A hom*ogeneous Rayleigh quotient with applications in gradient methods (2024)

Skip Abstract Section

1. [1] Parlett B.N., The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 1998.Google Scholar
2. [2] Stewart G.W., Matrix Algorithms. Vol. II, SIAM, Philadelphia, PA, 2001.Google Scholar
3. [3] Ferrandi G., Hochstenbach M.E., Krejić N., A harmonic framework for stepsize selection in gradient methods, Comput. Opt. Appl. 85 (2023) 75–106.Google Scholar
4. [4] Morgan R.B., Computing interior eigenvalues of large matrices, Linear Algebra Appl. 154 (1991) 289–309.Google Scholar
5. [5] Barzilai J., Borwein J.M., Two-point step size gradient methods, IMA J. Numer. Anal. 8 (1) (1988) 141–148.Google Scholar
6. [6] Vömel C., A note on harmonic Ritz values and their reciprocals, Numer. Linear Algebra Appl. 17 (1) (2010) 97–108.Google Scholar
7. [7] Stewart G.W., Sun J.G., Matrix Perturbation Theory, Academic Press Inc., Boston, MA, 1990.Google Scholar
8. [8] Stewart G.W., Gershgorin theory for the generalized eigenvalue problem A x = λ B x , Math. Comp. (1975) 600–606.Google Scholar
9. [9] Dedieu J.-P., Condition operators, condition numbers, and condition number theorem for the generalized eigenvalue problem, Linear Algebra Appl. 263 (1997) 1–24.Google Scholar
10. [10] Dedieu J.-P., Tisseur F., Perturbation theory for hom*ogeneous polynomial eigenvalue problems, Linear Algebra Appl. 358 (2003) 71–94.Google Scholar
11. [11] Hochstenbach M.E., Notay Y., hom*ogeneous Jacobi–Davidson, Electron. Trans. Numer. Anal. 29 (2007) 19–30.Google Scholar
12. [12] Li S., Zhang T., Xia Y., A family of Barzilai–Borwein steplengths from the viewpoint of scaled total least squares, 2022, Preprint arXiv:2212.05658.Google Scholar
13. [13] Li S., Xia Y., A new Barzilai–Borwein steplength from the viewpoint of total least squares, 2021, Preprint arXiv:2107.06687.Google Scholar
14. [14] Sleijpen G.L.G., vanden Eshof J., On the use of harmonic Ritz pairs in approximating internal eigenpairs, Linear Algebra Appl. 358 (1–3) (2003) 115–137.Google Scholar
15. [15] Sleijpen G.L.G., vanden Eshof J., Smit P., Optimal a priori error bounds for the Rayleigh–Ritz method, Math. Comp. 72 (242) (2003) 677–684.Google Scholar
16. [16] DiSerafino D., Ruggiero V., Toraldo G., Zanni L., On the steplength selection in gradient methods for unconstrained optimization, Appl. Math. Comput. 318 (2018) 176–195.Google Scholar
17. [17] Raydan M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim. 7 (1) (1997) 26–33.Google ScholarDigital Library
18. [18] Dai Y.H., Liao L.Z., R-linear convergence of the Barzilai and Borwein gradient method, IMA J. Numer. Anal. 22 (1) (2002) 1–10.Google Scholar
19. [19] Raydan M., On the Barzilai and Borwein choice of steplength for the gradient method, IMA J. Numer. Anal. 13 (3) (1993) 321–326.Google Scholar
20. [20] Dai Y.H., Alternate step gradient method, Optimization 52 (4–5) (2003) 395–415.Google Scholar
21. [21] Andrei N., An unconstrained optimization test functions collection, Adv. Model. Optim. 10 (1) (2008) 147–161.Google Scholar
22. [22] Moré J.J., Garbow B.S., Hillstrom K.E., Testing unconstrained optimization software, ACM Trans. Math. Software 7 (1) (1981) 17–41.Google Scholar
23. [23] Frassoldati G., Zanni L., Zanghirati G., New adaptive stepsize selections in gradient methods, J. Ind. Manag. 4 (2) (2008) 299–312.Google Scholar

Index Terms

1. A hom*ogeneous Rayleigh quotient with applications in gradient methods

   1. Applied computing

      1. Physical sciences and engineering

   2. Mathematics of computing

      1. Mathematical analysis

         1. Differential equations

            1. Partial differential equations

         2. Mathematical optimization

         3. Numerical analysis

   3. Theory of computation

      1. Design and analysis of algorithms

         1. Mathematical optimization

• Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems

  The restrictively preconditioned conjugate gradient (RPCG) method for solving large sparse system of linear equations of a symmetric positive definite and block two-by-two coefficient matrix is further studied. In fact, this RPCG method is essentially ...

  Read More

  See Also

  Properties of Logarithms (Product, Quotient and Power Rule)
• Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems

  By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian splitting (PSS), and then establish a ...

  Read More

• A generalized variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems

  Based on the variant of the deteriorated positive-definite and skew-Hermitian splitting (VDPSS) preconditioner developed by Zhang and Gu (BIT Numer. Math. 56:587---604, 2016), a generalized VDPSS (GVDPSS) preconditioner is established in this paper by ...

  Read More