The complexity of the matrix eigenproblem
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
On zeros of polynomial and vector solutions of associated polynomial system from Vieta theorem
Applied Numerical Mathematics
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Computations with quasiseparable polynomials and matrices
Theoretical Computer Science
Proceedings of the forty-second ACM symposium on Theory of computing
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
Journal of the ACM (JACM)
On the boolean complexity of real root refinement
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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Surprisingly simple corollaries from the Courant--Fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set S interleaving the set E of the eigenvalues of an n X n real symmetric tridiagonal (rst) matrix Tn (as well as a point that splits E into two subsets of comparable cardinalities). Furthermore, we extend this algorithm so as to approximate all the n eigenvalues of Tn at nearly optimal sequential and parallel cost, that is, at the cost of staying within polylogarithmic factors from the straightforward lower bounds. The resulting improvement of the known processor bound in NC algorithms for the rst-eigenproblem is roughly by factor n. Our approach extends the previous works [M. Ben-Or and P. Tiwari, J. Complexity, 6(1990), pp. 417--442] and [M. Ben-Or et al., SIAM J. Comput., 17(1988), pp. 1081--1092] for the approximation of the zeros of a polynomial having only real zeros, and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem.