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2011

Volume 32, Issue 4, pp. 1079-1559


Hessenberg Matrix Properties and Ritz Vectors in the Finite-Precision Lanczos Tridiagonalization Process

Christopher C. Paige and Ivo Panayotov

SIAM. J. Matrix Anal. & Appl. 32, pp. 1079-1094 (16 pages)

Online Publication Date: October 04, 2011

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We derive some properties of complex Hessenberg matrices and use the relevant normal matrix cases to examine the lengths of Ritz vectors in the rounding error analysis of the Lanczos tridiagonalization process. This question is important for the computational use of the process and has already been studied for the real symmetric matrix case, but because of its intricate and unedifying nature, part of the theory was never submitted to scientific journals. We develop a new and more palatable theory which also applies to Lanczos processes adapted to any form of normal matrix with collinear eigenvalues such as a Hermitian or skew-Hermitian matrix. The nonnormal matrix properties are intended to help in the analysis of the unsymmetric Lanczos process.

Shifted Power Method for Computing Tensor Eigenpairs

Tamara G. Kolda and Jackson R. Mayo

SIAM. J. Matrix Anal. & Appl. 32, pp. 1095-1124 (30 pages)

Online Publication Date: October 04, 2011

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Recent work on eigenvalues and eigenvectors for tensors of order $m \ge 3$ has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form $\boldsymbol{\mathscr{A}}\mathbf{x}^{m-1} = \lambda \mathbf{x}$ subject to $\|\mathbf{x}\|=1$, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs.

Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations

Marco Donatelli, Matteo Semplice, and Stefano Serra-Capizzano

SIAM. J. Matrix Anal. & Appl. 32, pp. 1125-1148 (24 pages)

Online Publication Date: November 01, 2011

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In this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computational cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its convergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution.

Low-Rank Matrix Approximation with Weights or Missing Data Is NP-Hard

Nicolas Gillis and François Glineur

SIAM. J. Matrix Anal. & Appl. 32, pp. 1149-1165 (17 pages)

Online Publication Date: November 01, 2011

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Weighted low-rank approximation (WLRA), a dimensionality reduction technique for data analysis, has been successfully used in several applications, such as in collaborative filtering to design recommender systems or in computer vision to recover structure from motion. In this paper, we prove that computing an optimal WLRA is NP-hard, already when a rank-one approximation is sought. In fact, we show that it is hard to compute approximate solutions to the WLRA problem with some prescribed accuracy. Our proofs are based on reductions from the maximum-edge biclique problem and apply to strictly positive weights as well as to binary weights (the latter corresponding to low-rank matrix approximation with missing data).

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Nicola Guglielmi and Michael L. Overton

SIAM. J. Matrix Anal. & Appl. 32, pp. 1166-1192 (27 pages)

Online Publication Date: November 01, 2011

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The $\varepsilon$-pseudospectral abscissa and radius of an $n\times n$ matrix are, respectively, the maximal real part and the maximal modulus of points in its $\varepsilon$-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue decompositions of order $n$, making them impractical when $n$ is large. We present new algorithms based on computing only the spectral abscissa or radius of a sequence of matrices, generating a sequence of lower bounds for the pseudospectral abscissa or radius. We characterize fixed points of the iterations, and we discuss conditions under which the sequence of lower bounds converges to local maximizers of the real part or modulus over the pseudospectrum, proving a locally linear rate of convergence for $\varepsilon$ sufficiently small. The convergence results depend on a perturbation theorem for the normalized eigenprojection of a matrix as well as a characterization of the group inverse (reduced resolvent) of a singular matrix defined by a rank-one perturbation. The total cost of the algorithms is typically only a constant times the cost of computing the spectral abscissa or radius, where the value of this constant usually increases with $\varepsilon$, and may be less than 10 in many practical cases of interest.

Minimizing Condition Number via Convex Programming

Zhaosong Lu and Ting Kei Pong

SIAM. J. Matrix Anal. & Appl. 32, pp. 1193-1211 (19 pages)

Online Publication Date: November 01, 2011

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In this paper we consider minimizing the spectral condition number of a positive semidefinite matrix over a nonempty closed convex set $\Omega$. We show that it can be solved as a convex programming problem, and moreover, the optimal value of the latter problem is achievable. As a consequence, when $\Omega$ is positive semidefinite representable, it can be cast into a semidefinite programming problem. We then propose a first-order method to solve the convex programming problem. The computational results show that our method is usually faster than the standard interior point solver SeDuMi [J. F. Sturm, Optim. Methods Softw., 11/12 (1999), pp. 625–653] while producing a comparable solution. We also study a closely related problem, that is, finding an optimal preconditioner for a positive definite matrix. This problem is not convex in general. We propose a convex relaxation for finding positive definite preconditioners. This relaxation turns out to be exact when finding optimal diagonal preconditioners.

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

L. M. Carvalho, S. Gratton, R. Lago, and X. Vasseur

SIAM. J. Matrix Anal. & Appl. 32, pp. 1212-1235 (24 pages)

Online Publication Date: November 01, 2011

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This work is concerned with the development and study of a minimum residual norm subspace method based on the generalized conjugate residual method with inner orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of nonsymmetric or non-Hermitian linear systems. First we recall the main features of flexible generalized minimum residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.

Further Results for Perron–Frobenius Theorem for Nonnegative Tensors II

Qingzhi Yang and Yuning Yang

SIAM. J. Matrix Anal. & Appl. 32, pp. 1236-1250 (15 pages)

Online Publication Date: November 08, 2011

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For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.

A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix

P. G. Martinsson

SIAM. J. Matrix Anal. & Appl. 32, pp. 1251-1274 (24 pages) | Cited 1 time

Online Publication Date: November 08, 2011

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Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient but have off-diagonal blocks that are; specifically, the class of so-called hierarchically semiseparable (HSS) matrices. HSS matrices arise frequently in numerical analysis and signal processing, particularly in the construction of fast methods for solving differential and integral equations numerically. The HSS structure admits algebraic operations (matrix-vector multiplications, matrix factorizations, matrix inversion, etc.) to be performed very rapidly, but only once the HSS representation of the matrix has been constructed. How to rapidly compute this representation in the first place is much less well understood. The present paper demonstrates that if an $N\times N$ matrix can be applied to a vector in $O(N)$ time, and if individual entries of the matrix can be computed rapidly, then provided that an HSS representation of the matrix exists, it can be constructed in $O(N\,k^{2})$ operations, where $k$ is an upper bound for the numerical rank of the off-diagonal blocks. The point is that when legacy codes (based on, e.g., the fast multipole method) can be used for the fast matrix-vector multiply, the proposed algorithm can be used to obtain the HSS representation of the matrix, and then well-established techniques for HSS matrices can be used to invert or factor the matrix.

Error Estimate for the Ensemble Kalman Filter Analysis Step

Andrey Kovalenko, Trond Mannseth, and Geir Nævdal

SIAM. J. Matrix Anal. & Appl. 32, pp. 1275-1287 (13 pages)

Online Publication Date: November 08, 2011

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The ensemble Kalman filter (EnKF) is an ensemble-based Monte Carlo formulation of the Kalman filter. In most practical cases it is based on a low-rank approximation of a covariance matrix from a moderately sized ensemble. Sampling errors lead to artificial effects, such as spurious correlations, deteriorating the estimates and the forecasts of the system states. Using random matrix theory, we derive the distribution of an energy norm of the EnKF sampling error for the estimate of the mean, assuming noiseless data. Despite this restriction, the obtained distribution should improve the understanding of the EnKF reliability. The distribution depends on ensemble size, model dimension, and observation locations. We demonstrate the use of the distribution on an example.

Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems

Daniel Kressner and Christine Tobler

SIAM. J. Matrix Anal. & Appl. 32, pp. 1288-1316 (29 pages)

Online Publication Date: November 10, 2011

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We consider linear systems $A(\alpha) x(\alpha) = b(\alpha)$ depending on possibly many parameters $\alpha = (\alpha_1,\ldots,\alpha_p)$. Solving these systems simultaneously for a standard discretization of the parameter range would require a computational effort growing drastically with the number of parameters. We show that a much lower computational effort can be achieved for sufficiently smooth parameter dependencies. For this purpose, computational methods are developed that benefit from the fact that $x(\alpha)$ can be well approximated by a tensor of low rank. In particular, low-rank tensor variants of short-recurrence Krylov subspace methods are presented. Numerical experiments for deterministic PDEs with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach.

CALU: A Communication Optimal LU Factorization Algorithm

Laura Grigori, James W. Demmel, and Hua Xiang

SIAM. J. Matrix Anal. & Appl. 32, pp. 1317-1350 (34 pages)

Online Publication Date: November 10, 2011

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Since the cost of communication (moving data) greatly exceeds the cost of doing arithmetic on current and future computing platforms, we are motivated to devise algorithms that communicate as little as possible, even if they do slightly more arithmetic, and as long as they still get the right answer. This paper is about getting the right answer for such an algorithm. It discusses CALU, a communication avoiding LU factorization algorithm based on a new pivoting strategy, that we refer to as tournament pivoting. The reason to consider CALU is that it does an optimal amount of communication, and asymptotically less than Gaussian elimination with partial pivoting (GEPP), and so will be much faster on platforms where communication is expensive, as shown in previous work. We show that the Schur complement obtained after each step of performing CALU on a matrix $A$ is the same as the Schur complement obtained after performing GEPP on a larger matrix whose entries are the same as the entries of $A$ (sometimes slightly perturbed) and zeros. More generally, the entire CALU process is equivalent to GEPP on a large, but very sparse matrix, formed by entries of $A$ and zeros. Hence we expect that CALU will behave as GEPP and it will also be very stable in practice. In addition, extensive experiments on random matrices and a set of special matrices show that CALU is stable in practice. The upper bound on the growth factor of CALU is worse than that of GEPP. However, there are Wilkinson-like matrices for which GEPP has exponential growth factor, but not CALU, and vice-versa.

Fast Calculation of Spectral Bounds for Hessian Matrices on Hyperrectangles

M. Mönnigmann

SIAM. J. Matrix Anal. & Appl. 32, pp. 1351-1366 (16 pages)

Online Publication Date: November 17, 2011

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This paper presents a fast method for the calculation of bounds on the spectra of Hessian matrix sets ${\cal H} \{\nabla^2 \varphi(x)|x \in S\}$ of nonlinear functions $\varphi : U\subset \mathbb{R}^n\to\mathbb{R}$ on hyperrectangles $S\subset U$. The new method differs from existing ones in that it deliberately does not use any interval matrices. Because interval matrices are never used, two interesting features result: (i) The new method requires only ${\cal O}(n) N(\varphi)$ operations (where $N(\varphi)$ denotes the number of operations necessary to evaluate $\varphi$ at a point in its domain), and (ii) for some (but not all) functions $\varphi$, the new method results in tighter eigenvalue bounds than the tight bounds for the interval Hessian matrix. This is surprising, since the fastest method for calculating the tight eigenvalue bounds for the interval Hessian requires ${\cal O}(2^n)$ operations. It is stressed, however, that it is easy to construct examples $\varphi$ for which the proposed method results in looser bounds than the tight bounds for the interval Hessian.

Discrete Eckart–Young Theorem for Integer Matrices

Matthew M. Lin

SIAM. J. Matrix Anal. & Appl. 32, pp. 1367-1382 (16 pages)

Online Publication Date: December 06, 2011

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The well-known Eckart–Young theorem asserts that the truncated singular value decomposition, obtained by discarding all but the first $k$ largest singular values and their corresponding left and right singular vectors, is the best rank-$k$ approximation in the sense of least squares to the original matrix. In other words, singular values alone serve well as unambiguous indicators of proximity to the data matrix. Unlike continuous data, the decomposition of a matrix with discrete data which is subject to the requirement that its approximations have the same type of data is a harder task and it is even harder when it comes to ranking these approximations. This work generalizes the notion of singular value decomposition via a sequence of variational formulations to discrete-type data. The process itself can guarantee neither the orthogonality, as is expected of discrete data, nor the ordering of best approximations. However, at the end of the undertaking, it is shown that a quantity analogous to the singular values and a truncated low rank factorization for discrete data analogous to the truncated singular value decomposition for continuous data are attainable. Our empirical study shows the applicability of our method to cluster analysis and pattern discovery using real-life data.

Structured Pseudospectra for Small Perturbations

Michael Karow

SIAM. J. Matrix Anal. & Appl. 32, pp. 1383-1398 (16 pages)

Online Publication Date: December 08, 2011

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In this paper we study the shape and growth of structured pseudospectra for small matrix perturbations of the form $A \leadsto A_\Delta=A+B\Delta C$, $\Delta \in \boldsymbol{\Delta}$, $\|\Delta\|\leq \delta$. It is shown that the properly scaled pseudospectra components converge to nontrivial limit sets as $\delta$ tends to 0. We discuss the relationship of these limit sets with $\mu$-values and structured eigenvalue condition numbers for multiple eigenvalues.

Krylov-Based Model Order Reduction of Time-delay Systems

Wim Michiels, Elias Jarlebring, and Karl Meerbergen

SIAM. J. Matrix Anal. & Appl. 32, pp. 1399-1421 (23 pages)

Online Publication Date: December 08, 2011

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We present a model order reduction method which allows the construction of a reduced, delay-free model of a given dimension for linear time-delay systems, whose characteristic matrix is nonlinear due to the presence of exponential functions. The method builds on the equivalent representation of the time-delay system as an infinite-dimensional linear problem. It combines ideas from a finite-dimensional approximation via a spectral discretization, on the one hand, and a Krylov–Padé model reduction approach, on the other hand. The method exhibits a good spectral approximation of the original model, in the sense that the smallest characteristic roots are well approximated and the nonconverged eigenvalues of the reduced model have a favorable location, and it preserves moments at zero and at infinity. The spectral approximation is due to an underlying Arnoldi process that relies on building an appropriate Krylov space for the linear infinite-dimensional problem. The preservation of moments is guaranteed, because the chosen finite-dimensional approximation preserves moments and, in addition, the space on which one projects is constructed in such a way that the preservation of moments carries over to the reduced model. The implementation of the method is dynamic, since the number of grid points in the spectral discretization does not need to be chosen beforehand and the accuracy of the reduced model can always be improved by doing more iterations. It relies on a reformulation of the problem involving a companion-like system matrix and a highly structured input matrix, whose structure are fully exploited.

Perturbation Theory and Optimality Conditions for the Best Multilinear Rank Approximation of a Tensor

Lars Eldén and Berkant Savas

SIAM. J. Matrix Anal. & Appl. 32, pp. 1422-1450 (29 pages)

Online Publication Date: December 08, 2011

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The problem of computing the best rank-$(p,q,r)$ approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analysis is performed using the Grassmann manifold framework. The analysis is illustrated in a few examples, and it is shown that the perturbation theory for the singular value decomposition is a special case of the tensor theory.

Blind Separation of Exponential Polynomials and the Decomposition of a Tensor in Rank-$(L_r,L_r,1)$ Terms

Lieven De Lathauwer

SIAM. J. Matrix Anal. & Appl. 32, pp. 1451-1474 (24 pages)

Online Publication Date: December 08, 2011

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We present a new necessary and sufficient condition for essential uniqueness of the decomposition of a third-order tensor in rank-$(L_r,L_r,1)$ terms. We derive a new deterministic technique for blind signal separation that relies on this decomposition. The method assumes that the signals can be modeled as linear combinations of exponentials or, more generally, as exponential polynomials. The results are illustrated by means of numerical experiments.

A Robust Two-Level Incomplete Factorization for (Navier–)Stokes Saddle Point Matrices

Fred W. Wubs and Jonas Thies

SIAM. J. Matrix Anal. & Appl. 32, pp. 1475-1499 (25 pages)

Online Publication Date: December 08, 2011

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We present a new hybrid direct/iterative approach to the solution of a special class of saddle point matrices arising from the discretization of the steady incompressible Navier–Stokes equations on an Arakawa C-grid. The two-level method introduced here has the following properties: (i) it is very robust, even close to the point where the solution becomes unstable; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively; (vi) the iteration takes place in the divergence-free space, so the method qualifies as a “constraint preconditioner”; (vii) the approach can also be applied to Poisson problems. This work is also relevant for problems in which similar saddle point matrices occur, for instance, when simulating electrical networks, where one has to satisfy Kirchhoff's conservation law for currents.

Perturbation of Matrices and Nonnegative Rank with a View toward Statistical Models

Cristiano Bocci, Enrico Carlini, and Fabio Rapallo

SIAM. J. Matrix Anal. & Appl. 32, pp. 1500-1512 (13 pages)

Online Publication Date: December 13, 2011

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In this paper we study how perturbing a matrix changes its nonnegative rank. We prove that the nonnegative rank can only increase in a neighborhood of a matrix with no zero columns. Also, we describe some special families of perturbations. We show how our results relate to statistics in terms of the study of maximum likelihood estimation for mixture models.

Computing the Stationary Distribution of a Finite Markov Chain Through Stochastic Factorization

André M. S. Barreto and Marcelo D. Fragoso

SIAM. J. Matrix Anal. & Appl. 32, pp. 1513-1523 (11 pages)

Online Publication Date: December 15, 2011

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This work presents an approach for reducing the number of arithmetic operations involved in the computation of a stationary distribution for a finite Markov chain. The proposed method relies on a particular decomposition of a transition-probability matrix called stochastic factorization. The idea is simple: when a transition matrix is represented as the product of two stochastic matrices, one can swap the factors of the multiplication to obtain another transition matrix, potentially much smaller than the original. We show in the paper that the stationary distributions of both Markov chains are related through a linear transformation, which opens up the possibility of using the smaller chain to compute the stationary distribution of the original model. In order to support the application of stochastic factorization, we prove that the model derived from it retains all the properties of the original chain which are relevant to the stationary distribution computation. Specifically, we show that (i) for each recurrent class in the original Markov chain there is a corresponding class in the derived model with the same period and, given some simple assumptions about the factorization, (ii) the original chain is irreducible if and only if the derived chain is irreducible and (iii) the original chain is regular if and only if the derived chain is regular. The paper also addresses some issues associated with the application of the proposed approach in practice and briefly discusses how stochastic factorization can be used to reduce the number of operations needed to compute the fundamental matrix of an absorbing Markov chain.

Reliable Eigenvalues of Symmetric Tridiagonals

Rui Ralha

SIAM. J. Matrix Anal. & Appl. 32, pp. 1524-1536 (13 pages)

Online Publication Date: December 20, 2011

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For the eigenvalues of a symmetric tridiagonal matrix $T$, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix $\widetilde{T}$ whose entries differ from the corresponding entries of $T$ by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than $\Vert T\Vert_{2}$ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of $T$. We present some numerical examples to illustrate the usefulness of our method.

Using the Restricted-denominator Rational Arnoldi Method for Exponential Integrators

Paolo Novati

SIAM. J. Matrix Anal. & Appl. 32, pp. 1537-1558 (22 pages)

Online Publication Date: December 20, 2011

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In this paper we investigate some practical aspects concerning the use of the restricted-denominator rational Arnoldi method for the computation of the core functions of exponential integrators for parabolic problems. We derive some useful a posteriori bounds together with hints for a suitable implementation inside the integrators. Numerical experiments arising from the discretization of sectorial operators are presented.
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SIAM. J. Matrix Anal. & Appl. 32, pp. 1559-1559 (1 page)

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A production error resulted in incorrect information regarding the affiliations of the authors of “Primitivity, the Convergence of the NQZ Method, and the Largest Eigenvalue for Nonnegative Tensors" (SIAM J. Matrix Anal. Appl., 32 (2011), pp. 806–819). None of the authors is connected with Nankai University.
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