×

Conjugate gradient methods for the Rayleigh quotient minimization of generalized eigenvalue problems. (English) Zbl 0788.65043

For the computation of extreme eigenvalues and corresponding eigenvectors of the generalized eigenvalue problem \(Ax= \lambda Bx\), where \(A\) is real symmetric and \(B\) is positive definite, the conjugate gradient method for minimization of the Rayleigh quotient by W. W. Bradbury and R. Fletcher [New iterative methods for solution of the eigenproblem. Numer. Math. 9, 259-267 (1966; Zbl 0202.435)] is discussed and modified. The author considers proper step sizes in the one-dimensional minimization of the Rayleigh quotient and discusses properties, in particular, the convergence of the modified conjugate gradient methods for the generalized eigenvalue problem.
Reviewer: Z.Mei (Toowoomba)

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Citations:

Zbl 0202.435
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bradbury, W. W., Fletcher, R.: New iterative methods for solution of the eigenproblem. Numer. Math.9, 259–267 (1966). · Zbl 0202.43502 · doi:10.1007/BF02162089
[2] Cullum, J., Willoughby, R. A.: Lanczos algorithms for large symmetric eigenvalue computations, I: Theory; II: Programs. Boston Basel Stuttgart: Birkhäuser 1985. · Zbl 0574.65028
[3] Daniel, J. W.: The approximate minimization of functionals. Englewood Cliffs: Prentice Hall 1971. · Zbl 0223.65014
[4] Faddeev, D. K., Faddeeva, V. N.: Computational methods of linear algebra. San Francisco London: Freeman 1963. · Zbl 0112.07503
[5] Fried, I.: Optimal gradient minimization scheme for finite element eigenproblems, J. Sound Vib.20, 333–342 (1972). · Zbl 0242.65041 · doi:10.1016/0022-460X(72)90614-1
[6] Geradin, M.: The computational efficiency of a new minimization algorithm for eigenvalue analysis. J. Sound Vib.19, 319–331 (1971). · Zbl 0225.65050 · doi:10.1016/0022-460X(71)90692-4
[7] Hackbusch, W.: On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multigrid method. SIAM J. Numer. Anal.16, 201–215 (1979). · Zbl 0403.65043 · doi:10.1137/0716015
[8] Longsine, D. E., McCormick, S. F.: Simultaneous Rayleigh-quotient minimization methods forAx={\(\lambda\)}Bx. Linear Algebra Appl.34, 195–234 (1980). · Zbl 0475.65021 · doi:10.1016/0024-3795(80)90166-4
[9] Mandel, J., McCormick, S.: A multilevel variational method forAu={\(\lambda\)}Bu on composite grids. J. Comput. Phys.80, 442–452 (1989). · Zbl 0674.65013 · doi:10.1016/0021-9991(89)90109-5
[10] Parlett, B. N.: The symmetric eigenvalue problem. Englewood Cliffs: Prentice Hall 1980. · Zbl 0431.65017
[11] Ruhe, A.: Computation of eigenvalues and eigenvectors. In: Baker, V. A. (Ed.) Sparse matrix techniques, pp. 130–184. Berlin Heidelberg New York: Springer 1977 (Lecture Notes on Mathmatics, Vol. 572). · Zbl 0355.65024
[12] Schwarz, H. R.: The method of coordinate overrelaxation for (A={\(\lambda\)}B)x=0. Numer. Math.23, 135–151 (1974). · doi:10.1007/BF01459947
[13] Schwarz, H. R.: Two algorithms for treatingAx={\(\lambda\)}Bx. Comput. Methods Appl. Mech. Eng.12, 181–199 (1977). · Zbl 0363.65026 · doi:10.1016/0045-7825(77)90011-1
[14] Schwarz, H. R.: Simultane Iterationsverfahren für große allgemeine Eigenwertprobleme. Ing.-Arch.50, 329–338 (1981). · Zbl 0472.73107 · doi:10.1007/BF00778428
[15] Schwarz, H. R.: Methode der finiten Elemente, and FORTRAN-programme zur Methode der finiten Elemente, 3rd ed. Stuttgart: Teubner 1991. · Zbl 0477.65074
[16] Stewart, G. W.: A bibliographical tour of the large, sparse generalized eigenvalue problem. In: Bunch, J. R., Rose, D. J. (Eds.) Sparse matrix computations, New York San Francisco London: Academic Press 1976, pp. 113–130. · Zbl 0343.65013
[17] Wilkinson, J. H.: The algebraic eigenvalue problem. Oxford: Oxford University Press 1965. · Zbl 0258.65037
[18] Wilkinson, J. H., Reinsch, C.: Handbook for automatic computation, Vol. II: Linear algebra. Berlin Heidelberg New York: Springer 1971. · Zbl 0219.65001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.