The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality. Associated with a desired set of eigenvalues is a maximum "reachable invariant subspace" that can be developed from the given starting vector. Convergence for this distinguished subspace is bounded in terms involving a polynomial approximation problem. Elementary results from potential theory lead to convergence rate estimates and suggest restarting strategies based on optimal approximation points (e.g., Leja or Chebyshev points); exact shifts are evaluated within this framework. Computational examples illustrate the utility of these results. Origins of superlinear effects are also described.
Let j{z) = Yu?-\ a }l( z~z ))> where z, ^ 0 and ^" J^l / k^l < oo. Then / c a n be realized as the complex conjugate of the gradient of a logarithmic potential or, for integral a } , as the logarithmic derivative of a meromorphic function. We investigate conditions on a } and z j that guarantee that / has zeros. In the potential theoretic setting, this asks whether certain logarithmic potentials with discrete mass distribution have equilibrium points.
ABSTRACT. Let wx and W2 be two linearly independent solutions to w" + Aw = 0, where A is a transcendental entire function of order p{A) < 1. We show that the exponent of convergence \{E) of the zeros of E = wxw¿ is either infinite or satisfies p(A)_1 + \{E)~X < 2. For p{A) = i, this answers a question of Bank.
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