The result of Greenbaum, Pt\'ak, and Strako\v{s} that for a given set of eigenvalues, any convergence curve is possible [SIMAX 1996] and the subsequent parameterization of such matrix-right-hand side pairs $(A,\mathbf{b})$ of Arioli, Pt\'ak, and Strako\v{s} [BIT 1998] demonstrated that the behavior of the GMRES could not be completely characterized by the eigenvalues of $A$ alone. In this paper, we consider how to use this theory to understand the admissible and attainable \gmres behavior for matrices with constrained structure, focussing on non-Hermitian (non-symmetric) tridiagonal Toeplitz matrices. We show that Toeptliz structure necessarily constrains the how the theory from these papers can manifest but that a continuum of admissible behaviors is still attainable.
We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\hat a(z) = (1-z)^{\alpha/2}(1-z^{-1})^{\alpha/2}$ for $\alpha \in (0,2)$. Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenva...
Jacek Wszoła· Linear Algebra and its Appli...· 0 citations
In previous work by several authors, the behavior of the condition numbers of banded Toeplitz matrices was studied as the matrix size tends to infinity. In the present contribution, two main directions are pursued. As a first step, we extend this study to block Toeplitz matrices with blocks of fixed size $N$. As in the...
M. Bogoya, A. Böttcher, Matteo Ferrari et al.· 0 citations
Extended Krylov subspace methods (EKSMs) are a well known approach for approximating the action of a matrix function on a vector $f(A)\mathbf{b}$, and their block versions are often used for solving Lyapunov and Sylvester matrix equations. EKSMs produce Laurent approximations to the underlying operator, which are gener...
We develop Toeplitz infinite GMRES for solving large sparse analytic parameterized systems $A(s)x(s)=z$ at many parameter values. The method exploits a block upper triangular Toeplitz structure in the companion Krylov sequence to construct the Arnoldi process and recover solution approximations without storing full Arn...
It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-h...
Gao-Feng Huang, F. Kutzschebauch, Son Nam Tran et al.· 0 citations
We study two-step recurrences with positive step sizes and nonnegative momentum coefficients whose terminal residual is the Chebyshev minimax polynomial on a prescribed spectral interval $[\mu,L]$. These recurrences have the same terminal error on a fixed quadratic, but their first-order responses to stepwise Hessian p...
Dmitry Pasechnyuk-Vilensky, Martin Takác· 0 citations
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