zbMATH — the first resource for mathematics

Applications of the Lanczos method. (English) Zbl 0798.65073
Summary: We are concerned with the application of the unsymmetric two sided Lanczos method to systems of first order differential equations and its associated problems; namely generalized eigenproblems and linear systems of equations. The unsymmetric Lanczos method is first used to generate two sets of vectors; the left and right Lanczos vectors. These Lanczos vectors are used directly in a method of weighted residual to reduce the differential equations to a small unsymmetric tridiagonal system. The advantage of this method is that the reduced system can be solved directly thus eliminating the usual computation of the complex eigenpairs.
An algorithm is then derived by simplifying the two sided Lanczos method for systems of equations with a symmetric matrix pair. By appropriate choice of the starting vectors we obtain an implementation of the Lanczos method that is generalized to the case with indefinite matrix coefficients. This modification results in a simple relation between the left and right Lanczos vectors, a symmetric tridiagonal, and a diagonal matrix. The system of differential equations can then be reduced to one with a symmetric tridiagonal and diagonal coefficient matrices. The modified algorithm is also suitable for eigenproblems with symmetric indefinite matrices and for solution of systems of equations using indefinite preconditioning matrices.
This approach is used to evaluate the vibration response of a damped beam problem and a space mast structure with a symmetric damping matrix arising from velocity feedback control forces. In both problems, accurate solutions were obtained with as few as 20 Lanczos vectors.

65L05 Numerical methods for initial value problems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
93B52 Feedback control
Full Text: DOI
[1] Craig, R.R., Structural dynamics — an introduction to computer methods, (1981), Wiley New York
[2] Craig, R.R., ()
[3] Hughes, T.J.R., The finite element method, (1987), Prentice-Hall Englewood Cliffs, New Jersey
[4] Lanczos, C., J. res. nat. bur. standards, 45, 255, (1950)
[5] Nour-Omid, B.; Parlett, B.N.; Taylor, R.L., Inter. J. num. meth. eng., 19, 859, (1983)
[6] Nour-Omid, B.; Clough, R.W., Earthquake eng. and structural dynamics, 12, No. 4, (July 1984)
[7] Nour-Omid, B.; Clough, R.W., Earthquake eng. and structural dynamics, 13, 271, (1985)
[8] Nour-Omid, B., J. num. meth. eng., 24, 251, (1987)
[9] Nour-Omid, B.; Parlett, B.N.; Ericsson, T.; Jensen, P.S., Math. comput., 48, No. 178, 663, (1987)
[10] Parlett, B.N., The symmetric eigenvalue problem, (1980), Prentice-Hall Englewood Cliffs · Zbl 0431.65016
[11] Parlett, B.N.; Scott, D., Math. comput., 33, No. 145, 217, (1979)
[12] Parlett, B.N.; Nour-Omid, B., Lin. alg. appl., 68, 179, (1985)
[13] B.N. Parlett and D.R. Taylor, Report no. PAM-43, University of California, Berkeley.
[14] Simon, H.D., Math. comp., 42, No. 165, 115, (1984)
[15] D.R. Taylor, Report no. PAM-108, University of California, Berkeley.
[16] Wilkinson, J.H., Comput. J., 148, (1958)
[17] Wilson, E.H.; Penzien, J., Inter. J. num. meth. eng., 4, 5, (1972)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.