## Pseudospectra of linear operators (1997)

Venue: | SIAM Rev |

Citations: | 114 - 8 self |

### BibTeX

@ARTICLE{Trefethen97pseudospectraof,

author = {Lloyd N. Trefethen},

title = {Pseudospectra of linear operators},

journal = {SIAM Rev},

year = {1997},

pages = {383--406}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.

### Citations

835 |
Semigroups of Linear Operators and Applications to Partial Differential Equations
- Pazy
- 1983
(Show Context)
Citation Context ...per we shall not discuss details of functional analysis; we only note that in all of what follows, we assume that A is a closed linear operator in a Hilbert space and that it generates a C0 semigroup =-=[35]-=-. (This setting applies in particular to our Theorems 1–5.) The Hilbert space norm is denoted by ‖·‖, and for our examples that come from physical problems, we always arrange matters so that this norm... |

427 |
Numerical Methods for Large Eigenvalue Problems
- Saad
- 1992
(Show Context)
Citation Context ...y of these matrices are nonsymmetric, which usually means nonnormal, and research into improved methods for computing their eigenvalues, especially iterative methods, is actively underway [28], [31], =-=[48]-=-. To date, most such computations generate just numbers, not pictures, and most of the people who carry them out are not in the habit of investigating nonnormality. As a result, it is likely that situ... |

347 |
Introduction to the theory of linear nonselfadjoint operators
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- 1969
(Show Context)
Citation Context ... decomposition. For any matrix or operator A we have (3) Λ ɛ(A) = {z ∈ C : σmin(zI − A) ≤ ɛ}, where σmin denotes the smallest singular value in the matrix case or the smallest snumber for an operator =-=[17]-=-. In stating (3), we have made use of the assumption that A acts in a Hilbert space; in a Banach space setting, we have (1) and (2) but not (3) (see [12]). If A is normal, then Λɛ(A) is exactly the se... |

241 |
Functional Analysis and Semi-groups
- Hille, Phillips
- 1957
(Show Context)
Citation Context ...he significance of exp(tA) isthatitis the solution operator for the linear, autonomous problem du/dt = Au; in the theory of semigroups, {exp(tA)} is the semigroup and A is its infinitesimal generator =-=[22]-=-, [35]. However, it should be remembered that other functions of operators besides the exponential are also of interest in applications. Examples are A n for discrete evolution388 LLOYD N. TREFETHEN ... |

185 |
Difference methods for initial-value problems", Interscience Tracts in pure and applied
- Richtmyer
- 1957
(Show Context)
Citation Context ...ompellingly described the pitfalls of eigenvalue analysis and the uses of the resolvent as an alternative. Some of Kreiss’s ideas were presented shortly afterwards in the text by Richtmyer and Morton =-=[45]-=-. A narrower question has to do with the history of pseudospectra—that is, of the explicit investigation of the sets bounded by level curves of the norm of the resolvent. The first mention of this ide... |

184 |
Hydrodynamic Stability
- Drazin, Reid
- 1981
(Show Context)
Citation Context ...r plot. Example 8: Orr–Sommerfeld operator (see [41]). With our final three examples, we move to a topic in the field of fluid mechanics: the instability of incompressible flows in pipes and channels =-=[13]-=-. In the past five years it has become clear that nonnormality plays a crucial role in destabilizing these flows. The problem of explaining why high-speed flows are invariably turbulent is more than a... |

131 |
Hydrodynamic stability without eigenvalues
- Trefethen, Trefethen, et al.
- 1993
(Show Context)
Citation Context ...12]). If A is normal, then Λɛ(A) is exactly the set of points in C at distance ≤ ɛ from Λ(A). If A is not normal, however, it may be much larger. Here is a physical interpretation of this observation =-=[57]-=-. Consider a time-dependent driven system du/dt = Au + eztf, where f is a fixed function in the Hilbert space under study. The solution to this problem is u(t) =ezt(zI − A) −1f. If z ∈ Λɛ(A), this mea... |

113 | Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators
- Trefethen, Embree
- 2005
(Show Context)
Citation Context ...lope of the ‖exp(tA)‖ curve is infinite at t = 0, though this is not visible in the figure. This example is the result of an unpublished joint work with André Weideman. Further details will appear in =-=[56]-=-.PSEUDOSPECTRA OF LINEAR OPERATORS 397 Example 7: Wiener–Hopf operator (see [38]). Among nonnormal matrices, the class whose pseudospectra are best understood are the Toeplitz matrices [43]. We can g... |

87 |
Linear Operators
- Dunford, Schwartz
- 1958
(Show Context)
Citation Context ... illustrating the curious behavior of the spectra of nonnormal operators. Treatments in monograph form include the book by Gohberg and Kreĭn [17] and the three-volume treatise of Dunford and Schwartz =-=[15]-=-. The use of the resolvent in the study of nonnormal operators has also been standard for many years, at least among the theoretically inclined. Of numerous contributions, I shall mention two that are... |

84 |
A Hilbert Space Problem
- Halmos
- 1982
(Show Context)
Citation Context ...ury. In a sense, it has been one of the central topics of functional analysis ever since, and its elements are386 LLOYD N. TREFETHEN widely known. For example, A Hilbert Space Problem Book by Halmos =-=[21]-=- contains numerous examples illustrating the curious behavior of the spectra of nonnormal operators. Treatments in monograph form include the book by Gohberg and Kreĭn [17] and the three-volume treati... |

83 |
Perturbation Theory for Linear Operators, 2nd ed
- Kato
- 1976
(Show Context)
Citation Context ...mong the theoretically inclined. Of numerous contributions, I shall mention two that are especially relevant to this paper. One is the remarkable book Perturbation Theory for Linear Operators by Kato =-=[24]-=-, in which all kinds of questions of matrix and operator theory are beautifully treated by resolvent techniques. The other is the work by Kreiss over the years in the field of finite difference method... |

63 |
Three-dimensional optimal perturbations in viscous shear flow
- Butler, Farrell
- 1992
(Show Context)
Citation Context ...ciliation was a beautiful work by Boberg and Brosa [4]. That paper, however, went largely unnoticed for several years, leaving it to the independent and equally impressive paper by Butler and Farrell =-=[9]-=- to communicate such ideas widely. Following closely after [9] were the third and fourth members of what now appear as a four-paper set, [39] and [57]. It was mentioned above that if the pseudospectra... |

59 |
Eigenvalues and pseudo-eigenvalues of Toeplitz matrices
- Reichel, Trefethen
- 1992
(Show Context)
Citation Context ...appear in [56].PSEUDOSPECTRA OF LINEAR OPERATORS 397 Example 7: Wiener–Hopf operator (see [38]). Among nonnormal matrices, the class whose pseudospectra are best understood are the Toeplitz matrices =-=[43]-=-. We can generalize these to operators of infinite dimension by making the domain unbounded, which gives an infinite Toeplitz matrix; by making it continuous, which gives a Wiener–Hopf integral operat... |

46 |
1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels
- Reynolds
(Show Context)
Citation Context ...rmality plays a crucial role in destabilizing these flows. The problem of explaining why high-speed flows are invariably turbulent is more than a century old. The foundations were laid by O. Reynolds =-=[44]-=-, who investigated flow of water through a long circular pipe. At low speeds, a laminar flow is observed,398 LLOYD N. TREFETHEN FIG. 7. The Volterra Wiener–Hopf integral operator (5) with kernel k(x)... |

40 |
Energy growth in viscous channel flows
- Reddy, Henningson
- 1993
(Show Context)
Citation Context ...ependent and equally impressive paper by Butler and Farrell [9] to communicate such ideas widely. Following closely after [9] were the third and fourth members of what now appear as a four-paper set, =-=[39]-=- and [57]. It was mentioned above that if the pseudospectra of an operator A protrude significantly into the right halfplane, then there must be a transient hump in the curve of ‖exp(tA)‖. One precise... |

38 |
Pseudospectra of the Orr-Sommerfeld operator
- Reddy, Schmid, et al.
- 1993
(Show Context)
Citation Context ...e 1992, various people have begun to compute pseudospectra of differential and integral operators, beginning with the outstanding paper on the Orr–Sommerfeld operator by Reddy, Schmid, and Henningson =-=[41]-=-. Such computations are often feasible on today’s workstations, and they are certainly feasible on today’s supercomputers. The purpose of this paper is to present ten examples of pseudospectra of oper... |

34 |
Pseudospectra of matrices, in Numerical Analysis
- TREFETHEN
- 1991
(Show Context)
Citation Context ...plot pseudospectra. On the workstations of 1990, it was possible to compute pseudospectra of 32×32 matrices in a few minutes, and in 1992 I published a paper presenting thirteen examples of this kind =-=[55]-=-. (The names given to the examples were Jordan block, Limaçon, Grcar, Wilkinson, Frank, Kahan, Demmel, Lenferink–Spijker, Companion, Gauss–Seidel, Chebyshev spectral, Random, and Random upper-triangul... |

31 | An evaluation of software for computing eigenvalues of sparse nonsymmetric matricies
- Lehoucq, Scott
- 1996
(Show Context)
Citation Context ...atrices. Many of these matrices are nonsymmetric, which usually means nonnormal, and research into improved methods for computing their eigenvalues, especially iterative methods, is actively underway =-=[28]-=-, [31], [48]. To date, most such computations generate just numbers, not pictures, and most of the people who carry them out are not in the habit of investigating nonnormality. As a result, it is like... |

27 | Computing the field of values and pseudospectra using the Lanczos method with continuation
- Braconnier, Higham
- 1996
(Show Context)
Citation Context ...9]. Half a dozen other studies of iterative methods for computing pseudospectra have also been carried out in the past year or so. For one such approach, with references to several of the others, see =-=[8]-=-. If A is a differential or integral operator, we must, of course, discretize it. There is little general literature on this at present, but a variety of methods have proven successful. The obvious th... |

27 |
Energy growth of three-dimensional disturbances in plane Poiseuille flow
- Gustavsson
- 1991
(Show Context)
Citation Context ...ortex → streak mechanism involved in transition to turbulence and in turbulence itself. Contours at ɛ =10−2,10−2.5, 10−3 , 10−3.5 . (From Trefethen, Trefethen, Reddy, and Driscoll [57]; see also [9], =-=[20]-=-, [39].) 2000. The pseudospectra, illustrated in Figure 10, give some indication of how this can happen. They look much the same as in the last example, protruding substantially into the right halfpla... |

26 | Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems
- Meerbergen, Roose
- 1996
(Show Context)
Citation Context ...s. Many of these matrices are nonsymmetric, which usually means nonnormal, and research into improved methods for computing their eigenvalues, especially iterative methods, is actively underway [28], =-=[31]-=-, [48]. To date, most such computations generate just numbers, not pictures, and most of the people who carry them out are not in the habit of investigating nonnormality. As a result, it is likely tha... |

25 | Pseudospectra of the convection-diffusion operator
- Reddy, Trefethen
- 1994
(Show Context)
Citation Context ...normality of A. The height of the steps, R, is also equal to κ(V ), the condition number of a normalized infinite matrix (Riesz basis) of eigenfunctions. Example 5: convection–diffusion operator (see =-=[42]-=-). Nonnormal differential operators arise most familiarly in problems mixing first and second derivatives, e.g., convection and diffusion. Specifically, consider the operator A = d/dx + d 2 /dx 2 acti... |

25 |
Matrix Eigensystem Routines—EISPACK Guide
- SMITH, BOYLE, et al.
- 1976
(Show Context)
Citation Context ...ntial items of numerical software over the years has been EISPACK, a collection of Fortran subroutines for matrix eigenvalue computations used around the world since its introduction in the mid 1970s =-=[50]-=-. Eigenvalues are useful for three reasons. The algorithmic reason is that if a matrix or linear operator can be diagonalized, transforming the problem to a basis of eigenfunctions, the solution of va... |

23 |
Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren
- Kreiss
- 1962
(Show Context)
Citation Context ...autifully treated by resolvent techniques. The other is the work by Kreiss over the years in the field of finite difference methods for partial differential equations. A landmark 1962 paper by Kreiss =-=[26]-=-, containing what became known as the Kreiss matrix theorem, compellingly described the pitfalls of eigenvalue analysis and the uses of the resolvent as an alternative. Some of Kreiss’s ideas were pre... |

21 |
Linear stability analysis in the numerical solution of initial value problems
- Dorsselaer, Kraaijevanger, et al.
- 1993
(Show Context)
Citation Context ...x case or the smallest snumber for an operator [17]. In stating (3), we have made use of the assumption that A acts in a Hilbert space; in a Banach space setting, we have (1) and (2) but not (3) (see =-=[12]-=-). If A is normal, then Λɛ(A) is exactly the set of points in C at distance ≤ ɛ from Λ(A). If A is not normal, however, it may be much larger. Here is a physical interpretation of this observation [57... |

21 | Computation of pseudospectra by continuation
- LUI
- 1997
(Show Context)
Citation Context ...d to Hessenberg or triangular form, then this form can be preserved in computations of σmin(zI − A) for various values z by inverse iteration; the result is often a speedup by a factor of ten or more =-=[29]-=-. Half a dozen other studies of iterative methods for computing pseudospectra have also been carried out in the past year or so. For one such approach, with references to several of the others, see [8... |

18 |
Pseudospectra and singular values of large convolution operators
- BÖTTCHER
- 1994
(Show Context)
Citation Context ...atrix A we have (2) Λ ɛ(A) = {z ∈ C : z ∈ Λ(A +∆A) for some ∆A with ‖∆A‖ ≤ɛ}, and if A is an operator, the equivalence of (1) and (2) still holds if we take the closure of the set defined by (2) (see =-=[5]-=-, [47]). Thus, a number z is in the interior of the ɛ-pseudospectrum of A if and only if it is in the spectrum of some perturbed operator A +∆Awith ‖∆A‖ <ɛ. A third equivalent definition, closer to co... |

18 |
A counterexample for two conjectures about stability
- Demmel
- 1987
(Show Context)
Citation Context ...vosibirsk, which pursued various ideas in this vein; others in this group include Bulgakov, Kirilyuk, and Malyshev. The first computer-generated plot of pseudospectra of which I am aware is by Demmel =-=[11]-=-. Meanwhile, other related contributions were made in the 1970s and 1980s by various people, including Chatelin, Hinrichsen, Pritchard, Varah, and Wilkinson. My own first use of pseudospectra (the ide... |

17 | On the stability of streamwise streaks and transition thresholds in plane channel flows
- Reddy, Schmid, et al.
- 1998
(Show Context)
Citation Context ...se then undergo further evolution in a manner dependent on the nonlinear terms in the Navier–Stokes equations [2]. The elucidation of the details of this process is an active area of current research =-=[40]-=-. What about the bizarre spectrum depicted in Figure 10? For the infinite channel, there were two unbounded dimensions and, thus, two continuous Fourier parameters, giving rise to a spectrum that was ... |

16 |
On the spectrum of C0-semigroups
- Prüss
- 1984
(Show Context)
Citation Context ...itrary operators A. For each ɛ ≥ 0, define the ɛ-pseudospectral abscissa of A by α ɛ(A) = sup Rez. z∈Λɛ (A) The following result may be called Prüss’s theorem, as it was first established by Prüss in =-=[37]-=-, generalizing earlier work by Gearhart and others. We remind the reader that390 LLOYD N. TREFETHEN FIG. 2. The Hille–Phillips operator, the original example illustrating that γ(A) may exceed α(A). T... |

16 | From the Buffon needle problem to the Kreiss matrix theorem - WEGERT, TREFETHEN - 1994 |

15 |
Onset of turbulence in a pipe
- Boberg, Brosa
- 1988
(Show Context)
Citation Context ...cent years it was not appreciated how such effects could be reconciled with eigenvalue analysis. The first paper with a complete view of such a reconciliation was a beautiful work by Boberg and Brosa =-=[4]-=-. That paper, however, went largely unnoticed for several years, leaving it to the independent and equally impressive paper by Butler and Farrell [9] to communicate such ideas widely. Following closel... |

15 |
On the Stability for Three-Dimensional Disturbances of Viscous Flow Between Parallel Walls
- Squire
- 1933
(Show Context)
Citation Context ...lest Reynolds number R at which an eigenvalue crosses into the right halfplane. It was proved by Squire that this first crossing occurs for a two-dimensional flow; i.e., β =0 (Squire’s theorem, [13], =-=[52]-=-). The eigenvalue problem reduces in this case to a fourth-order ordinary differential equation, the Orr–Sommerfeld equation. The first high-accuracy computations of Orr–Sommerfeld eigenvalues were ca... |

14 |
Optimal energy density growth in Hagen-Poiseuille flow
- Schmid, Henningson
- 1994
(Show Context)
Citation Context ...he boundaries of the ɛ-pseudospectra (just those portions to the right of the spectrum) for ɛ =10 −2 ,10 −2.5 , 10 −3 , 10 −3.5 . (From Trefethen, Trefethen, and Schmid [53]; see also [3], [4], [33], =-=[49]-=-.) streaks; these then undergo further evolution in a manner dependent on the nonlinear terms in the Navier–Stokes equations [2]. The elucidation of the details of this process is an active area of cu... |

13 |
A mostly linear model of transition to turbulence, Phys
- Baggett, Driscoll, et al.
- 1995
(Show Context)
Citation Context ... (From Trefethen, Trefethen, and Schmid [53]; see also [3], [4], [33], [49].) streaks; these then undergo further evolution in a manner dependent on the nonlinear terms in the Navier–Stokes equations =-=[2]-=-. The elucidation of the details of this process is an active area of current research [40]. What about the bizarre spectrum depicted in Figure 10? For the infinite channel, there were two unbounded d... |

12 | The rate at which energy decays in a string damped at one end, Indiana University Mathematics Journal n o 44
- Cox, Zuazua
- 1995
(Show Context)
Citation Context ...y is reflected and we have a nilpotent process of duration 2π. Forδ̸= 1, some energy is reflected; we have an imperfectly absorbing boundary condition. This problem has been studied by Cox and Zuazua =-=[10]-=-, Rideau [46], and Veselić [59], among others.PSEUDOSPECTRA OF LINEAR OPERATORS 393 FIG. 4. A wave operator with a nearly absorbing boundary condition. The steps correspond to waves bouncing back and... |

12 |
C: Pseudospectra of Wiener-Hopf integral operators and constant coefficient differential operators
- Reddy
- 1993
(Show Context)
Citation Context ...(tA)‖ depends linearly on t. The following theorem is established in [1]. THEOREM 2. ‖exp(tA)‖ = eγt for all t ≥ 0 if and only if αɛ(A) =γ+ɛfor all ɛ>0. Example 3: differentiation operator (see [14], =-=[38]-=-). Our third example, presented in Figure 3, is perhaps a more familiar one. Let A be the derivative operator d/dx acting in L2 [0,d] with boundary condition u(d) = 0. The evolution process associated... |

11 |
An accurate solution of the Orr-Sommerfeld equation
- ORSZAG
- 1971
(Show Context)
Citation Context ...oblem reduces in this case to a fourth-order ordinary differential equation, the Orr–Sommerfeld equation. The first high-accuracy computations of Orr–Sommerfeld eigenvalues were carried out by Orszag =-=[32]-=-, who found that the critical Reynolds number is R ≈ 5772, with α ≈ 1.02. Thus, plane Poiseuille flow is mathematically stable for R<5772 and unstable for R>5772. The unstable mode for R>5772 has been... |

10 |
On Szegő’s eigenvalue distribution theorem and non-Hermitian kernels
- LANDAU
- 1975
(Show Context)
Citation Context ... of pseudospectra—that is, of the explicit investigation of the sets bounded by level curves of the norm of the resolvent. The first mention of this idea in print that I have found is by H. J. Landau =-=[27]-=-, who used the term “ɛ-spectrum.” The first sketch of a pseudospectrum of which I know is a hand-drawn figure by Kostin and Razzakov [25] (spectral portrait). These authors were members of a group led... |

10 |
C ∗ -algebra techniques in numerical analysis
- Roch, Silbermann
- 1996
(Show Context)
Citation Context ... A we have (2) Λ ɛ(A) = {z ∈ C : z ∈ Λ(A +∆A) for some ∆A with ‖∆A‖ ≤ɛ}, and if A is an operator, the equivalence of (1) and (2) still holds if we take the closure of the set defined by (2) (see [5], =-=[47]-=-). Thus, a number z is in the interior of the ɛ-pseudospectrum of A if and only if it is in the spectrum of some perturbed operator A +∆Awith ‖∆A‖ <ɛ. A third equivalent definition, closer to computat... |

9 | Spectra and Pseudospectra of Waveform Relaxation Operators
- LUMSDAINE, WU
- 1995
(Show Context)
Citation Context ... of an infinite Toeplitz matrix to those of the infinite matrix. An excellent introduction to this circle of ideas is found in [6]. For related developments in the context of waveform relaxation, see =-=[30]-=-; for Abel integral operators, see [36]. The behavior of Toeplitz and Wiener–Hopf operators can be quite wild, but pseudospectra of the wilder examples have not yet been computed. Figure 7 shows an ex... |

8 | Spectra and pseudospectra for pipe Poiseuille flow
- Trefethen, Trefethen, et al.
- 1999
(Show Context)
Citation Context ...sion m; see [12], [26], [45], [58]. In particular cases, however, upper bounds generally can be derived via contour integrals such as that of Theorem 4. Example 10: pipe Poiseuille flow operator (see =-=[53]-=-). Our final example looks very different yet is physically almost the same. We now consider Reynolds’ original problem of flow through a circular pipe instead of an infinite channel, following larges... |

6 |
Infinite matrices and projection methods
- Böttcher
- 1996
(Show Context)
Citation Context ...ives a Wiener–Hopf integral operator; or both, which gives a Wiener–Hopf integral operator on an unbounded domain. Pseudospectra of such operators have been studied by Reddy [38] and by Böttcher [5], =-=[6]-=-, Böttcher and Wolf [7], and Roch and Silbermann [47]. The last group have elaborated powerful C ∗ -algebra techniques for analysis of pseudospectra and have proved a number of theorems regarding, for... |

5 |
Optimal growth of small disturbances in pipe Poiseuille flow. Phys. Fluids A
- Bergstrom
- 1993
(Show Context)
Citation Context ...ash curves are the boundaries of the ɛ-pseudospectra (just those portions to the right of the spectrum) for ɛ =10 −2 ,10 −2.5 , 10 −3 , 10 −3.5 . (From Trefethen, Trefethen, and Schmid [53]; see also =-=[3]-=-, [4], [33], [49].) streaks; these then undergo further evolution in a manner dependent on the nonlinear terms in the Navier–Stokes equations [2]. The elucidation of the details of this process is an ... |

5 | Pseudospectra of the wave operator with an absorbing boundary
- Driscoll, Trefethen
- 1996
(Show Context)
Citation Context ...g ‖exp(tA)‖ depends linearly on t. The following theorem is established in [1]. THEOREM 2. ‖exp(tA)‖ = eγt for all t ≥ 0 if and only if αɛ(A) =γ+ɛfor all ɛ>0. Example 3: differentiation operator (see =-=[14]-=-, [38]). Our third example, presented in Figure 3, is perhaps a more familiar one. Let A be the derivative operator d/dx acting in L2 [0,d] with boundary condition u(d) = 0. The evolution process asso... |

5 |
Do the pseudospectra of a matrix determine its behavior
- Greenbaum, Trefethen
- 1993
(Show Context)
Citation Context ...EAR OPERATORS 387 Along the way we shall present a few theorems that partially answer the question, what do pseudospectra tell us about the behavior of an operator? (The full answer is not known; see =-=[18]-=-.) This paper is in every way a sequel to [55], to which I hope all readers have access. 4. Computation of pseudospectra. Before presenting the examples, we must make a few remarks about how pseudospe... |

4 |
Pseudospectra of an operator of Hille and
- Baggett
- 1994
(Show Context)
Citation Context ...a is αɛ(A) = 1 + ɛ for each ɛ > 0 and, thus, 1 in the limit ɛ → 0. The contours in the upper plot are boundaries of ɛ-pseudospectra Λɛ (A) for ɛ =10−1,10−2, 10−4 , 10−6 , 10−8 , 10−10 . (From Baggett =-=[1]-=-, based on [60].) processes, A −1 for systems of equations, and polynomial or rational functions p(A) or r(A) for iterations such as conjugate gradients, Lanczos, or GMRES. Though we do not discuss it... |

4 |
Transient growth in circular pipe flow, 1: Linear disturbances
- O’Sullivan, Breuer
- 1994
(Show Context)
Citation Context ... are the boundaries of the ɛ-pseudospectra (just those portions to the right of the spectrum) for ɛ =10 −2 ,10 −2.5 , 10 −3 , 10 −3.5 . (From Trefethen, Trefethen, and Schmid [53]; see also [3], [4], =-=[33]-=-, [49].) streaks; these then undergo further evolution in a manner dependent on the nonlinear terms in the Navier–Stokes equations [2]. The elucidation of the details of this process is an active area... |

4 | Private communication
- Kato
- 2008
(Show Context)
Citation Context ...tially for every z inside the critical parabola Rez = −(Imz) 2 . The dashed curve marks this critical parabola; the boundary of the numerical range is the same curve shifted left by π 2 /d 2 ≈ 0.0062 =-=[61]-=- (From Reddy and Trefethen [42]; see also [24].) with solution ( ) uy(t, ·) ut(t, ·) ( ) f = exp(tA) . g Thus, the data in the evolution problem are block 2-vectors, and the solution operators are blo... |

3 |
Spectral approximation for Segal–Bargmann space Toeplitz operators
- BÖTTCHER, WOLF
- 1997
(Show Context)
Citation Context ...gral operator; or both, which gives a Wiener–Hopf integral operator on an unbounded domain. Pseudospectra of such operators have been studied by Reddy [38] and by Böttcher [5], [6], Böttcher and Wolf =-=[7]-=-, and Roch and Silbermann [47]. The last group have elaborated powerful C ∗ -algebra techniques for analysis of pseudospectra and have proved a number of theorems regarding, for example, convergence o... |