2,186 research outputs found
Convergence of the Iterative Rational Krylov Algorithm
The Iterative Rational Krylov Algorithm (IRKA) of [8] is an interpolatory
model reduction approach to the optimal approximation problem.
Even though the method has been illustrated to show rapid convergence in
various examples, a proof of convergence has not been provided yet. In this
note, we show that in the case of state-space symmetric systems, IRKA is a
locally convergent fixed point iteration to a local minimum of the underlying
approximation problem
A nested Krylov subspace method to compute the sign function of large complex matrices
We present an acceleration of the well-established Krylov-Ritz methods to
compute the sign function of large complex matrices, as needed in lattice QCD
simulations involving the overlap Dirac operator at both zero and nonzero
baryon density. Krylov-Ritz methods approximate the sign function using a
projection on a Krylov subspace. To achieve a high accuracy this subspace must
be taken quite large, which makes the method too costly. The new idea is to
make a further projection on an even smaller, nested Krylov subspace. If
additionally an intermediate preconditioning step is applied, this projection
can be performed without affecting the accuracy of the approximation, and a
substantial gain in efficiency is achieved for both Hermitian and non-Hermitian
matrices. The numerical efficiency of the method is demonstrated on lattice
configurations of sizes ranging from 4^4 to 10^4, and the new results are
compared with those obtained with rational approximation methods.Comment: 17 pages, 12 figures, minor corrections, extended analysis of the
preconditioning ste
From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation
We propose a new method for the approximate solution of the Lyapunov equation
with rank- right-hand side, which is based on extended rational Krylov
subspace approximation with adaptively computed shifts. The shift selection is
obtained from the connection between the Lyapunov equation, solution of systems
of linear ODEs and alternating least squares method for low-rank approximation.
The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure
A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions
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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