T. Braconnier, F. Chatelin, and V. Frayss'e. The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra. Technical Report TR-PA-94/07, CERFACS, Toulouse, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of the Kronecker canonical form, by for instance Kagstrom and Van Dooren, this still needs much further research. Also the computation of invariant subspaces of highly nonnormal matrices is still in its infancy, notwithstanding useful contributions by, for instance, Chaitin Chatelin et al. [11, 7] and Lee [36] It is also necessary that we get effcient tools for checking the condition of (partial) eigensystems, angles between invariant subspaces, etcetera. There is a need for efficient algorithms for the computation of a (partial) SVD for very large sparse matrices. Several special types ....

T. Braconnier, F. Chatelin, and V. Frayss'e. The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra. Technical Report TR-PA-94/07, CERFACS, Toulouse, 1994.

....and the outliers are real or complex. The number of QMR iterations required appears to be more sensitive to changes in outliers than to the changes in nonnormality which we use. For additional discussions of the effects of nonnormality on iterative methods for equation(1) see for example [3, 5, 20]. 6.3. Outliers and Their Effect on the Convergence of QMR We want to study the effect of different eigenvalue distributions upon the convergence rate of QMR. We consider both normal and nonnormal test matrices. For each set of tests in this section we fix the eigenvector matrix X as defined in ....

T. Braconnier, F. Chatelin and V. Fraysse, The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra, CERFACS Technical Report TR-PA-94-07 (1994). CERFACS, Toulouse, France.

....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 thesis ....

.... For some related results see [29] and also [9] and [58] Similar dependence of accuracy of iterative algorithms for computing eigenvalues and solving linear systems on the properties of the problem was described experimentally by Chatelin, Godet Thobie, Frayss e and Braconnier in [17] 16] [11], 10] Remark 4.5. In the previous text we compared the norms of the true and Arnoldi residuals. However, in practice the norm of the Arnoldi residual is computed as the absolute value of the bottom element of the right hand side vector e 1 transformed by application of the Givens rotations ....

T. Braconnier, F. Chatelin, V. Frayss'e, The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra, submitted to Lin. Alg. Appl.

....computation of the Kronecker canonical form, by for instance Kagstrom and Van Dooren, this still needs further research. Also the computation of invariant subspaces of highly nonnormal matrices is still in its infancy, notwithstanding useful contributions by, for instance, Chaitin Chatelin et al. [17, 12] and Lee [77] For recent references, see [4] Van Dooren described, in papers published in 1981, how the Kronecker form can be used in system control problems (input output systems) 139, 138] Related to eigendecompositions is the singular value decomposition. Let A be a real m by n matrix, ....

T. Braconnier, F. Chatelin, and V. Frayss'e. The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra. Technical Report TR-PA-94/07, CERFACS, Toulouse, 1994.

....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 ....

....case . For some related results see [13] and also [5] and [31] Similar dependence of accuracy of iterative algorithms for computing eigenvalues and solving linear systems on the properties of the problem was described experimentally by Chatelin, GodetThobie, Frayss e and Braconnier in [9] 10] [6], 7] NUMERICAL STABILITY OF GMRES 17 REMARK: In the previous text we compared the norms of the true and Arnoldi residuals. In practice, however, the norm of the Arnoldi residual is computed as an absolute value n 1 of the bottom element of the right hand side vector e 1 transformed by ....

T. Braconnier, F. Chatelin, V. Frayss'e, The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra, submitted to Lin. Alg. Appl. 24 J. DRKO SOV ' A, A. GREENBAUM, M. ROZLO ZN ' IK AND Z. STRAKO S

....a (0) 3.2. 4 The computation of k(A Gamma zI) Gamma1 k 2 We want to compute k(A Gamma zI) Gamma1 k 2 = min (H a (z) Since the Lanczos algorithm converges faster to the largest eigenvalues, we compute 1 provided that fi j 1 jy (j) i j kHa (0)k2 machineprecision : See Bennani and Braconnier (1993b) max (H a (z) Gamma1 ) instead of min (H a (z) Step a) of the algorithm previously described, becomes r j H a (z) Gamma1 q j : Actually, we do not explicitly invert the matrix H a (z) but we solve the system H a (z)r j = q j exploiting its block structure. In other words, we ....

T. Braconnier, F. Chatelin, and V. Frayss' e, (1993), The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra.

....of the Kronecker canonical form, by for instance Kagstrom and Van Dooren, this stiil needs much further research. Also the computation of invariant subspaces of highly nonnormal matrices is still in its infancy, notwithstanding useful contributions by, for instance, Chaitin Chatelin et al. [23, 7] and Lee [36] It is also necessary that we get effcient tools for checking the condition of (partial) eigensystems, angles between invariant subspaces, etcetera. There is a need for efficient algorithms for the computation of a (partial) SVD for very large sparse matrices. Several special types ....

T. Braconnier, F. Chatelin, and V. Frayss'e. The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra. Technical Report TR-PA-94/07, CERFACS, Toulouse, 1994.

....symmetric Lanczos recursion, We would like to be able to answer the question: What spectral properties of A control the convergence of each of these methods We have not answered this question but describe several results which might be useful in such studies. See for example, the related work [2, 3, 4, 5, 6, 7, 8, 27, 28, 29]. In section 2 we outline briefly the Arnoldi and the nonsymmetric Lanczos eigenvalue procedures we are considering. In section 3 we exhibit a certain relationship between these two methods. We prove that given any matrix A and any application of a nonsymmetric Lanczos procedure to A, there exists ....

T. Braconnier, F. Chatelin and V. Fraysse, The influence of large nonnormality on the quality of convergence of iterative methods in linear algebra, CERFACS Technical Report TR-PA-94-07 (1994). CERFACS, Toulouse, France.

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