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Adaptation of the Lanczos and Arnoldi methods to the spectrum, or why the two Krylov subspace methods are powerful

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... Figure 3.4 shows the convergence rate of the Extended Krylov Subspace Method, together with our asymptotic estimate, where the optimal parameter a has once again been determined numerically. The agreement is sufficiently satisfactory, taking into account that on this problem adaptation of the method to the spectrum can be observed [35]. ...
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For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd.
... Then we derive a general convergence estimate for the RKSM approximation. We expect the Galerkin approximation associated with the rational Krylov subspace to be better than ADI, when using the same poles, because of the projection process, which allows the method to improve adaptation to the spectrum [34] . However , as we have already seen in Theorem 3.4, the two methods may be equivalent under certain choices of the shifts. ...
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For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for a long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit (ADI) recurrence. In this paper we develop a convergence analysis of the rational Krylov subspace method (RKSM) based on the Kronecker product formulation and on potential theory. Moreover, we propose new enlightening relations between this approach and the ADI method. Our results provide solid theoretical ground for recent numerical evidence of the superiority of RKSM over ADI when the involved parameters cannot be computed optimally, as is the case in many practical application problems.
... We want to point out that, in many cases, as for instance dealing with discretizations of elliptic operators, the RAM exhibits a very fast convergence and the a priori error bounds turn out to be pessimistic. As it is well known this fact often occurs in the application of Krylov subspace methods and it is due to their " good " adaptation to the spectrum (see [18]). Thus, in order to detect the actual behavior of the RAM, a posteriori error estimates could be more suitable. ...
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Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy–Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix.
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