Abstract
For computing the smallest eigenvalue and the corresponding eigenvector of a Hermitian matrix, by introducing a concept of perfect Krylov subspace, we propose a class of perfect Krylov subspace methods. For these methods, we prove their local, semilocal, and global convergence properties, and discuss their inexact implementations and preconditioning strategies. In addition, we use numerical experiments to demonstrate the convergence properties and exhibit the competitiveness of these methods with a few state-of-the art iteration methods such as Lanczos, rational Krylov sequence, and Jacobi-Davidson, when they are employed to solve large and sparse Hermitian eigenvalue problems.
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References
Bai, Z.-Z., Miao, C.-Q.: On local quadratic convergence of inexact simplified Jacobi-Davidson method. Linear Algebra Appl. 520, 215–241 (2017)
Bai, Z.-Z., Miao, C.-Q.: On local quadratic convergence of inexact simplified Jacobi-Davidson method for interior eigenpairs of Hermitian eigenproblems. Appl. Math. Lett. 72, 23–28 (2017)
Berns-Müller, J., Graham, I.G., Spence, A.: Inexact inverse iteration for symmetric matrices. Linear Algebra Appl. 416, 389–413 (2006)
Druskin, V., Knizhnerman, L.: Extended Krylov subspaces: approximation of the matrix square root and related functions. SIAM J. Matrix Anal. Appl. 19, 755–771 (1998)
Freitag, M.A., Spence, A.: Convergence of inexact inverse iteration with application to preconditioned iterative solves. BIT 47, 27–44 (2007)
Freitag, M.A., Spence, A.: A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems. IMA J. Numer. Anal. 28, 522–551 (2008)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)
Golub, G.H., Ye, Q.: An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems. SIAM J. Sci. Copmut. 24, 312–334 (2002)
Jagels, C., Reichel, L.: The extended Krylov subspace method and orthogonal Laurent polynomials. Linear Algebra Appl. 431, 441–458 (2009)
Jagels, C., Reichel, L.: Recursion relations for the extended Krylov subspace method. Linear Algebra Appl. 434, 1716–1732 (2011)
Jones, M.T., Patrick, M.L.: Bunch-Kaufman factorization for real symmetric indefinite banded matrices. SIAM J. Matrix Anal. Appl. 14, 553–559 (1993)
Knizhnerman, L., Simoncini, V.: A new investigation of the extended Krylov subspace method for matrix function evaluations. Numer. Linear Algebra Appl. 17, 615–638 (2010)
Morgan, R.B., Scott, D.S.: Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems. SIAM J. Sci. Comput. 14, 585–593 (1993)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (2000)
Parlett, B.N.: The Symmetric Eigenvalue Problem. SIAM, Philadelphia (1998)
Ruhe, A.: Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl. 58, 391–405 (1984)
Ruhe, A.: Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils. SIAM J. Sci. Comput. 19, 1535–1551 (1998)
Saad, Y.: Numerical Methods for Large Eigenvalue Problems, 2nd edn. SIAM, Philadelphia (2011)
Saad, Y.: On the rates of convergence of the Lanczos and the block-Lanczos methods. SIAM J. Numer. Anal. 17, 687–706 (1980)
Shank, S.D., Simoncini, V.: Krylov subspace methods for large-scale constrained Sylvester equations. SIAM J. Matrix Anal. Appl. 34, 1448–1463 (2013)
Sleijpen, G.L.G., Van der Vorst, H.A.: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401–425 (1996)
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The authors are very much indebted to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper.
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Supported by The National Natural Science Foundation (No. 11671393) and The National Natural Science Foundation for Creative Research Groups (No. 11321061), People’s Republic of China.
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Bai, ZZ., Miao, CQ. Computing eigenpairs of Hermitian matrices in perfect Krylov subspaces. Numer Algor 82, 1251–1277 (2019). https://doi.org/10.1007/s11075-018-00653-y
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DOI: https://doi.org/10.1007/s11075-018-00653-y