M. Bennani and T. Braconnier, (1994), Stopping criteria for eigensolvers, Tech. Rep. TR/PA/94/22, CERFACS.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in its efficiency and robustness. 4. 5 The stopping criteria Each of the codes uses different stopping criteria, which adds to the difficulties associated with trying to compare their performance (see Section 5) Helpful discussions of stopping criteria for iterative eigensolvers are given by Bennani and Braconnier (1994) and Scott (1995) Throughout this section, ffl denotes a user defined tolerance. EB13 follows Stewart (1978) and bases its stopping criterion on demanding that AX r X r T r : The difficulty of choosing appropriate stopping criteria was recognised during the development of EB13 (Scott, 1995) and ....

....on the Ritz estimate. Moreover, since only the last component of s is needed, ARPACK does not compute the full eigenvectors of Hm at each iteration. The computation is terminated on the first iteration that r Ritz values all satisfy kf m k je T m sj j jffl: Recent work by Chatelin (1993) and Bennani and Braconnier (1994) suggests that when A is highly non normal, there can be a significant difference between the Ritz estimate and the eigenvector residual. Because of this potential difference, ARNCHEB computes both the scaled Ritz estimate and the direct backward error given by kf mk je T m sj kAk F kyk 2 ....

M. Bennani and T. Braconnier. Stopping criteria for eigensolvers. Technical report, CERFACS, November 1994.

....of the modified Gram Schmidt algorithm. 4. 5 The stopping criteria The codes all use different stopping criteria, which adds to the difficulties associated with trying to compare their performance (see Section 5) Useful discussions of stopping criteria for iterative eigensolvers are given by Bennani and Braconnier (1994) and Scott (1995) Throughout this section, ffl denotes a user defined tolerance. For subspace iteration, the dominant eigenvalues converge most rapidly so the (j 1) st eigenvalue needs to be tested only once the jth one has converged. In the code LOPSI, a column of Xm is accepted as an ....

M. Bennani and T. Braconnier. Stopping criteria for eigensolvers. Technical report, CERFACS, November 1994.

....kF , is large, the matrix A is considered highly non normal. Assuming that A is diagonalizable, a large Henrici number implies that the basis of eigenvectors is ill conditioned [8] Bennani and Braconnier compare the use of the Ritz estimate and direct residual kAx Gamma x k in Arnoldi algorithms [4]. They suggest normalizing the Ritz estimate by the norm of A resulting in a stopping criteria based on the backward error. The backward error is defined as the smallest, in norm, perturbation DeltaA such that the Ritz pair is an eigenpair for A DeltaA. Scott [33] presents a lucid account of ....

M. Bennani and T. Braconnier, Stopping criteria for eigensolvers, technical report, November 1994. Submitted to Jour. Num. Lin. Alg. Appl.

....kf m k je T m sj j jffl: We remark that since k(AXm ) j k 2 jjAjj 2 and j j jjAjj 2 , ARPACK, EB12, and SRRIT also base their stopping criterion on the backward error. Moreover, the user should consider the size of j j when selecting ffl for these codes. Recent work by Chatelin (1993) and Bennani and Braconnier (1994) suggests that when A is highly non normal, there can be a significant difference between the Ritz estimate and the eigenvector residual. Because of this potential difference, ARNCHEB computes both the scaled Ritz estimate and the direct backward error given by kf m k je T m sj kAk F kyk 2 ....

M. Bennani and T. Braconnier. Stopping criteria for eigensolvers. Technical report, CERFACS, November 1994.

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M. Bennani and T. Braconnier, (1994), Stopping criteria for eigensolvers, Tech. Rep. TR/PA/94/22, CERFACS.

....the order of machine precision ffl M ; it is said to be backward unstable otherwise. We recall that for nonnormal matrices, the quality of the orthogonality of the computed basis is crucial to maintain the backward stability of eigensolvers such as subspace iteration or Arnoldi type methods (see Bennani and Braconnier (1994), Braconnier (1995) Chaitin Chatelin and Frayss e (1996) In the experiments reported in section 6, we have used a robust implementation of the subspace iteration based on the iterative modified Gram Schmidt algorithm. Alternative strategies could be the Householder method and the Givens ....

M. Bennani and T. Braconnier, (1994), Stopping criteria for eigensolvers, Tech. Rep. TR/PA/94/22, CERFACS.

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M. Bennani and T. Braconnier, (1994b), Stopping criteria for eigensolvers, Tech. Rep. TR/PA/94/22, CERFACS.

....having a 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; ....

....normwise backward 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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M. Bennani and T. Braconnier, (1993), Stopping criteria for eigensolvers.

....AA k of 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) ....

....matrices : 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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M. Bennani and T. Braconnier, (1993), Stopping criteria for eigensolvers.

....2 kBk 2 kxk 2 ; where r = Bx Gamma x. For any Ritz vector y i and its corresponding Ritz value i we can compute j without forming r, because, from (3.1) we find that krk 2 = jt m 1;m jjs mi j, where Tm = S diag( i )S is a spectral decomposition. As suggested by Bennani and Braconnier [2], our stopping criterion for Algorithm 1 is that the Lanczos backward error j L is less than or equal to the unit roundoff, where j L = jt m 1;m jjs mi j kBk 2 Table 3.1: Chebyshev interval and z 0 . Case d; a z 0 1 d = 2 m ) 2, a = max (d Gamma 2 ; m Gamma d) z 0 = Vm s 1 2 d = ....

Maria Bennani and Thierry Braconnier. Stopping criteria for eigensolvers. Technical Report TR/PA/94/22, CERFACS, Toulouse, France, 1994.

....k 2 , we obtain ffl in the left hand side j d i : the direct backward error, ffl in the right hand side jA i : the Arnoldi backward error. Contrarily to what is usually done [17] we recommend to use the Arnoldi backward error jA i as the stopping criterion rather than the Arnoldi residual (see [4], 19] which may provide misleading information as n increases with a fixed m, when kAk F is large. To solve the generalized problem, we must check in the previous presentation, the matrice A by the matrice C = B Gamma1 A. 2.4 Determination of the ellipse (Step 7) We have to determine the ....

M. Bennani and T. Braconnier. Stopping criteria for eigensolvers, November 1993. Submitted to Jour. Num. Lin. Alg. Appl.

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Bennani M. and Braconnier T., (1994b), Stopping criteria for eigensolvers, Tech. Rep.

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M. Bennani and T. Braconnier, (1993b), Stopping criteria for eigensolvers. Submitted to J. Num. Lin. Alg. Appl.

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M. Bennani and T. Braconnier, (1993b), Stopping criteria for eigensolvers.

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