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143
Nineteen Dubious Ways to Compute the Exponential of a Matrix, TwentyFive Years Later
, 2003
"... In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indica ..."
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Cited by 422 (0 self)
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In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.
The mathematics of eigenvalue optimization
 MATHEMATICAL PROGRAMMING
"... Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemp ..."
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Cited by 117 (11 self)
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Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.
Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 86 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
Structured Pseudospectra For Polynomial Eigenvalue Problems With Applications
 SIAM J. MATRIX ANAL. APPL
, 2001
"... Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. W ..."
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Cited by 64 (8 self)
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Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrLx equation and thereby factorizes the quadratic hmatrLx. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques.
A Block Algorithm for Matrix 1Norm Estimation, with an Application to 1Norm Pseudospectra
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. The matrix 1norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. However, it has the limitations that it offers the user no control over the accuracy and reliability of the estimate and that it is based on level 2 BLA ..."
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Cited by 49 (23 self)
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Abstract. The matrix 1norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. However, it has the limitations that it offers the user no control over the accuracy and reliability of the estimate and that it is based on level 2 BLAS operations. A block generalization of the 1norm power method underlying the estimator is derived here and developed into a practical algorithm applicable to both real and complex matrices. The algorithm works with n × t matrices, where t is a parameter. For t = 1 the originalalgorithm is recovered, but with two improvements (one for realmatrices and one for complex matrices). The accuracy and reliability of the estimates generally increase with t and the computationalkernels are level 3 BLAS operations for t>1. The last t−1 columns of the starting matrix are randomly chosen, giving the algorithm a statistical flavor. As a byproduct of our investigations we identify a matrix for which the 1norm power method takes the maximum number of iterations. As an application of the new estimator we show how it can be used to efficiently approximate 1norm pseudospectra.
An aggregationbased algebraic multigrid method
, 2008
"... An algebraic multigrid (AMG) method is presented to solve large systems of linear equations. The coarsening is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the nu ..."
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Cited by 40 (9 self)
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An algebraic multigrid (AMG) method is presented to solve large systems of linear equations. The coarsening is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the number of variables by a factor slightly less than four. The matching algorithm favors the strongest negative coupling(s), inducing a problem dependant coarsening. This aggregation is combined with piecewise constant (unsmoothed) prolongation, ensuring low setup cost and memory requirements. Compared with previous aggregationbased multigrid methods, the scalability is enhanced by using a socalled Kcycle multigrid scheme, providing Krylov subspace acceleration at each level. Numerical results on second order discrete scalar elliptic PDEs indicate that the proposed method may be significantly more robust than the classical AMG method as implemented in the code AMG1R5 by K. Stüben. The parallel implementation is also discussed. Satisfactory speedups are obtained on a 24 nodes processors cluster with relatively high communication latency, providing that the number of unknowns per processor is kept significant.
Computing absolute and essential spectra using continuation
, 2007
"... A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantage of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected part ..."
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Cited by 35 (13 self)
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A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantage of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability. We also discuss the implementation and usage of this approach with the software package auto and provide example computations for the FitzHugh–Nagumo and the complex Ginzburg–Landau equation.
Optimization and pseudospectra, with applications to robust stability
 SIAM Journal on Matrix Analysis and Applications
, 2003
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Robust stability and a crisscross algorithm for pseudospectra
 IMA Journal of Numerical Analysis
, 2003
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ENSO suppression due to weakening of the North Atlantic thermohaline circulation
 J. Climate
, 2005
"... Changes of the North Atlantic thermohaline circulation (THC) excite wave patterns that readjust the thermocline globally. This paper examines the impact of a freshwaterinduced THC shutdown on the depth of the Pacific thermocline and its subsequent modification of the El Niño–Southern Oscillation (E ..."
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Cited by 31 (7 self)
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Changes of the North Atlantic thermohaline circulation (THC) excite wave patterns that readjust the thermocline globally. This paper examines the impact of a freshwaterinduced THC shutdown on the depth of the Pacific thermocline and its subsequent modification of the El Niño–Southern Oscillation (ENSO) variability using an intermediatecomplexity global coupled atmosphere–ocean–sea ice model and an intermediate ENSO model, respectively. It is shown by performing a numerical eigenanalysis and transient simulations that a THC shutdown in the North Atlantic goes along with reduced ENSO variability because of a deepening of the zonal mean tropical Pacific thermocline. A transient simulation also exhibits abrupt changes of ENSO behavior, depending on the rate of THC change. The global oceanic wave adjustment mechanism is shown to play a key role also on multidecadal time scales. Simulated multidecadal global sea surface temperature (SST) patterns show a large degree of similarity with previous climate reconstructions, suggesting that the observed panoceanic variability on these time scales is brought about by oceanic waves and by atmospheric teleconnections. 1.