L. N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., Chapman and Hall, 1990, pp. 336--360.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we concentrate ourselves in the T. Chan circulant preconditioners here, the convergence results in Theorems 1 and 2 can easily be extended to include other circulant preconditioners. For instance, results for Strang s circulant preconditioners can be obtained if we replace Lemma 1 by theorems in [24]. In particular, using Theorem 6 there, we can show that if the quotient function h(z) is a rational function of type ( then our method converges in at most (1 2 maxf; g ) steps for large n. 14 ....

L. Trefethen, Approximation Theory and Numerical Linear Algebra, Algorithms for Approximation II, J. Mason and M. Cox, eds., Chapman and Hall, London, 1990.

....works for a slightly smaller class of functions (see (30) than T. Chan s preconditioner does. In the conjugate gradient method, an estimate of the number of iterations required for convergence can be obtained by studying the precise rate at which jje q jj goes to zero in (1) Trefethen [18] first proved that if f is a positive rational function of type ( then the preconditioned system S n A n has at most 1 2 maxf; g distinct eigenvalues. Hence the conjugate gradient method, when applied to the preconditioned system, converges in at most 1 2 maxf; g steps. He also proved ....

L. Trefethen, Approximation Theory and Numerical Linear Algebra, Algorithms for Approximation II, M. Cox and J. Mason, eds, 1989.

....of k are 11, 12, and 11. The Legendre approximation of the eigenvalues is superior to the two Chebyshev approximations. The best approximation displayed in Fig. 3.3 is that shown in (f) where (a, 3) 3, 1) gives N = 0.43333 and k = 13. It is of interest to note that recent work by Trefethen [10] suggests that since D is not normal accuracy should be assessed by considering pseudo eigenvalues rather than exact eigenvalues. This point deserves further examination. I I I I 5 10 15 20 25 30 (e) log0[eigenvaluel N=32 o o [ I I I lO 15 20 25 30 (f) FIG. 3.3. Continued. 3.3. ....

....7.92(11) 2.13( 11) 5.70( 11) 1.74( 11) 4.43( 11) 7.11( 10) 2.44( 9) 7.36( 10) 1.97(9) marginally better performance in approximating the eigenvalues of the differential operator. As stated at the end of 3. 2, it is possible that accuracy assessment based on exact eigenvalues might be misleading [10]. 5. Comments. We have considered pseudospectral methods for global polynomial approximation of functions and of solutions to differential equations. Generalized quadrature rules have been presented which identify interior collocation points for the pseudospectral solution of certain model ....

L. N. TREFETHEN, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., Chapman, London, 1990.

.... [76, Chap.7] Recently, the problem has been studied for more general matrices [93] and its solution involves Faber polynomials [45, 46, 47, 114] or generalizations of Chebyshev polynomials [104, 9] The connections between numerical linear algebra and approximation theory have been reviewed in [116]. In this paper, we shall consider the application of a semi iterative method to an arbitrary sequence of iterates not necessarily produced by a stationary iterative method and we shall also make use of a different minimization criterion based on the residual vector ae n instead of the error ....

L.N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J.C. Mason and M.G. Cox eds., Chapman and Hall, London, 1990, pp. 336-- 360.

....This phenomenon is known as superlinear convergence of the CG method. Once one knows more about the spectrum of A, one may exploit the link to some polynomial extremal problem (see (2. 12) below) and obtain sharper error estimates using techniques from approximation theory, see for instance [DTT98, Gr97, GrTr94, Tre90, TrBa97] and the references therein. Indeed, according to a well known observation in numerical linear algebra, superlinear convergence behavior means that some of the Ritz values approach very well some eigenvalues, see for instance [vSvV86, Gre79, SlvS96, DTT98] However, it seems that an analytic ....

L.N. Trefethen, Approximation theory and numerical linear algebra, in: Algorithms for Approximation II, J.C. Mason and M.G. Cox, eds, Chapman & Hall, London, 1990.

....by the National Science Foundation under Grants CCR 95 03126 and CCR 97 32022. y Applied Mathematics and Scienti c Computing Program, University of Maryland, College Park, MD 20742 (iaz cs.umd.edu) 1 characteristics of the coecient matrix such as the eld of values [3] and pseudo spectrum [20]. However, none of these bounds is immune to the problems of inaccuracy and computational complexity. More recently Greenbaum and her colleagues discovered that eigenvalues alone cannot explain GMRES behavior [8, 7] However, in this paper, as well as in an accompanying manuscript [23] we ....

L. N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. Mason and L. Cox, eds., Chapman and Hall, London, 1990.

....This phenomenon is known as superlinear convergence of the CG method. Once one knows more about the spectrum of A, one may exploit the link to some polynomial extremal problem (see (2. 12) below) and obtain sharper error estimates using techniques from approximation theory, see for instance [DTT98, Gr97, GrTr94, Tre90, TrBa97] and the references therein. Indeed, according to a well known observation in numerical linear algebra, superlinear convergence behavior means that some of the Ritz values approach very well some eigenvalues, see for instance [vSvV86, Gre79, SlvS96, DTT98] However, it seems that an analytic ....

L.N. Trefethen, Approximation theory and numerical linear algebra, in: Algorithms for Approximation II, J.C. Mason and M.G. Cox, eds, Chapman & Hall, London, 1990.

.... of I M T transpose of the matrix M 2 C N ThetaN M H hermitian transpose of M (M ) spectrum of M F (M) field of values of M , F (M) fv H Mv : v 2 C N ; kvk = 1g ffl (M) ffl pseudospectrum of M , ffl (M) fz 2 C : k(zI Gamma M) Gamma1 k ffl Gamma1 g, for all ffl 0 [74] (M) condition number of M , M) kMkkM Gamma1 k for nonsingular M , M) 1 otherwise n set of all polynomials of degree at most n with value one at the origin 5 6 1 Introduction One of the most important tasks in scientific computing is to solve systems of linear equations, Ax = ....

L. N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for approximation II, J. Mason and M. Cox, eds., Chapman and Hall, London, 1990, pp. 336--360.

....by H. J. Landau in 1975 [24] and has been subsequently employed by others including Varah [39] Demmel [6] and especially Godunov, Kostin, Malyshev, and their colleagues in Novosibirsk [13] We and our coauthors have applied pseudospectra to problems in numerical linear algebra [12] 27] [35], the numerical solution of differential equations [8] 20] 31] and fluid dynamics [30] 38] For introductions to these ideas, see [29] 33] 36] 37] The results of this paper can be summarized as follows. Though the spectrum of L is a subset of the negative real axis, its ....

....to (7.7) 27] When (M) is large and ffl is not too small, a comparison of (7.7) and (7.8) suggests that when L (N) is far from normal, the parameters of a nonsymmetric matrix iteration may be more appropriately based on the pseudospectra than on the spectrum. Experiments confirm this prediction [35]. A related example is the phenomenon of direction dependent convergence considered in [4] 11] 19] a Gauss Seidel or SOR iteration for a convection diffusion problem may converge in far fewer steps if one sweeps with the convection rather than pseudospectra of the convection diffusion ....

L. N. Trefethen, Approximation theory and numerical linear algebra, in J. C. Mason and M. G. Cox, eds., Algorithms for Approximation II, Chapman and Hall, London, 1990.

....factor (Z) in (7) may be quite large. See, for instance, 13] for some interesting physical examples. In such cases, the bound (7) may be a large overestimate of the actual residual. To deal with such problems, Trefethen has introduced bounds based on the ffl pseudo eigenvalues of the matrix [16]. There are two equivalent definitions of the ffl pseudo spectrum of a matrix A: Definition 1. The ffl pseudo spectrum of A, denoted ffl (A) is the set of points z ffl C such that k(zI Gamma A) Gamma1 k ffl Gamma1 . Definition 2. The ffl pseudo spectrum of A is the set of points z ffl ....

L. N. Trefethen, "Approximation Theory and Numerical Linear Algebra, " in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., Chapman and Hall, London, 1990.

....Lecture Notes Comp. Sc. Springer Verlag. 266 BIBLIOGRAPHY L. B. Rall, 1991) Tools for mathematical computation, in Computer aided proofs in analysis, K. Meyer and D. Schmidt, eds. vol. 28, IMA Volumes in Mathematics and its Appl. Springer Verlag, New York, Berlin, 217 228. S. C. Reddy and L. N. Trefethen, 1990) Lax stability of fully discrete spectral methods via stability regions and pseudo eigenvalues, Comp. Meth. Appl. Mech. Eng. 80, 147 164. S. C. Reddy and L. N. Trefethen, 1992) Stability of the method of lines, Numer. Math. 62, 235 267. S. C. Reddy, L. N. Trefethen, and D. ....

....vol. 28, IMA Volumes in Mathematics and its Appl. Springer Verlag, New York, Berlin, 217 228. S. C. Reddy and L. N. Trefethen, 1990) Lax stability of fully discrete spectral methods via stability regions and pseudo eigenvalues, Comp. Meth. Appl. Mech. Eng. 80, 147 164. S. C. Reddy and L. N. Trefethen, 1992) Stability of the method of lines, Numer. Math. 62, 235 267. S. C. Reddy, L. N. Trefethen, and D. Pathria, 1993) Pseudospectra of the convection diffusion operator, Tech. Rep. CTC93 TR126, Cornell Theory Center. S. C. Reddy, 1991) Pseudospectra of operators and discretization ....

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L. N. Trefethen, (1990), Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and e. M. G. Cox, eds., London, Chapman and Hall, 336--359.

....matrix but on its eigenvalues. For arbitrary f 2 C 2 of the form (2.12) we were not able to prove properties concerning the distribution of the eigenvalues of AN (f)M N (f) Gamma1 . But for special generating functions, namely rational functions, we obtain the following result (see also [17]) Theorem 2.7 Let f be a rational function of order (s 1 ; s 2 ) s 1 s 2 6= 0) given by f(z) p(z) q(z) p 0 p 1 z : p s1 z s1 q 0 q 1 z : q s2 z s2 : Define M N (f) by (2:8) with grid points satisfying (2:13) if f(e it j ) 0 (j = 1; m) Then AN (f)M ....

N. Trefethen. Approximation Theory and Numerical Linear Algebra. In J. Mason and M. Cox, editors, Algorithms for Approximation II, Chapman and Hall, London, 1990.

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L. N. Trefethen, Approximation theory and numerical linear algebra, in J. C. Mason and M. G. Cox, eds., Algorithms for Approximation II, Chapman and Hall, London, 1990, pp. 336-360.

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L. N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., London, 1990, Chapman and Hall.

.... is the Kreiss Matrix Theorem, which contains a constant eN that does not reduce cleanly to 1 as 0 [10, 17] Another is the bound on a polynomial norm kp(A)k, of immediate relevance to iterations such as GMRES, that can be obtained by integrating p(z) over the boundary contour(s) of (A) [19]. For new results comparing such contour integral techniques to other approaches, see [5] 2. Sixteen Theorems. Our rst theorem states that ill conditioning is equivalent to the existence of a pseudoeigenvalue near the origin. The result has been attributed to Gastinel (see [18, pp. 120, 133] ....

L. N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., London, 1990, Chapman and Hall.

....Khoruzhenko, Silvestrov, and Sommers, among others. Pseudospectra and related quantities for nonnormal matrices and operators were rst investigated in the 1970s and 1980s [21,35,55] and became a standard tool in the 1990s [8,48,49,51] with applications in uid mechanics [54] numerical analysis [29,42,46,47], operator theory [2,5] control theory [30] Markov chains [31] di erential equations [9,10,43] and integral equations [40,41] In all of these elds it has been found that in cases of pronounced nonnormality, eigenvalues and eigenvectors alone do not always reveal much about the aspects of the ....

....ff 1g; 1g. The solid dots, at z = 1, are the eigenvalues. The contours, from outside to inside, are the boundaries of (A) for = 10 2 ; 10 6 ; 10 10 ; 10 14 ; 10 30 . matrix obtained by a perturbation of norm . The equivalence of (i) iv) is discussed in [13] 45] and [47], and much more extensive related material can be found in [32] Figure 1 shows pseudospectra of two matrices from bidiag N ff 1g; 1g, one with N = 100 and one with N = 10,000. 1 The pictures reveal that the nonnormality of these matrices is pronounced. The resolvent norm grows exponentially in ....

L. N. Trefethen, Approximation theory and numerical linear algebra, in J. C. Mason and M. G. Cox, eds., Algorithms for Approximation II, Chapman and Hall, London, 1990, pp. 336-360.

....CGNR, GMRES, CGS, Bi CGSTAB, and QMR iterations. Sometimes rational approximation problems also arise, notably in the analysis of ADI iterations, circulant preconditioned Toeplitz iterations, and Krylov subspace algorithms via Pad e approximation. Recent references on these matters include [4,8,16,25]. The approximation problems that are discussed in the linear algebra literature almost invariably involve scalar functions defined on subsets of the complex plane, or, if the matrix A is symmetric, the real axis. The set in question is the spectrum (A) or an estimate of the spectrum. If A is ....

....the approximation problem and the convergence of the matrix iteration there is a gap of size (V ) the condition number of a matrix of eigenvectors of A. When (V ) is large, predictions based on the approximation problem may have little bearing on the actual convergence of the matrix algorithm [16,25]. The purpose of this paper is to explore a different kind of approximation problem that can also be associated with iterative linear algebra, involving matrices instead of scalars. Instead of asking how small a polynomial p(z) can be on the set (A) we ask how small the norm kp(A)k can be. ....

L. N. Trefethen, Approximation theory and numerical linear algebra, in J. C. Mason and M. G. Cox, eds., Algorithms for Approximation II, Chapman and Hall, London, 1990.

....of approximations. The degree of nonnormality of a matrix a notion we shall not attempt to define precisely can vary arbitrarily. In particular, most estimates such as (STEP3) that are based on the eigenvalues of a matrix can fail to be sharp to an arbitrary degree, for certain matrices [75]. As an elementary example, consider the norms of powers #A n # for an N N Jordan matrix of the form (15) A = 0 B B B B 02 02 02 02 0 1 C C C C A . For n Nwe have #A n # =2 n , even though z n is identically zero on the spectrum of A (the origin) Nor does this ....

....which shrink substantially as # decreases. A connection can be made between pseudospectra and convergence of Krylov subspace iterations via the contour integral p(A) 1 2#i) R # (zI A) 1 p(z) dz, valid for any polynomial p and any simple closed integration contour # enclosing #(A) 53] 71] [75]. If p(z) is small on a region where #(zI A) 1 # is not too large, then taking absolute values in this formula may give a reasonable bound on #p(A)#. For example, POTENTIAL THEORY AND MATRIX ITERATIONS 565 if # is taken as the boundary of # # (A) for some #,weget (20) #p(A)## 1 2# L# ....

L. N. TREFETHEN, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., Chapman and Hall, London, 1990, pp. 336--360.

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L. N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., Chapman and Hall, 1990, pp. 336--360.

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L. N. Trefethen, 1990. Approximation theory and numerical linear algebra. In J. C. Mason and M. G. Cox, editors, Algorithms for approximation II. Chapman and Hall, London.

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L. N. Trefethen, Approximation theory and numerical linear algebra, in Algorithms for Approximation II, J. C. Mason and M. G. Cox, eds., London, 1990, Chapman and Hall.

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L.N. TREFETHEN, Approximation theory and numerical linear algebra, in: Algorithms for Approximation II, J.C. Mason and M.G. Cox, eds, Chapman & Hall, London, 1990.

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L. Trefethen, Approximation Theory and Numerical Linear Algebra, Algorithms for Approximation II, M. Cox and J. Mason, eds., Chapman and Hall, London, 1990.

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N. Trefethen. Approximation Theory and Numerical Linear Algebra. In J. Mason and M. Cox, editors, Algorithms for Approximation II, Chapman and Hall, London, 1990.

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L. Trefethen, Approximation Theory and Numerical Linear Algebra, Algorithms for Approximation II, M. Cox and J. Mason, eds, 1989.

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