## Matrix Iterations: The Six Gaps Between Potential Theory and Covergence |

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achieved actual algorithm annihilated Appl Applied approximation associated asymptotic convergence factor becomes begin behavior bound boundary Chebyshev close complex computed conformal map conjugate gradients connected consider constant continuous continuum convergence curve corresponding defined derivation discrete distance effect eigenvalues En(S equal equations estimated asymptotic convergence exact example explained exterior Faber Figure Figure 17 finite follows Frank H.T. Rhodes function GAP4 GMRES Greenbaum illustrates important inequality initial interval isolated points Iterative Methods known Krylov subspace iterations Lanczos limit linear systems Math mathematically matrix iterations minimal multiple non-normality nonsymmetric normal norms Numer operator optimal origin orthogonal orthogonal matrix outlier P(A)roll plot polynomial potential theory practice problem pseudospectra random ratio reason recurrence region require residual result roll rounding errors semidefinite programming sense sequence shows SIAM side Solution solving spectrum stages step symmetric Trefethen typically vectors w-plane zeros