Braconnier, T. (1993). The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems. Tech. Report TR/PA/93/25, CERFACS, Toulouse.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and Bennani (1990) Garratt, Moore, and Spence (1991) Garratt (1991) Sadkane (1991a, 1991b) and Sorensen (1992) In conjunction with the published papers, several codes have been developed which implement variations of Arnoldi s method. These include codes by Sadkane (1991a, 1991b) and Braconnier (1993). These codes may be regarded as experimental codes since their use requires details of Arnoldi s method to be understood before appropriate values can be given to the input parameters. At present there is no code which implements an Arnoldi based method for the unsymmetric eigenvalue problem ....

....the accuracy of the desired i basis vectors. Choosing a to be the norm of the residual Ay l y favours the basis vectors which i i i converge slowly. Thus slow convergence is off set by a starting vector which is richer in the corresponding basis vector. This choice for the a s is used by Braconnier (1993). Further details and i discussion may be found in Saad (1980,1984) An alternative approach for computing several eigenvalues is to employ a locking technique. The idea behind locking, which is sometimes termed implicit deflation (see Saad 1989) is to exploit the fact that the initial ....

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Braconnier, T. (1993). The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems. Tech. Report TR/PA/93/25, CERFACS, Toulouse.

....in our study, the codes must be available either in the public domain or under licence. There are currently (to the authors knowledge) three such packages (we apologise if there are any other packages which meet our criteria but which we are not aware of) These are the ARNCHEB package of Braconnier (1993), the ARPACK package (Lehoucq, Sorensen, Vu and Yang, 1995) and the Harwell Subroutine Library code EB13 (Scott, 1995) The reports and papers which accompany each of these codes provide limited numerical results illustrating their use but results comparing their performances have not been ....

....(ira iteration) is that it avoids the need to restart the reduction from scratch at each iteration. 3 Arnoldi iteration software In this section we briefly review the software packages ARNCHEB, ARPACK, and EB13, which implement restarted Arnoldi iterations. 3. 1 ARNCHEB The ARNCHEB package of Braconnier (1993) provides subroutine ARNOL that implements an explicitly restarted Arnoldi method. The code is based on the algorithms of Saad (1980, 1984) and may be used to compute either the eigenvalues of largest or smallest real parts, or those of largest imaginary part. In ARNCHEB, the computation of the ....

T. Braconnier. The Arnoldi--Tchebycheff algorithm for solving large nonsymmetric eigenproblems. Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

....related results were addressed e.g. in [9] 58] or [29] substantially less is known in the the nonsymmetric case. Several nonsymmetric iterative algorithms for computing eigenvalues and for solving linear systems were studied experimentally by Chatelin, Godet Thobie, Frayss e and Braconnier in [10], 11] 16] and [17] They illustrated with several examples how a large departure from normality can affect the backward error of the approximate solutions. The work summarized in the following thesis was motivated by open questions related to these papers. 1.1 Organization of the thesis This ....

....orthogonalization, the matrix V n is close to V n , k V n Gamma V n k i 4 n 3=2 N : 3.12) Formulas (3.4) and (3. 5) suggest an explanation for the deterioration effects of rounding errors to the Krylov subspace methods which were observed for very ill conditioned matrices in [16] [10], for example. Suppose that for some actual finite precision run the size of kF n k in (3.4) is close to the given bound (3.5) If, at the same time, the entries in H n 1;n are much less in magnitude than kAk, then any method, for which the construction of the approximate solution is based on ....

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T. Braconnier, The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems, Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

....vectors. With either form of deflation, the eigenvalues of A j are i Gamma oe i for i j and i otherwise and both forms leave the Schur vectors unchanged. This motivates Saad to suggest that an approximate Schur basis should be incrementally built as Ritz vectors of A j converge. Braconnier [6] employs the Wielandt variant and discusses the details of deflating a converged Ritz value that has nonzero imaginary part in real arithmetic. We now compare our locking scheme to the Schur Wielandt deflation techniques. We shall assume that AU j = U j R j is a real partial Schur form of order j ....

T. Braconnier, The Arnoldi--Tchebycheff algorithm for solving large nonsymmetric eigenproblems, Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

....observations, but, unfortunately, it does not give clear, well justified conclusions. Backward stability of several related iterative methods for both solution of linear systems and computing eigenvalues was studied experimentally by Chatelin, Frayss e, Godet Thobie and Braconnier [9] 10] 6] [7]. Our work was partially motivated by the open questions related to these papers. For concepts of stability and many other relevant results we refer to a recent paper by Higham and Knight [14] Our paper is organized as follows. In Section 2, the Arnoldi recurrence for the quantities actually ....

....on the details of the arithmetic ( 30] 1] 4] We assume, for simplicity, i j 1; j = 1; 2; 6: Formulas (2.2) and (2. 3) suggest an explanation for the deterioration effects of rounding errors to the Krylov subspace methods which were observed for very ill conditioned matrices in [10] [7], for example. Suppose that for some actual finite precision run the size of kF n k in (2.2) is close to the given bound (2.3) If, at the same time, the entries in H n 1;n are much less in magnitude than kAk, then any method, for which the construction of the approximate solution is based on the ....

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T. Braconnier, The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems, Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

....10. Numerical Results. Lehoucq and Scott [17] presented a software survey of large scale eigenvalue methods and comparative results. The Arnoldi based software included the following three packages, which are available either in the public domain or under licence. These are the ARNCHEB package [4], the ARPACK [19] software package, and the Harwell Subroutine Library code EB13 [37] The ARNCHEB package provides the subroutine ARNOL, which implements an explicitly restarted Arnoldi iteration. The code is based on the algorithm given in Table 9.1 without the use of locking. It uses Chebyshev ....

T. Braconnier, The Arnoldi--Tchebycheff algorithm for solving large nonsymmetric eigenproblems, Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

.... have been a number of research developments in the numerical solution of large scale eigenvalue problems [25] 13] 20] 6] 23] 19] 15] 1] 5] The state of the art has advanced considerably and general purpose numerical software has begun to emerge for the nonsymmetric problem [8] [4], 14] 3] 22] 12] General purpose software for the nonsymmetric problem was virtually nonexistent until very recently. However, none of these new software packages are able to effectively utilize a preconditioned iterative solver to mimic the shift and invert spectral transformation to ....

T. Braconnier. The Arnoldi--Tchebycheff algorithm for solving large nonsymmetric eigenproblems. Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

....into numerical methods for computing selected eigenvalues and eigenvectors of (1. 1) This has led to the development of new software and to papers and reports describing the usefulness of the software for solving practical problems (see, for example, Duff and Scott, 1993; Bai and Stewart, 1992; Braconnier, 1993; Sorensen, 1995; Scott, 1995) However; the published numerical results are extremely limited, and, in general, authors of software have provided few results comparing the performance of their software with that of rival software. The recent books by Saad (1992) and Chatelin (1993) consider the ....

.... sparse nonsymmetric eigenproblem (era denotes explicitly restarted Arnoldi, and ira denotes implicitly restarted Arnoldi) Code Method Authors Year Availability LOPSI Subspace Stewart and Jennings 1981 TOMS SRRIT Subspace Stewart and Bai 1993 ftp EB12 Subspace Duff and Scott 1991 HSL ARNCHEB era Braconnier 1993 ftp EB13 era Scott 1993 HSL ARPACK ira Sorensen, Lehoucq, and Vu 1995 netlib In Table 1, TOMS denotes the ACM Transactions on Mathematical Software; ftp indicates that the code is available by anonymous ftp; HSL denotes the Harwell Subroutine Library (1996) and, finally, netlib indicates that ....

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T. Braconnier. The Arnoldi--Tchebycheff algorithm for solving large nonsymmetric eigenproblems. Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

.... there have been a number of research developments in the numerical solution of large scale eigenvalue problems [21] 11] 17] 6] 19] 16] 13] 1] 5] The state of the art has advanced considerably and general purpose numerical software is emerging for the nonsymmetric problem [8] [4], 12] 3] 18] 10] The development of this new general purpose software for the nonsymmetric problem is a welcomed advance. However, the methods in these packages are not able to effectively utilize a preconditioned iterative solver to implement a shift and invert spectral transformation to ....

T. Braconnier. The Arnoldi--Tchebycheff algorithm for solving large nonsymmetric eigenproblems. Technical Report TR/PA/93/25, CERFACS, Toulouse, France, 1993.

.... 1360 1725 UMIST Influence of Orthogonality on the Backward Error and the Stopping Criterion for Krylov Methods T. Braconnier Numerical Analysis Report No. 281 December 1995 Manchester Centre for Computational Mathematics Numerical Analysis Reports DEPARTMENTS OF MATHEMATICS Reports available from: Department of Mathematics University of Manchester Manchester M13 9PL England And over the World Wide Web from URLs ....

....of the Krylov subspace Vm = span[v 1 ; vm ] of size n Theta m with m n and a Hessenberg matrix Hm of size m Theta m such that AVm = VmHm hm 1;m vm 1 e T m ; where e m is the mth unit vector. More details can be found in Saad (1992) Bennani (1991) Sorensen (1992) Chatelin (1993) Braconnier (1993) and Scott (1993) The previous equality can be written in matrix form as AVm = Vm 1H; where H is the (m 1) Theta m matrix 0 Hm 0 0 0 : hm 1;m 1 A : Following Paige (1994) s suggestion, we denote by the Arnoldi process the Hessenberg factorization of the matrix A using the Arnoldi ....

T. Braconnier, (1993), The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems, Tech. Rep. TR/PA/93/25, CERFACS.

....large Henrici number. 1 (A) is invariant if A becomes ffA. 2.2 Backward errors In the previous section, we have seen that if a matrix A is highly nonnormal then kAk is large. Therefore, for iterative eigensolvers, we should use stopping criteria which take this into account (see Bennani and Braconnier (1993)) In this section, we define the backward errors associated with the two problems we want to solve. 2.2.1 Backward error associated with Ax = x We recall the following Definition 2.3 The normwise backward error associated with the approximate eigenpair ( x) is j = minfffl 0; ....

....error associated with the approximate eigenpair ( x) is j = kaek kAkk xk ; where ae = A x Gamma x. Remarks : 1. Deif (1989) proved this theorem with the 2 norm, but it holds for any matrix norm induced by a vector norm. The proof of theorem 2. 1 can be found in Bennani and Braconnier (1993). 2. Geurts (1982) proved that the componentwise backward error associated with the eigenvalue problem is j = max 1in jaej (jAjj xj) i ; where jAj = fja ij jg i;j=1: n . 2.2.2 Backward error associated with an invariant subspace computation We can also define the backward error associated ....

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T. Braconnier, (1993), The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems, Tech. Rep. TR/PA/93/25, CERFACS.

....a matrix A can dramatically affect the convergence of iterative methods. This fact has already been pointed out for iterative solvers for linear systems : in Chatelin and Frayss e (1993) for successive iterations, in Trefethen (1990) for Richardson method and for some eigensolvers in Bennani and Braconnier (1993). We found the use of the normalized term kAk F in the backward error formulation necessary to take into account for matrices with large norms, which is the case when the departure from normality is large : kA A Gamma AA k 2kAk 2 (see Chatelin and Frayss e (1993) and Chatelin (1992) ....

....ffl QR, ffl subspace iterations, ffl Arnoldi Tchebycheff. We only present results for the eigenvalue problem Ax = x. Similar results for iterative methods that compute an approximate invariant subspace associated with a desired number of eigenvalues are obtained and can be found in Bennani and Braconnier (1993) and Braconnier (1994) 5.1 Symmetric eigensolvers 5.1.1 Jacobi algorithm 0 100 200 300 400 500 600 700 16 14 12 10 8 6 4 Iteration Number Figure 2: Jacobi : max i=1;150 j i versus the iteration number. The Jacobi method allows the computation of all the eigenvalues of a matrix A. It ....

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T. Braconnier, (1993), The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems, Tech. Rep. TR/PA/93/25, CERFACS.

....can be seen as particular case of the standard problems. To solve the standard problem Ax = x when the matrix A is large, real and nonsymmetric, we have implemented the code arncheb, the iterative incomplete Arnoldi method, associated with the Tchebycheff acceleration (for more details, see [5]) Basically, this method requires 1. to build a Krylov subspace using a starting vector, 2. to project the matrix A in this subspace to obtain a upper Hessenberg matrix of smaller order, 3. to build an ellipse containing the unwanted computed spectrum, 4. to perform a Tchebycheff acceleration ....

....and stopping criterion (Step 4 and 6) For the Arnoldi Tchebycheff algorithm, the residual is crucial since it drives the stopping criterion. In the standard real case and to conform with the theory of backward analysis, the stopping criterion has been chosen as the backward error (see [5]) j d i = kAx i Gamma i x i k 2 kAk F kx i k 2 : To completely agree with the theory, one should use kAk 2 , but the upper bound kAk F is cheaper to compute. Consequently, j d i is a lower bound of the actual backward error associated with ( i ; x i ) We found the use of the term kAk F ....

[Article contains additional citation context not shown here]

T. Braconnier. The Arnoldi-Tchebycheff algorithm for solving large nonsymmetric eigenproblems. Tech. Rep. TR/PA/93/25, CERFACS, 1993.

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