Backward perturbation analysis and relative algorithms for nonsymmetric linear systems with multiple right-hand sides. (English) Zbl 1513.65123
Keywords:
backward perturbation; Kronecker product; minimum joint backward perturbation; global Arnoldi; matrix Krylov subspace; block methods; iterative methods; nonsymmetric linear systems; multiple right-hand sidesReferences:
| [1] | K. Jbilou, A. Messaoudi and H. Sadok, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math. 31 (1999), 49-63. · Zbl 0935.65024 |
| [2] | E. M. Kasenally and V. Simoncini, Analysis of a minimum perturbation algorithm for nonsymmetric linear systems, SIAM J. Numer. Anal. 34(1) (1997), 48-66. · Zbl 0873.65029 |
| [3] | W. Arnoldi, The principle of minimised iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9 (1951), 17-29. · Zbl 0042.12801 |
| [4] | B. Vital, Etude de quelques méthodes de résolution de problémes linéaires de grande taille sur multiprocesseur, Ph. D. Thesis, Univérsité de Rennes, Rennes, France, 1990. |
| [5] | R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van Der Vorst, Template for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA, 1993. · Zbl 0814.65030 |
| [6] | A. Ben-Israel and T. N. E. Greville, Generalised Inverses: Theory and Applications, Wiley-Interscience, New York, 1974. · Zbl 0305.15001 |
| [7] | J. W. Brewer, Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits and Systems, IEEE Trans. Circuits and Systems 25 (1978), 772-781. · Zbl 0397.93009 |
| [8] | P. E. Gill and W. Murray, The computation of Lagrange multiplier estimates for constrained minimisation, Math. Program. 17 (1979), 32-60. · Zbl 0423.90073 |
| [9] | G. H. Golub and C. F. Van Loan, An analysis of the total least squares problem, SIAM J. Numer. Anal. 17 (1980), 883-893. · Zbl 0468.65011 |
| [10] | G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., The John Hopkins University Press, Baltimore, MD, 1990. |
| [11] | E. M. Kasenally, GMBACK: a generalised minimum backward error algorithm for nonsymmetric linear systems, SIAM J. Sci. Comput. 16 (1995), 698-719. · Zbl 0823.65029 |
| [12] | Backward Perturbation Analysis and Relative Algorithms … 987 |
| [13] | N. M. Nachtigal, L. Reichel and L. N. Trefethen, A hybrid GMRES algorithm for nonsymmetric matrix iterations, SIAM J. Matrix Anal. Appl. 13 (1992), 796-825. · Zbl 0757.65035 |
| [14] | A. Nikishin and A. Yeremin, Variable block CG algorithms for solving large sparse symmetric positive definite linear systems on parallel computers, I: general iterative scheme, SIAM J. Matrix Anal. Appl. 16 (2006), 1135-1153. · Zbl 0837.65029 |
| [15] | L. Reichel and L. N. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Linear Algebra Appl. 162/164 (1992), 153-185. · Zbl 0748.15010 |
| [16] | Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp. 37 (1981), 105-126. · Zbl 0474.65019 |
| [17] | Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, U.K., 1992. · Zbl 0991.65039 |
| [18] | Y. Saad and M. H. Schultz, GMRES: a generalised minimum residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986), 856-869. · Zbl 0599.65018 |
| [19] | V. Simonconi and E. Gallopoulos, An iterative method for nonsymmetric systems with multiple right-hand sides, SIAM J. Sci. Comput. 15 (1995), 917-933. · Zbl 0831.65037 |
| [20] | X. D. Yang and W. H. Huang, Backward error analysis of the matrix equations for Sylvester and Lyapunov, J. Sys. Sci. Math. Sci. 28(5) (2008), 524-534. · Zbl 1174.65392 |
| [21] | G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990. · Zbl 0706.65013 |
| [22] | L. N. Trefethen, Approximation Theory and Numerical Linear Algebra, Chapman and Hall, London, 1990. · Zbl 0747.41005 |
| [23] | S. Van Huffel and J. Vandwalle, Algebraic connections between the least squares and total least squares problems, Numer. Math. 55 (1989), 431-449. · Zbl 0663.65038 |
| [24] | S. Van Huffel and J. Vandwalle, The total least squares problem, computational aspects and analysis, Frontiers in Applied Mathematics, Volume 9, SIAM, Philadelphia, PA, 1991. · Zbl 0789.62054 |
| [25] | J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, London, 1965. · Zbl 0258.65037 |
| [26] | P. N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Statist. Comput. 12 (1991), 58-78. · Zbl 0719.65022 |
| [27] | Zhanshan Yang and Xilan Liu 988 |
| [28] | T. Chan and W. Wang, Analysis of projection methods for solving linear systems with multiple right-hand sides, SIAM J. Sci. Comput. 18 (1997), 1698-1721. · Zbl 0888.65033 |
| [29] | J. Cullum and W. E. Donath, A block Lanczos algorithm for computing the q algebraically largest eigenvalues and a corresponding eigenspace of large, sparse, real symmetric matrices, Proc. of IEEE Conf. Decision and Control, Phoenix, 1974. |
| [30] | G. H. Golub and R. R. Underwood, The block Lanczos method for computing eigenvalues, J. R. Rice, ed., Mathematical Software, Vol. 3, Academic Press, New York, 1977, pp. 364-377. · Zbl 0407.68040 |
| [31] | M. Malhotra, R. Freund and P. M. Pinsky, Iterative solution of multiple radiation and scattering problems in structural acoustics using a block quasi-minimal residual algorithm, Comput. Methods Appl. Mech. Engrg. 146 (1997), 173-196. · Zbl 0901.76059 |
| [32] | D. P. O’Leary, The block conjugate gradient algorithm and related methods, Linear Algebra Appl. 29 (1980), 293-322. · Zbl 0426.65011 |
| [33] | B. Parlett, A new look at the Lanczos algorithm for solving symmetric systems of linear equations, Linear Algebra Appl. 29 (1980), 323-346. · Zbl 0431.65016 |
| [34] | Y. Saad, On the Lanczos method for solving symmetric linear systems with several right-hand sides, Math. Comp. 48 (1987), 651-662. · Zbl 0615.65038 |
| [35] | Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Press, New York, 1995. · Zbl 1002.65042 |
| [36] | M. Sadkane, Block Arnoldi and Davidson methods for unsymmetric large eigenvalue problems, Numer. Math. 64 (1993), 687-706. · Zbl 0791.65021 |
| [37] | C. Smith, A. Peterson and R. Mittra, A conjugate gradient algorithm for treatment of multiple incident electromagnetic fields, IEEE Trans. Antennas Propagation 37 (1989), 1490-1493. |
| [38] | V. Simoncini and E. Gallopoulos, Convergence properties of block GMRES and matrix polynomial, Linear Algebra Appl. 247 (1994), 97-119. · Zbl 0861.65023 |
| [39] | B. Y. Wang and F. Z. Zang, Schur complements and matrix inequalities of Hadamard products, Linear Multilinear Algebra 43 (1997), 315-326. · Zbl 0893.15008 |
| [40] | P. Lancaster, An explicit solutions of linear matrix equations, SIAM Rev. 12 (1970), 544-566. · Zbl 0209.06502 |
| [41] | Backward Perturbation Analysis and Relative Algorithms … 989 |
| [42] | B. Y. Wan and F. Z. Zang, Trace and eigenvalue inequalities for ordinary and Hadamard product of positive semidefinite Hermitian matrices, SIAM J. Matrix Anal. Appl. 16(4) (1995), 1173-1183. · Zbl 0855.15009 |
| [43] | R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, England, 1991, pp. 239-293. · Zbl 0729.15001 |
| [44] | J. G. Sun and S. F. Xu, Perturbation analysis of the maximal solution of the matrix equation, Π, Linear Algebra Appl. 362 (2003), 211-228. · Zbl 1020.15012 |
| [45] | J. G. Sun, Perturbations Analysis of Matrix, 2nd Press, Peking, 2001. |
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