G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595--633.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....involve the solution of the eigenvalue problem, A x i = i x i for the extreme (largest or smallest) eigenvalues, i , and eigenvectors, x i , of a large, sparse, symmetric matrix A. One such method that has attracted attention in recent years is the Jacobi Davidson (JD) method [26, 25]. This method constructs an orthonormal basis of vectors V that span a subspace K from which the approximate Ritz values, i , and Ritz vectors x i are computed at each iteration. These approximations and the residual r i = Ax i x i are then used to solve the correction equation: I x i x i ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36(3):595--633, 1996.

....section we show how we applied the cluster algorithm to a block Jacobi Davidson method for solving eigenvalue problems. 3. A multigrain, block Jacobi Davidson method for use in Grid like environments Jacobi Davidson (JD) is a popular method for computing eigenvalues of large, sparse matrices [19, 18]. A block version is also possible that follows the general iterative model of figure 1. At each iteration the method computes the current approximate eigenvalues i , the approximate eigenvectors x i , and the associated residuals r i = Ax i x i which are then used to solve the correction ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36(3):595--633, 1996.

....and projecting the nonlinear eigenvalue problem onto this subspace. Then the projected problem is solved with methods for dense nonlinear eigenvalue problems. A similar Jacobi Davidson type approach was already introduced by Sleijpen and van der Vorst for quadratic eigenvalue problems (cf. [30]) But they did not work out strategies to find several eigenvalues with the Jacobi Davidson approach without the problem of convergence against old eigenvalues that were already extracted in previous steps of the algorithm. This shall be done here. The idea of Jacobi Davidson for the nonlinear ....

....(5.29) Here, #, u) is the current approximation for an eigenpair of T ( p denotes the vector p = T # (#)u and r denotes the residual r = T (#)u. If T (#) #I A we arrive at the correction equation 3.18 for the linear case. This extension was already proposed by Sleijpen and van der Vorst in [30]. We want to perform a similar analysis for equation (5.29) as in chapter 3.4. The correction equation (5.29) can be written as T (#)t #p = where # must be chosen such that t u. The last equation leads to #T (#) 1 p since r = T (#)u. By substituting p with T # (#) 1 u we arrive ....

G.L.G. Sleijpen, J.G.L. Booten, D.R. Fokkema, and H.A. van der Vorst. JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36:595--633, 1996.

.... j. This amounts to the linear system Q( e ) Q 2ex v = Gamma 42 where r = Q( e )ex is the residual and Q ( e ) 2 e M C. Since e and e x are computed by a projection method we have that e x Q( e )ex = 0. Further manipulations lead to the correction equation [130] 132] [131] ( e )exex Q( e ) e xex v = r: 6.6) The new basis vector v k 1 is obtained by orthonormalizing v against the previous columns of V k . The Jacobi Davidson method has been successfully used to compute the most unstable eigenvalues related to the incompressible attachment line ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. Van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595--633.

....of P( 0 onto the subspace spanned by the selected eigenvectors. The eigenvectors can be chosen to correspond to parts of the spectrum of interest and can be computed using the Arnoldi process on the companion form pencil (F, G) or directly on with the Jacobi Davidson method or its variants [26] [35]. In the latter case, the matrLx V is built during the Davidson process. 3.3.2. Direct approach. This approach consists of approximating IIP) 11 at each grid point z. Techniques analogous to those used for single matrices can be applied, such as the Lanczos method applied to P(z) P(z) or its ....

G. L. G. SLEIJPEN, A. G. L. BOOTEN, D. R. FOKKEM^, ^ND H. A. V^N DER VORST, Jacobi- Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595-633.

....involve the solution of the eigenvalue problem, A x i = i x i for the extreme (largest or smallest) eigenvalues, i , and eigenvectors, x i , of a large, sparse, symmetric matrix A. One such method that has attracted attention in recent years is the Jacobi Davidson (JD) method [32, 31]. This method constructs an orthonormal basis of vectors V that span a subspace K from which the approximate Ritz values, i , and Ritz vectors x i are computed at each iteration. These approximations and the residual r i = Ax i x i are then used to solve the correction equation: I x i x i ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36(3):595-633, 1996.

....section we show how we applied the cluster algorithm to a block Jacobi Davidson method for solving eigenvalue problems. 4 3 A multigrain, block Jacobi Davidson method for use in Grid like environments Jacobi Davidson (JD) is a popular method for computing eigenvalues of large, sparse matrices [19, 18]. A block version is also possible that follows the general iterative model of figure 1. At each iteration the method computes the current approximate eigenvalues i , the approximate eigenvectors x i , and the associated residuals r i = Ax i x i which are then used to solve the correction ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36(3):595--633, 1996.

....correction equation as was shown in Figure 2. Before the correction phase, each solve group must obtain its respective coarse grain x i and r i vectors. The preconditioned inner solver is then applied by each solve group to solve for a different correction vector using the technique described in [27] for applying a preconditioner of A to the projected matrix in 1. For the difficult problems in our experiments, 10 20 iterations of the inner solver are usually needed to obtain timely convergence of the target eigenvectors. After the corrections have been computed they must be transitioned back ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36(3):595--633, 1996.

....is based on the works of Cullum and Willoughby and the fast Fourier transform. In this paper we present a Jacobi Davidson type algorithm for computing a few eigenpairs of a complex symmetric matrix that exploit the structure of the matrices. For the original Jacobi Davidson algorithm see [22] [21], 8] In contrast to the complex symmetric methods mentioned before, our Jacobi Davidson algorithm can be transcribed quite easily into a solver for the generalized eigenvalue problem (1.2) The paper is organized as follows. In Section 2 we investigate how the Rayleigh quotient is best defined ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595--633.

....is based on the works of Cullum and Willoughby and the fast Fourier transform. In this paper we present a Jacobi Davidson type algorithm for computing a few eigenpairs of a complex symmetric matrix that exploit the structure of the matrices. For the original Jacobi Davidson algorithm see [22] [21], 8] In contrast to the complex symmetric methods mentioned before, our Jacobi Davidson algorithm can be transcribed quite easily into a solver for the generalized eigenvalue problem (1.2) The paper is organized as follows. In Section 2 we investigate how the Rayleigh quotient is best de ned ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595-633.

....Newton iterations [15] 19] For a good review of such methods, we refer to Ruhe [21] More recently, Guillaume [10] developed a new method based on the derivative of the function x( Q( b where b is a given vector. For large sparse problems, Jacobi Davidson techniques have been investigated [22]. The usual way of dealing with the QEP (3.1) is to transform it into a generalized eigenvalue problem of twice the order. There are several possible ways to carry out such a transformation. The most commonly used transformation is to companion form, given by GammaB GammaC ; 3.2) In ....

Gerard L. G. Sleijpen, Albert G. L. Booten, Diederik R. Fokkema, and Henk A. Van der Vorst. Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36(3):595--633, 1996.

....may be used as well. The algorithm is continued until the norm of the residual satisfies some preset tolerance (krk 2 tol) The Jacobi Davidson algorithm has recently gained wide popularity and the method and generalizations have been used to solve several hard eigenvalue problems, see e.g. [1, 12, 16, 15], In this paper we focus on the two problems mentioned in the abstract. The purpose of this paper is not to compare the Jacobi Davidson algorithm with other Department of Computer Science, University of Illinois at Urbana Champaign (sturler uiuc.edu) 1 2 Eric de Sturler eigensolvers nor to ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. Van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (

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G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (

....eigenproblem Ax = Bx. It turns out that much of our discussion for the standard eigenproblem carries over to the generalized eigenproblem. 2. The generalized eigenproblem. The Jacobi Davidson approach can also be followed for computing a few selected eigenpairs of generalized eigenproblems [16, 3] of the form Ax Gamma Bx = 0: 2.1) Here we suggest to follow a Petrov Galerkin method for the construction of approximate solutions. An approximate solution in a search subspace spanned by v 1 ; v k is tested against a test subspace spanned by vectors w 1 ; w k : AV k s Gamma ....

....considered as approximation for an eigenpair of A. The value j will be called a Petrov value and u j is a Petrov vector. Of course, the test subspace could have been chosen to be equal to the search subspace, but linear combinations of AV k and BV k , seem to be more effective, see x4 and [16, 3], We define the residual r j as r j j Gamma(A Gamma j B)u j . The search subspace is expanded by the solution t j of the Jacobi Davidson correction equation (I Gamma q The selected Petrov vector u j and the test vector q j are assumed to be normalized. For ....

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G.L.G. Sleijpen, J.G.L. Booten, D.R. Fokkema, and H.A. Van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36:3 (1996), pp. 595--633.

....preconditioner can be identified. Note that the operator A Gamma I will be indefinite in general, so that one has to be careful with incomplete decomposition techniques. The operator restricted to the subspace orthogonal to the Ritz vector corresponding to , however, is not indefinite, and in [67] it is shown how available preconditioners for A Gamma I can be restricted to that subspace. 6 A novel extension for the Jacobi Davidson method In some circumstances the Jacobi Davidson method has apparent disadvantages with respect to Arnoldi s method. For instance, in many cases we see rapid ....

G.L.G. Sleijpen, J.G.L. Booten, D.R. Fokkema, and H.A. van der Vorst. Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems: Part I. Technical Report 923, University Utrecht, Department of Mathematics, 1995.

....(R) and f 2 L 2( Omega Gamma6 and u satisfies suitable boundary conditions. An example of (4) is, for instance ; 5) where Omega is some domain in R and u = 0 on (see also Section 8) Guided by the known approaches for the linear system (cf. 25, 29, 7] and the eigenproblem (cf. [28, 27]) we will define accelerated Inexact Newton schemes for more general nonlinear systems. This leads to a combination of Krylov subspace methods for Inexact Newton (cf. 16, 4] and also [8] with acceleration techniques (as in [2] and offers us an overwhelming choice of techniques for further ....

....efficiently and the projections have been included correctly. The new methods have been called Jacobi Davidson methods (Jacobi took proper care of the projections, but did not build a search subspace as Davidson did (see [28] for details and further references) The analysis and results in [3, 27] show that these Jacobi Davidson methods can also be effective for solving generalized eigenproblems, even without any matrix inversion. The Jacobi Davidson methods allow for a variety of choices that may improve efficiency of the steps and speed of convergence and are good examples of AIN methods ....

[Article contains additional citation context not shown here]

G. L. G. Sleijpen, J. G. L. Booten, D. R. Fokkema, and H. A. Van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems: Part I, Preprint 923, Dept. Math., University Utrecht, Utrecht, The Netherlands, 1995.

.... t represents some second order correction (cf. 11] 19] Ignoring this contribution results in = r m (A Gamma m I n )t Gamma u m : 4) Consider some subspace W such that 2 W Vm . With W, a matrix of which the columnvectors form an orthonormal basis for W , we decompose (4) cf. [14]) in = WW (A Gamma m I n )t Gamma u m ; in W , and in )r = I n Gamma WW ) A Gamma m I n )t r m ; 5) orthogonal to W . Thenew direction t will be used to expand the subspace Vm to Vm 1 . An approximation ( m 1 ; is computed with respect to Vm 1 . Because W ....

G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36:595--633, 1996.

....reason, and also since the standard problem allows for a less complicated description, we have chosen to consider both situations in detail. Our algorithms are based on the Jacobi Davidson method described in [20] and adapted for generalized eigenproblems (and other polynomial eigenproblems) in [18]. We have chosen the Jacobi Davidson approach for the computation of a partial Schur form for the standard eigenproblem, and for a partial generalized Schur form for the generalized eigenproblem. The partial Schur forms have been chosen mainly for numerical stability, since they involve orthogonal ....

....method, and by a proper selection of the approximate eigenpair for the correction equation, the process can be guided to find eigenpairs close to a given target value. More details will be given in x2.1 and x3.1, and for a complete description of the Jacobi Davidson method we refer to [20] [18]. A problem in the Jacobi Davidson method is that convergence towards a specific eigenvalue is favored, and for efficient computation of several eigenvalues, one has to apply the usual restart with a different target. Because of memory limitations one may also be forced to restart, even before an ....

[Article contains additional citation context not shown here]

G. L. G. Sleijpen, J. G. L. Booten, D. R. Fokkema, and H. A. Van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems: Part I, Preprint 923, Departement of Mathematics, Utrecht University, Utrecht, Revised version, November 1995.

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G. L. G. Sleijpen, G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, Jacobi Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595--633.

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G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595--633.

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G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595-633.

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G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. Van der Vorst, Jacobi-- Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36 (1996), pp. 595--633.

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G. L. G. Sleijpen, A. G. L. Booten, D.R. Fokkema and H. A. van der Vorst, JacobiDavidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36 (1996), no. 3, 595--633. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, E-mail address: dattab@math.niu.edu E-mail address: sarkiss@math.niu.edu

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G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, H. A. van der Vorst, Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36 (3) (1996) 595-633.

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G. Sleijpen, J. Booten, D. Fokkema and H. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT, 36(1996):593-633. 35

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