## Matrix iterations: the six gaps between potential theory and covergence |

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algorithm Appl Arnoldi iteration associated asymptotic convergence factor behavior Bi-CGSTAB boundary coefficient Comp complex plane computed condition number conformal map conjugate gradients constant continuous charge distribution convergence curve convergence rate curve of Figure discrete effect ellipse estimated asymptotic convergence example exterior of S0 Faber polynomials Figure 10(b Figure 17 Figure 22 finite floating-point arithmetic GAP2 GAP3 GMRES GMRES curve Green's function Greenbaum illustrates inequality initial vector ro interval isolated points iteration proceeds Iterative Methods Krylov subspace iterations L. N. Trefethen Lanczos iterations level curves linear equations Math mathematically Matrix Anal matrix iterations minimal norms minimax polynomial minimizes rn normal normal matrix nth root Numer optimal polynomials origin orthogonal outlier eigenvalue p€Pn phenomenon potential theory preconditioner problem pseudospectra region rounding errors S0 is connected Schwarz-Christoffel map sequence SIAM six gaps solving linear Theorem three-term recurrences Toeplitz Toeplitz matrix typically unit disk upper curve z-plane zeros