F. Chatelin. Valeurs propres de matrices. Masson (1988).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....une ou plusieurs des valeurs propres cherch ees [8, 6] elle risque de manquer une ou plusieurs valeurs propres. En arithm etique exacte, la convergence de la m ethode des it erations simultann ees est assur ee si la condition de s eparation et celle d information compl ete sont v eri ees [4, 7]. Ainsi, les exp erimentations num eriques se d eroulent de la mani ere suivante en 4 phases successives: CERFACS et IRISA INRIA, 42 avenue Gustave Coriolis, 31057 Toulouse Cedex. E mail: zaoui cerfacs.fr CERFACS. E mail: travies cerfacs.fr Phase 1. Utilisation de la m ethode ....

F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....algorithms [1] and [11] We note that block methods are usually effective for relatively dense matrices. In this paper, we are interested in large Sylvester matrix equations (the matrix A is large and B is of moderate size (s n) Such problems arise in the solution of large eigenvalue problems [4] and in the boundary value problems. In Section 2, we present the block Arnoldi Sylvester method and give some theoretical results. Section 3 is devoted to the block GMRES Sylvester method. We first show how to derive the new method and consider the practical problem of solving the linear matrix ....

F. Chatelin, Valeurs Propres de Matrices, Masson, Paris 1988.

....On these questions, see, for example, Davis [40] Laurent [88] or Cryer [39] The following general result should also be reminded If P k 6= 0, then kP k k 1. kP k k = 1 if and only if P k is an orthogonal projection. Other results about projections can be found, for example, in Chatelin [34, 35]. The connection with Galerkin approximation is studied in [117, pp. 291ff] see also subsection 2.5 below. The subject is also closely related to biorthogonality as explained in [19] some other results will be given in subsection 2.4. All the preceding results will be useful in the sequel. 2.2 ....

F. Chatelin, Valeurs Propres de Matrices, Masson, Paris, 1988.

....of A in the orthonormal basis Wm , is a Hessenberg form of A when m = n. But the most used form of this method is when m n. Arnoldi himself hinted that the process could give good approximations to some eigenvalues if stopped before completion [1] This version is called by some authors [2] the incomplete Arnoldi method (because of m n) This appellation is used by some other author [12] to design another variant of Arnoldi s method based onto an incomplete reorthogonalization of the Krylov vectors of K m;v subspace. For m n, we call the above algorithm Basic Arnoldi Projection ....

F. Chatelin, Valeurs Propres de Matrices, Masson, Paris, 1988.

....i;j ) is an Hessenberg representation of A in the orthonormal basis Wm of K m;v when m = n. Arnoldi hinted that the process could give good approximations to some eigenvalues if stopped before completion [1] i.e. when m n. Today, it is the most common use of the method. This version is called [3] the incomplete Arnoldi method (because of m n) But this appellation is used by some other author [20] to design another variant of Arnoldi s method based on an incomplete re orthogonalization of the Krylov vectors of K m;v . The matrices Hm and Wm issued from AR algorithm and the matrix A ....

F. Chatelin, Valeurs Propres de Matrices, Masson, Paris, 1988.

....or more if a variant by blocks is employed, is added to the basis. Convergence is then achieved as the basis size, m, increases and it usually happens with m n. The algorithms of Lanczos and Arnoldi are also attractive because they do not perform modi cations on the matrices of the problem. See [6, 14, 24, 25, 26] for a detailed presentation of these methods. The study of collective motions of molecules provides useful insights into the large amplitude conformational changes the molecules experiment during chemical reactions. Such motions are likely to play an important role in enzymatic activities [4, ....

F. Chatelin. Valeurs Propres de Matrices. Masson, Paris, France, 1988.

....eigenelements produced by Arnoldi, ensuring that no eigenvalue is missed near . 1.1 Description of the hybrid eigensolver ISABeL The block Arnoldi method and Subspace Iterations are well known methods to compute a few of the dominant eigenvalues of a large matrix A. The reader is referred to ([6, 7, 8]) The block version of Arnoldi allows for the computation of multiple eigenvalues (multiplicity size of block) It is used with shift and invert, that is on the matrix C = A B) 1 B to compute the eigenvalues near of the pencil (A; B) IRISA and CERFACS, 42 avenue Gaspard Coriolis, ....

....ISABeL 2 Even though Arnoldi has proved extremely fast and ecient to compute prescribed eigenvalues ( 9] there is no proof of convergence for the restarted version which is most widely used. On the contrary, the convergence of Subspace Iterations is fully understood in exact mathematics ([6, 7, 8]) Keeping these two facts in mind, the idea is to produce a hybrid method which would retain the speci c qualities of each method (speed for Arnoldi, convergence for Subspace Iterations) and compensate for their respective de ciencies. The result is the hybrid eigensolver ISABeL to solve the ....

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F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....of the eigenvalue alone. Indeed, it is often the case that the eigenvalue converges faster than the eigenvector and therefore, k ) reaches the prescribed tolerance earlier than ( k ; u k ) Since the Power method can be viewed as a Newton method which minimises the Rayleigh quotient [4], we expect that the accuracy of the matrix vector product z = Au k governs the convergence in the sense that more and more accuracy is required while approaching convergence. In order to illustrate this point numerically, we introduce random perturbations on the matrix vector products as ....

F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....sensitivity of eigenvalues to changes in the original matrix elements is much lower. To avoid amplifying the sensitivity of eigenvalues, many state of the art numerical methods use unitary similarity transformations to simplify the matrix, and then compute the eigenvalues of the simplified matrix [3, 4, 5, 9, 14, 17, 18, 27]. In [20] Wielandt proposes two methods: one to compute the characteristic polynomial of a complex matrix A and a second one to locate the roots of a polynomial with complex coefficients. Wielandt s objective was to develop methods for locating eigenvalues that are simpler, more efficient, and ....

F. Chatelin. Valeurs Propres de Matrices. Masson, Paris, 1986.

....of origin, the convergence of a (k) nn to an eigenvalue is asymptotically quadratic. The study of the convergence of the Arnoldi method is far less sufficient than that of the Lanczos method, since the theory of the uniform approximation on a compact set in the complex plane is not so advanced [2]. 5 Numerical Experiments This chapter reports the results of the numerical experiments of our new method and evaluates its performance. The experiments are performed on a HP9000 720, using double precision. We start from the decision of each element of the matrix given in the problem. In this ....

F. Chatelin, Valeurs Propres de Matrices. Paris: Masson, 1988.

....and the iterative modified Gram Schmidt algorithm. 1 Introduction The departure from normality of a matrix plays an essential role in numerical matrix computations. The bad numerical behaviour of highly nonnormal matrices has been known for a long time (Henrici (1962) Van der Sluis (1975) Chatelin (1988)) But this first effect of high nonnormality, i.e. the increase of the spectral instability was long considered by practitioners to be a mathematical oddity, since such matrices were not often encountered in practice. It appears that now, more and more matrices that model physical problems at ....

....0 = Q 0 R 0 2. Do until convergence U j = AQ j Gamma1 ; U j = Q j R j and j = j 1: iteratively builds an orthogonal basis Qm of size n Theta m (m n) of an approximate invariant subspace associated with the m eigenvalues having the largest moduli (see Golub and Van Loan (1989) Chatelin (1988)) We computed the m desired eigenvalues of the matrix A using the QR algorithm on the matrix of size m Theta m resulting from the projection of the original problem onto the computed invariant subspace. This method can be associated with a Tchebycheff acceleration to improve the speed of ....

F. Chatelin, (1988), Valeurs propres de matrices, Masson, Paris.

....For such a matrix, one normally uses the Arnoldi algorithm (see Arnoldi (1951) Let A be an n Theta n matrix. The Arnoldi method builds an orthogonal basis of the Krylov subspace Vm = fv i g i=1: m of size n Theta m with m n and a Hessenberg matrix Hm of size m Theta m (see for example Chatelin (1988)) such that AVm = VmHm hm 1;m vm 1 e T m ; 1) where e T m denotes the transpose of the m th vector of the canonical basis. We have the following : Lemma 3.1 If ( x) is an approximate eigenpair for the matrix A and ( y) is the associated eigenpair for the matrix Hm , then r ....

....m, m n; of an approximate invariant subspace associated with the m eigen15 0 100 200 300 400 500 600 700 800 900 20 19.5 19 18.5 18 17.5 17 16.5 16 15.5 Index of the eigenvalues of A Figure 4: QR : j i i=1: 989 versus the index of the eigenvalues. values having the largest moduli (see Chatelin (1988), Golub and Van Loan (1989) We computed the m eigenvalues of the matrix A using the QR algorithm on the m Theta m matrix resulting from the projection of the original problem onto the computed invariant subspace. This method may use a Tchebycheff acceleration to improve the speed of ....

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F. Chatelin, (1988), Valeurs propres de matrices, Masson, Paris.

....3. or total reorthogonalization. The eigenvalues of the projected matrix are good approximations of the original eigenvalues situated at the periphery of the spectrum of A. This is why we propose to compute the eigenvalues having the largest real or imaginary parts or the smallest real parts (see [7] and [8] Because of storage limits, this method can only permit to compute a small number of eigenvalues compared with the order of A. To increase the number of the computed eigenvalues, a Wielandt deflation has been implemented (see for example [17] The Laboratoire Central de Recherche ....

....to step 3 . All steps of both algorithms for large, real nonsymmetric matrices are descripted in [6] Here are some references for the different steps for both algorithms : ffl for the choices of the initial parameters (Steps 1 and 1 ) see [3] ffl for the Arnoldi process (Step 2) see [3] [7], 1] 17] ffl for the reorthogonalization step (Step 2) see [3] ffl for the Tchebycheff acceleration (Step 9) see [16] 17] 12] In the following paragraphs, we will present the choice of the stopping criteria (Step 6) the ellipse determination (Step 7) and the Wielandt deflation (Step ....

F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....the fluid dynamics and the structural analysis, there are a number of cases where a few eigenvalues with the largest real parts of a nonsymmetric matrix are required. In economic modeling, the stability of a model is interpreted in terms of the dominant eigenvalues of a large nonsymmetric matrix [1, 7]. Several methods have been proposed for this problem. The method proposed by Arnoldi in 1951 and the subspace iteration method due to Rutishauser , which are variants of the projection method, have been the most effective for this purpose. The Arnoldi method, however, has a drawback of the ....

F. Chatelin, Valeurs Propres de Matrices. Paris: Masson, 1988.

.... (if not implemented carefully) 2] 3] even stable algorithms such as QR give wrong approximations for eigenvalues due to the huge spectral ill conditioning [3] 6] Moreover, we still have no access to a condition number for clustered eigenvalues; only upper bounds exist in the literature [5], 6] In this context, a computed eigenvalue may loose its meaning if no information is known about its stability. The trend today is to consider not an eigenvalue by itself but its topological neighbourhood. Such an approach permits to understand the influence of perturbations on the spectrum of ....

F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

.... of molecules (hinge bending motions) in structural engineering, dynamic properties of a given model (natural vibration frequencies and mode shapes) and in nuclear power plants, neutron fluxes (if the dominant eigenvalue is greater than one the behaviour of the plant is super critical) See [4] and [29] for different cases and meanings. Depending on the level of the discretization of a continuous problem or the precision required for the results, A and B can reach dimensions of many thousands. Although in practical analyses only a few eigenpairs ( x) are taken into account, either in ....

F. Chatelin. Valeurs Propres de Matrices. Masson, Paris, France, 1988.

....multiplicity one is called simple. A nonsimple eigenvalue is called multiple. Definition 2.1.14 An eigenvalue of A is said to have the geometric multiplicity fl, if the maximum number of independent eigenvalues associated with is fl. For the proofs of the following two theorems, see Chatelin [10] and Halmos [26] Lemma 2.1.1 For every integer l and each eigenvalue i , we have N(A 0 i I) l 1 ae N(A 0 i I) l : 2.10) In a finite dimensional space, there is a smallest integer l i such that N(A 0 i I) l i 1 = N(A 0 i I) l i : 2.11) Here, l i is called the index of i . The ....

....holds. If we denote by x 1 and y 1 the right and left eigenvectors of 1 , the sequence u 6= 0; q 0 = u k u k 2 ; q k = Aq k01 k Aq k01 k 2 ; k 1 (3.2) is such that j(Aq k ; q k ) 0 1 j = O fi fi fi fi 2 1 fi fi fi fi k (3.3) if and only if y T 1 u 6= 0. Proof. See Chatelin [10] or Wilkinson [77] for example. 2 3.1.2 The Inverse Iteration Let oe be an approximation to a simple eigenvalue , with right eigenvector x. If oe is not close to the eigenvalues of A, other than , then the dominant eigenvalue of (A 0 oeI) 01 is 1= 0 oe) This fact is exploited in the ....

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F. Chatelin, Valeurs Propres de Matrices, Masson, Paris, 1988.

....or more if a variant by blocks is employed, is added to the basis. Convergence is then achieved as the basis size, m, increases and it usually happens with m n. The algorithms of Lanczos and Arnoldi are also attractive because they do not perform modifications on the matrices of the problem. See [6, 14, 24, 25, 26] for a detailed presentation of these methods. The study of collective motions of molecules provides useful insights into the large amplitude conformational changes the molecules experiment during chemical reactions. Such motions are likely to play an important role in enzymatic activities [4, 10, ....

F. Chatelin. Valeurs Propres de Matrices. Masson, Paris, France, 1988.

....of A in the orthonormal basis Wm , is a Hessenberg form of A when m = n. But the most used form of this method is when m n. Arnoldi himself hinted that the process could give good approximations to some eigenvalues if stopped before completion [1] This version is called by some authors [2] the incomplete Arnoldi method (because of m n) This appellation is used by some other author [12] to design another variant of Arnoldi s method based onto an incomplete reorthogonalization of the Krylov vectors of K m;v subspace. For m n, we call the above algorithm Basic Arnoldi Projection ....

F. Chatelin, Valeurs Propres de Matrices, Masson, Paris, 1988.

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F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....on the size of DeltaA resulting from the backward error. On the other hand, homotopic perturbations where DeltaA = tE can be useful to study the theoretical convergence of iterative methods where the exact matrix A is replaced by an approximate matrix A 0 such that E = A Gamma A 0 is known [3, 4, 5]. Associated with the homotopic family of matrices A(t) A tE, which is such that A(0) A and A(1) A E, where A and B are known, we can consider the two problems [3, 4, 5] i) solve (A(t) Gamma zI)x(t) b for any z not an eigenvalue of A(t) ii) solve A(t) t) t) t) t) 6= 0. The ....

.... methods where the exact matrix A is replaced by an approximate matrix A 0 such that E = A Gamma A 0 is known [3, 4, 5] Associated with the homotopic family of matrices A(t) A tE, which is such that A(0) A and A(1) A E, where A and B are known, we can consider the two problems [3, 4, 5] i) solve (A(t) Gamma zI)x(t) b for any z not an eigenvalue of A(t) ii) solve A(t) t) t) t) t) 6= 0. The homotopic pseudospectrum contains all (t) for j t j 1, and the complementary set in l C is the set of all z such that x(t) can be computed by a converging Neumann series x(t) ....

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F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....d epasse de beaucoup le cadre etroit du calcul a pr ecision nie (sur ordinateur) dans lequel cette analyse est g en eralement con n ee. C est tout le calcul approch e, calcul par perturbations et calcul asymptotique, qui peut etre d ecrit dans le formalisme unique de l analyse inverse [4, 5, 6]. 3.1 Mesure de la qualit e de solutions approch ees Pour le probl eme spectral Ax = x, x 6= 0, on dispose de plusieurs mesures de qualit e de solutions approch ees selon l information consid er ee : 3 i) information pour approcher : la qualit e se mesure par la distance a la singularit e ....

....Krylov [7] 3.2 Calcul inexact et perturbations homotopiques Supposons que, quelque soit z dans l ensemble r esolvant de A, on sache e ectuer exactement l op eration x (A zI) 1 x. On s int eresse alors a B = A E et aux deux probl emes associ es (B zI)y = b; By = y; y 6= 0: Il est utile [4, 5, 6] de consid erer la famille A(t) A tE o u t 2 l C , j t j 1, avec A(0) A et A(1) B, et les deux probl emes associ es : A(t) zI)x(t) b; A(t)x(t) t)x(t) x(t) 6= 0 La famille A(t) est une famille de matrices d e nies a partir de A par addition de la perturbation homotopique tE, ....

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F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....a bound on the size resulting from the backward error. On the other hand, homotopic perturbations where DeltaA = tE can be useful to study the theoretical convergence of iterative methods where the exact matrix A is replaced by an approximate matrix A 0 such that E = A Gamma A 0 is known [3, 4, 5]. In such an analysis the arithmetic is, of course, supposed to be exact. 3 Illustration on the Arnoldi method 3.1 Backward error for the Arnoldi method in exact arithmetic In the Arnoldi method, at the iteration k, the eigenvalues of the Hessenberg matrix H k are supposed to approach at most k ....

F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

.... nonnormality, exact arithmetic, finite precision, reliability of numerical software 2 Is nonnormality a serious computational difficulty in practice 1 INTRODUCTION It has long been known that nonnormal matrices can exhibit spectral instability (Henrici (1962) van der Sluis (1975) Chatelin (1988, 1993) Despite this theoretical wisdom, nonnormality was considered in practice until recently as a mathematical curiosity only. However in the past few years, problems arising from high technology (Braconnier, Chatelin, and Dunyach (1995) or theoretical physics (Reddy (1991) Kerner (1989) ....

Chatelin F., (1988), Valeurs propres de matrices, Masson, Paris.

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F. Chatelin, (1988), Valeurs propres de matrices, Masson, Paris.

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F. Chatelin, (1988b), Valeurs propres de matrices, Masson, Paris.

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F. Chatelin, (1988b), Valeurs propres de matrices, Masson, Paris.

....cedex, France. Email : fraysse cerfacs.fr CERFACS Technical Report TR PA 94 07 ABSTRACT The departure from normality of a matrix plays an essential role in the numerical matrix computations. The bad numerical behaviour of highly nonnormal matrices has been known for a long time ( 14] 25] [5]) But this first effect of high nonnormality i.e. the increase of the spectral instability was considered by practionners as a mathematical oddity, since such matrices were not often encountered in practice. Even the most recent textbooks for engineers on eigenvalue computations, such as [19] do ....

F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....highly nonnormal matrices come from iii) the influence of nonnormality on numerical stability in exact arithmetic, iv) its influence on the reliability of Numerical Software. It has long been known that nonnormal matrices can exhibit spectral instability (Henrici (1962) van der Sluis (1975) Chatelin (1988, 1993) However, nonnormality was considered until recently as a mathematical curiosity only. But in the past few years, problems arising from high technology (Braconnier, Chatelin, and Dunyach (1995) or theoretical physics (Reddy (1991) Kerner (1989) have emerged which display a departure ....

....under the perturbations on A generated by finite precision computations. An example of diverging successive iterations on a matrix A such that ae(A) 0:41 in exact arithmetic is given in Chatelin (1993b) For Ax = x, the subspace iteration converges under a separation condition on the spectrum (Chatelin (1988, 1993) The influence of nonnormality on Tchebycheff subspace iteration is exposed in Bennani and Braconnier (1993a) 5 Conclusion We have shown that high nonnormality can occur in Physics and Technology whenever there is a strong coupling of phenomena giving rise to physical instabilities. ....

F. Chatelin, (1988), Valeurs propres de matrices, Masson, Paris.

.... the expansion of the resolvent map z R(z) A Gamma zI) Gamma1 in the neighborhood of its pole of order is, for z 6= R(z) Gamma P z Gamma Gamma Gamma1 X k=1 D k (z Gamma ) k 1 1 X k=0 (z Gamma ) k S k 1 : 1) The matrices P , D and S are defined in [7, 2, 5, 13]. P is the spectral projection and D the nilpotent (D Gamma1 6= 0 and D = 0) associated with . Using the expansion (1) what can be said about the sensitivity of the defective eigenvalue to perturbations in A What is the perturbation Delta on which results from a perturbation ....

....on the subject. We emphasize that the definition (3) when applied to the map A , is exactly the definition proposed by John Rice in [18] for a differentiable or Lipschitz continuous function. When is defective, clearly the above condition number (3) tends to infinity. It has been proposed in [7], to replace the definition (3) by that of a Holder condition number, since the map A is now Holder continuous, that is j Deltaj is proportional to jj DeltaAjj 1= See [3, 7, 2] Definition 2.1 C 1= lim ffi 0 max jj DeltaAjjffi j Deltaj jj DeltaAjj 1= is the Holder condition ....

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F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

....It has been checked (Frayss e (1992) that t 0 is a good estimate of the distance to singularity ffi. If K t is not infinite for t = t 0 , this is because a perturbation achieving the singularity is never realized in practice via random perturbations. This point is analysed in more detail in Chatelin and Frayss e (1993) At t = t 0 , the regularity of the process, as seen by the computer, changes so that when t t 0 , K t decreases according to the new (and weaker) degree of regularity. Our experience is that K t varies like t fl Gamma1 (which appears as a straight line of slope fl Gamma 1 on the plot) We give an ....

F. Chatelin, (1988b), Valeurs propres de matrices, Masson, Paris.

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F. Chatelin. Valeurs propres de matrices. Masson (1988).

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F. Chatelin, Valeurs propres de matrices, Masson, Paris, 1988.

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F. Chatelin. Valeurs Propres de Matrices. Masson, Paris, 1988.

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F. Chatelin, Valeurs Propres de Matrices, Masson, Paris, 1988.

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F. Chatelin. Valeurs Propres de Matrices. Masson, Paris, 1988.

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F. Chatelin, Valeurs Propres de Matrices, Masson, 1986.

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F. Chatelin, Valeurs Propres de Matrices, Masson, Paris, 1986.

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F. Chatelin. Valeurs Propres de Matrices. Masson, Paris, 1988.

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F. Chatelin, (1988), Valeurs propres de matrices, Masson, Paris.

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F. Chatelin. Valeurs propres de matrices. Masson, Paris, 1988.

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