Lloyd N. Trefethen. Pseudospectra of matrices. In D. F. Griths and G. A. Watson, editors, Numerical analysis 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which is in general incompatible with the boundary condition. Since (D) D = 0, we can write this solution exactly as a finite sum when k 6= 0: f = Gamma g (10) This equation, while exact, is extremely poorly conditioned for large N ; this constitutes the second difficulty [5]. By changing the polynomial basis functions from x to x =n , we see that the above formula defines an upper triangular Toeplitz matrix T with entries T ij = Gammak i Gammaj Gamma1 for j i. However, the similarity transform between the basis functions and x =n has the condition ....

L. N. Trefethen, Pseudospectra of matrices, in D. F. Griffiths and G. A. Watson, Numerical Analysis 1991.

....r. Define as in (13) Then the smallest singular value of R I satisfies the inequality, ffw min IIP11 I111 (15) In the case when R is diagonal then clearly min is simply the distance from to the spectrum of R. Note that the above inequality can be interpreted from a pseudospec trum viewpoint [7, 3]. It stipulates that the approximate eigenmlue is located inside the e pseudo spectrum level curve of R defined for e equal to the right hand side of (lS) In the inequality (14) we could use the fact that N is a nilpotent matrix, i.e. there exists an integer k not exceeding the dimension p of ....

L. N. Trefethen. Pseudospectra of matrices. In D. F. Grifiiths and G. A. Watson, editors, Numerical Analysis, 1991.

....in areas such as fluid mechanics, Markov chains, and control theory. Most of the existing work is for the standard eigenproblem, although attention has also been given to matrix pencils [4] 23] 33] 40] 46] The literature on pseudospectra is large and growing. We refer to Trefethen [41], 42] 43] for thorough surveys of pseudospectra and their computation for a single matrLx; see also the Web site [3] In this work we investigate pseudospectra for polynomial matrices (or A matrices) i.i) m 1 Ao, where Ak E C x, k = 0: m. We first define the e pseudospectrum and ....

....The left plot in Figure 4.1 shows the boundaries of e pseudospectra with perturbations measured in the absolute sense (i 1) with e between 10 3 and 10 8. The eigenvalues are plotted as dots. Another way of approximating a pseudospectrum is by random perturbations of the original matrices [41]. We generated 200 triples of complex random normal perturbation matrices (AA, AA2, AA3) with II AAjll2: 10 8, j: 1: 3. In the right plot of Figure 4.1 are superimposed as small dots the eigenvalues of the perturbed polynomials 2(A2 AA2) A AA) AA0 AA0. The solid curve marks the ....

L. N. TREFETHEN, Pseudospectra of matrices, in Numerical Analysis

....as an accelerated inner outer iteration scheme. Of course, much more has happened, and the above reflect only personal impressions. Other important developments and improvements were (the list is still personally colored and incomplete) ffl The concept and use of the pseudospectrum [71] as a means to make sensitivities in the spectrum of a nonnormal matrix easily visible. The pseudospectrum does not make classical perturbation theory superfluous, but it helps to detect situations that need further analysis, and it does so in a way that is easily understandable for non expert ....

....global level curves are too pessimistic. It may be the case that it is not realistic to assume equal perturbations for all matrix entries, but nevertheless the pseudospectrum points the attention to critical places in the spectrum. A nice introduction to the relevance of pseudospectra is given in [71], where for a number of matrices pseudospectra are actually computed and discussed. Due to the nature of pseudospectra, this useful tool is often restricted to matrices of moderate size. This poses another problem: if we study the pseudospectrum for a discretized PDE with rather course meshsize ....

L. N. Trefethen. Pseudospectra of matrices. In D. F. Griffiths and G. A. Watson, editors, Numerical Analysis 1991, pages 234--266. Longman, 1992.

....closed loop system generating a strict contraction semigroup with respect to the spectral norm. 1Introduction Trajectoriesofasymptotically stable linearsystems maymove far away from the origin before ultimately approaching it. This transient behavior has recently been studied by several authors [2, 3, 6, 7, 8], in particular its relation to the behavior of the spectrum of the system matrix under perturbations. There are several motivations for the analysis of the transient behavior of linear systems. One of these concerns the relation between a nonlinear system and its linearization at an ....

L. N. Trefethen, "Pseudospectra of matrices," In D. F. Gri#ths and G. A. Watson, eds., Numerical Analysis,volume 91, pages 234--266. Longmann, 1992.

.... MATLAB Test Matrix Toolbox functions ps (left) and pscont (right) where (A) denotes the spectrum of A, has become a tool for the investigation of the behavior of several (nonnormal) matrix dependent algorithms, ranging from iterative methods for large linear systems to time stepping algorithms [27,28]. We note, for instance, the inclusion of specific functions to that effect (ps and pscont) in the popular Test Matrix Toolbox of MATLAB [18] the former uses definition (1) and the latter definition (2) In general, however, computing the pseudospectrum is significantly more expensive than ....

....1 V 2 . The modules were coordinated by the LAPACK driver zgesvd. 3. 2 Test matrices For the numerical experiments we use nonnormal matrices from [18] i) the upper triangular matrix kahan with elements a kk = s and a kj = Gammas c when j k, where s n Gamma1 = 0:1 and s c = 1 [27]; ii) the pentadiagonal Toeplitz matrix grcar A = Toeplitz( Gamma1; 1; 1; 1; 1] where the underlined element is in the diagonal [15] iii) matrix smoke (complex) which has unit elements in the superdiagonal and in position (n; 1) powers of roots of unity along the diagonal and zero everywhere ....

L.N. Trefethen. Pseudospectra of matrices. In D.F. Griffiths and G.A. Watson, editors, Numerical Analysis 1991.

.... ffl g or, equivalently, in terms of the resolvent (zI Gamma A) as ffl (A) f z 2 C : k(zI Gamma A) k ffl g: Most published work on pseudospectra has dealt with the 2 norm and the utility of 2norm pseudospectra in revealing the effects of non normality is well appreciated [19], 20] 22] The 2 norm and any other p norm of an n Theta n matrix differ by a factor at most n. For small n, pseudospectra therefore do not vary much between different p norms. However, J onsson and Trefethen have shown [17] that in Markov chain applications the choice of norm for ....

Lloyd N. Trefethen. Pseudospectra of matrices. In Numerical Analysis 1991.

....power. The effect of rounding errors in this example is to cause the forward error curve in Figure 1 to level off near k = 100, instead of decaying to zero as it would in exact arithmetic. More insight into the initial behaviour of the errors can be obtained using the notion of pseudo eigenvalues [23]. To establish what we should try to prove, we review some normwise and componentwise backward error results and perturbation theory. If y is an approximate solution to Ax = b then the normwise (relative) backward error is j(y) minfffl : A DeltaA)y = b Deltab; k DeltaAk fflkAk; ....

....examples, they emphasise that while it is the spectral radius of the iteration matrix M N that determines the asymptotic rate of convergence, it is the norms of the powers of this matrix that govern the behaviour of the iteration in the early stages. This point is also elucidated by Trefethen [23], using the tool of pseudospectra. Dennis and Walker [7] obtain bounds for kx Gamma b x k 1 k=kx Gamma b x k k for stationary iteration as a special case of error analysis of quasi Newton methods for nonlinear systems. The bounds in [7] do not readily yield information about normwise or ....

Lloyd N. Trefethen. Pseudospectra of matrices. In D. F. Griffiths and G. A. Watson, editors, Numerical Analysis 1991.

....then [16, Theorem 4] 2 (X) 1 1=2 and it can be shown by a 2 Theta 2 example that minX 2 (X) can exceed Delta F (A) kAkF by an arbitrary factor [2, Section 8.1.2] 1, Section 4.2. 7] Another tool that can be used to bound the norms of powers is the pseudospectrum of a matrix [22]. The ffl pseudospectrum of A 2 C is defined for a given ffl 0 to be the set ffl (A) f z : z is an eigenvalue of A E for some E with kEk 2 ffl g; and it can also be represented, in terms of the resolvent (zI Gamma A) as ffl (A) f z : k(zI Gamma A) g: As Trefethen notes ....

.... [22] The ffl pseudospectrum of A 2 C is defined for a given ffl 0 to be the set ffl (A) f z : z is an eigenvalue of A E for some E with kEk 2 ffl g; and it can also be represented, in terms of the resolvent (zI Gamma A) as ffl (A) f z : k(zI Gamma A) g: As Trefethen notes [22], by using the Cauchy integral representation of A (which involves a contour integral of the resolvent) one can show that ae ffl (A) 2.7) where the ffl pseudospectral radius ae ffl (A) maxf jzj : z 2 ffl (A) g: 2.8) This bound is very similar in flavour to (2.4) The difficulty is ....

L. N. Trefethen, Pseudospectra of matrices, in Numerical Analysis

....in memory and to use it in computation. In asymptotic convergence rates of the methods, this waist is usually not visible. However, aiming for as little iterations as possible, these asymptotics hardly show up in the first place. In particular in non normal applications, it is by now well known [15] that one should be more worried about the initial phase of an iterative process. Therefore, in this paper, we concentrate on the start of Krylov subspace methods, as opposed to their asymptotic behavior. 1.2 Outline The outline of this paper is as follows. In Section 2 we recall iterative ....

....B has the same eigenvectors but with eigenvalues 1 2 and Gamma 1 2 . Let r 0 be the vector v 1 v 2 , then kr 0 k = 2ffl. Applying I Gamma B gives r 1 = 1 2 (v 1 Gamma v 2 ) In spite of the spectral radius being one half, the norm of the first residual is kr 1 k = 1. The pseudo spectrum [15] of M often gives a better bound on the norm of the powers of a matrix, although the relevant pseudo spectral radius is not easy to compute. 2.2 The Local Minimal Residual method A first effort to overcome some of the problems of classical iterative methods is the following. Having found a pair ....

L.N. Trefethen, Pseudo-spectra of matrices, in D.F. Griffiths and G.A. Watson, Numerical Analysis

....if ( is ill conditioned, r is large and the assumption 0 r may not be satis ed. Pseudospectra are a valuable tool for assessing the global sensitivity of matrix eigenvalues to perturbations in the matrix. The literature on pseudospectra is large and growing and we refer to Trefethen [15], 16] 17] Implicit Gamma Theorems (I) Pseudoroots and Pseudospectra 23 for thorough surveys of pseudospectra and their computation for a single matrix; see also the Web site [5] Pseudospectra have recently been de ned and characterized for square and rectangular matrix polynomials by ....

Lloyd N. Trefethen. Pseudospectra of matrices. In D. F. Griths and G. A. Watson, editors, Numerical Analysis

....typical: Chebyshev differentiation matrices are not antisymmetric (for all n 2) unlike the Fourier spectral and centered differences differentiation matrices on uniform periodic grids. This has been observed many times in the literature, usually without comment other than it being unfortunate [3, 11, 13, 14, 18, 34, 35, 36, 37, 41]. This failure to be antisymmetric is puzzling in view of the (formal) skew adjointness of x . Is the global nature of Chebyshev interpolation the problem Is the lack of translation symmetry of the grid at fault Is some key property of trigonometric functions missed by polynomials 1 ....

....well characterized by their eigenvalues. Instead of a fairly straightforward eigenvalue analysis [14, 18, 31] the stability analysis of schemes based on nonnormal matrices involves more sophisticated techniques based on the Kreiss matrix theorem [14, 16, 18, 28, 31, 30, 38, 39] or pseudospectra [32, 33, 36]. iii) Skew adjointness is a structural feature associated with the differential operator x . Such features should be preserved under discretization if we want schemes that are natural and robust in the face of changes in the grid, the solution, or the operator. The solution to the puzzle is ....

L. N. Trefethen. Pseudospectra of matrices. In D.F. Griffiths and G.A. Watson, editors, Numerical Analysis

....classical matrix theory, that predicts that the pseudospectrum will consist of the union of the disks of radius surrounding each eigenvalue of A. The pseudospectrum becomes of interest, on its own or as an alternative to standard eigenvalue analysis, when A is not normal (e.g. nonsymmetric) see [16, 18]. An important barrier in making pseudospectra a standard engineering tool is the expense involved in their calculation. In the sequel we assume that we use the spectral norm max(A) kAk2 ; we also de ne s(z) min(zI A) the minimum singular value of matrix zI A. It is known that at points ....

L.N. Trefethen. Pseudospectra of matrices. In Numerical Analysis 1991, Proc. 14th Dundee Conf., D. Griths and G. Watson, Eds. Essex, UK: Longman Sci. and Tech., 1991, pp. 234-266.

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L. N. Trefethen, Pseudospectra of matrices, in D. F. Griths and G. A. Watson, eds., Numerical Analysis

....magnitude. If the dimension of A is increased from 16 to 32 in the same example, the numbers increase to approximately 2:9 Theta 10 10 and 2:5 Theta 10 11 , respectively. The restatement of these observations in terms of pseudospectra runs as follows. For each ffl 0, the ffl pseudospectrum [19, 26, 27] of a matrix A is the compact subset of C defined by ffl (A) fz 2 C : k(zI Gamma A) Gamma1 k ffl Gamma1 g: For z 2 (A) we set k(zI Gamma A) Gamma1 k = 1. Equivalently, ffl (A) is the set of z 2 C that are eigenvalues of some matrix A E with kEk ffl. Now it is easy to verify ....

L.N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991, eds. D.F. Griffiths and G.A. Watson, Longman, 234--266.

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Lloyd N. Trefethen. Pseudospectra of matrices. In D. F. Griths and G. A. Watson, editors, Numerical analysis 1991.

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L.N. Trefethen. Pseudospectra of matrices. In D.F. Griffiths and G.A. Watson, editors, Numerical Analysis, volume 91, pages 234--266. Longmann, 1992.

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L. N. Trefethen, \Pseudospectra of matrices," In D. F. Griths and G. A. Watson, eds., Numerical Analysis, volume 91, pages 234-266. Longmann, 1992.

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L. N. Trefethen. Pseudospectra of matrices. In 1991.

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L. TREFETHEN, Pseudospectra of matrices, in Numerical Analysis 1991.

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L.N. Trefethen. Pseudospectra of matrices. In D.F. Gri#ths and G.A. Watson, editors, Numerical Analysis 1991.

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L. N. Trefethen, \Pseudospectra of matrices," In D. F. Griths and G. A. Watson, eds., Numerical Analysis, volume 91, pages 234-266. Longmann, 1992.

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L. N. Trefethen, Pseudospectra of matrices, in Numerical Analysis

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Lloyd N. Trefethen. Pseudospectra of matrices. In D. F. Gri#ths and G. A. Watson, editors, Numerical Analysis 1991, Proceedings of the 14th Dundee Conference, volume 260 of Pitman Research Notes in Mathematics, pages 234--266. Longman Scientific and Technical, Essex, UK, 1992.

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