Abstract
In this paper, we introduce a generalized Krylov subspace \({\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}\) based on a square matrix sequence {A j } and a vector sequence {u j }. Next we present a generalized Arnoldi procedure for generating an orthonormal basis of \({\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}\). By applying the projection and the refined technique, we derive a restarted generalized Arnoldi method and a restarted refined generalized Arnoldi method for solving a large-scale polynomial eigenvalue problem (PEP). These two methods are applied to solve the PEP directly. Hence they preserve essential structures and properties of the PEP. Furthermore, restarting reduces the storage requirements. Some theoretical results are presented. Numerical tests report the effectiveness of these methods.
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References
Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, 17–29 (1951)
Bai, Z., Sleijpen, G., Van der Vorst, H.: Quadratic eigenvalue problems (Section 9.2). In: Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.) Templates for the Solution of Algebraic Eigenvalue Problems: A Pratical Guide, pp. 281–290. SIAM, Philadelphia (2000)
Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26, 640–659 (2005)
Dedieu, J., Tisseur, F.: Perturbation theory for homogeneous polynomial eigenvalue problems. Linear Algebra Appl. 358, 71–94 (2003)
Demmel, J.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hoplins University Press, Baltimore (1996)
Guo, J., Lin, W., Wang, C.: Numerical solutions for large sparse quadratic eigenvalue problems. Linear Algebra Appl. 225, 57–89 (1995)
Higham, N.J., Tisseur, F.: More on pseudospectra for polynomial eigenvalue problems and applications in control theory. Linear Algebra Appl. 351/352, 435–453 (2002)
Hochstenbach, M.E., van der Vorst, H.A.: Alternatives to the Rayleigh quotient for the quadratic eigenvalue problem. SIAM J. Sci. Comput. 25, 591–603 (2003)
Hoffnung, L., Li, R.-C., Ye, Q.: Krylov type subspace methods for matrix polynomials. Linear Algebra Appl. 415, 52–81 (2006)
Holz, U.B., Golub, G., Law, K.H.: A subspace approximation method for the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26, 498–521 (2004/05)
Hwang, T.M., Lin, W.W., Liu, J.L., Wang, W.: Jacobi-Davidson methods for cubic eigenvalue problem. Numer. Linear Algebra Appl. 12, 605–624 (2005)
Hwang, T., Lin, W., Mehrmann, V.: Numerical solution of quadratic eigenvalue problems with structure-preserving methods. SIAM J. Sci. Comput. 24, 1283–1302 (2003)
Ishihara, K.: Descent iterations for improving approximate eigenpairs of polynomial eigenvalue problems with general complex matrices. Computing 68, 239–254 (2002)
Jia, Z.: Refined iterative algorithm based on Arnoldi’s process for large unsymmetric eigenproblems. Linear Algebra Appl. 259, 1–23 (1997)
Jia, Z.: A refined subspace iteration algorithm for large spares eigenproblems. Appl. Numer. Math. 32, 35–52 (2000)
Li, R.-C., Ye, Q.: A Krylov subspace method for quadratic matrix polynomials with application to constrained least squares problems. SIAM J. Matrix Anal. Appl. 25, 405–428 (2003)
Mehrmann, V., Watkins, D.: Polynomial eigenvalue problems with Hamiltonian structure. Electron. Trans. Numer. Anal. 13, 106–118 (2002)
Raeven, F.A.: A new Arnoldi approach for polynomial eigenproblems. In: Proceedings of the Copper Mountain Conference on Iterative Methods. http://www.mgnet.org/Conferences/CMCIM96/Psfiles/raeven.ps.gz (1996)
Saad, Y.: Variations on Arnoldi’s method for computing eigenproblems of large non-Hermitian matrices. Linear Algebra Appl. 34, 269–295 (1980)
Schmitz, H., Volk, K., Wendland, W.L.: On three-dimensional singularities of elastic fields near vertices. Numer. Methods Partial Differ. Equ. 9, 323–337 (1993)
Schwarz, H.R.: Methode der Finiten Elemente. Teubner, Stuttgart (1984)
Sleijpen, G.L.G., Booten, G.L., Fokkema, D.R., Van der Vorst, H.A.: Jacobi Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36, 595–633 (1996)
Sleijpen, G.L.G., van der Vorst, H.A., van Gijzen, M.B.: Quadratic eigenproblems are no problem. SIAM News 29, 8–9 (1996)
Tisseur, F.: Backward error and condition of polynomial eigenvalue problems. Linear Algebra Appl. 309, 339–361 (2000)
Tisseur, F., Higham, N.J.: Structured pseudospectra for polynomial eigenvalue problems, with applications. SIAM J. Matrix Anal. Appl. 23, 187–208 (2001)
Tisseur, F., Mearbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43, 235–286 (2001)
Zeng, Y., Li, Y.: Integrable Hamiltonian systems related to the polynomial eigenvalue problem. Math. J. Phys. 31, 2835–2839 (1990)
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Yiqin Lin is supported by Scientific Research Startup Foundation of Hunan University of Science and Engineering.
Yimin Wei is supported by the National Natural Science Foundation of China and Shanghai Education Committee.
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Bao, L., Lin, Y. & Wei, Y. Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems. Numer Algor 50, 17–32 (2009). https://doi.org/10.1007/s11075-008-9214-7
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DOI: https://doi.org/10.1007/s11075-008-9214-7
Keywords
- Polynomial eigenvalue problem
- Generalized Krylov subspace
- Generalized Arnoldi procedure
- Projection technique
- Refined technique
- Restarting