider), then the latter deterministic cost bound can be replaced by the randomized bound O(K.a(2n) + n* + (n log’n) log b) where Ka(2n) denotes the cost of the computation of the 2n 1 vectors A’v, i = 1,. ,2n 1, maximized over all n-dimensional vectors v; Ka(2n) = O(M(n)logn), for M(n) = o(n2.3’6) denoting the arithmetic complexity of n x 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 eigenprohlem for the important special classes of matrices. In particular, we prove the bound O(n’ log n + (n log’ n) log b) on the randomized arithmetic complexity of the eigenproblem for generic matrices of the classes of n x n Toeplitz, Hank& Toeplitzlike, Hank&like and Toeplits-likeplus-Hank&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 Caucby-like, Vandermonde-like and sparse matrices. All our complexity estimates for the eigenproblem improve the known ones by order of magnitude.

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