A FORTRAN IV algorithm is presented for determining sets of dominant eigenvalues and corresponding eigenvectors of symmetric matrices. It is also extended to the solution of the equations of natural vibration of a structure for which symmetric stiffness and mass matrices are available. The matrices are stored and processed in variable bandwidth form, thus enabling advantage to be gained from sparseness in the equations. Some of the procedures may also be used to solve symmetric positive definite equations such as those arising from the static analysis of structures loaded within the elastic range.

A simultaneous iteration algorithm for symmetric eigenvalue problems

A single procedure is presented for the determination of eigenvalue and eigenvector derivatives of general eigensystems, including those arising in flutter and divergence analysis. The method requires only the knowledge of the eigensolution under consideration and no information about the transposed problem. In the procedure, the eigenproblem is not presented as such, but as a nonlinear algebraic system of equations with the eigenvalue and the eigenvector as unknowns and the eigenderivatives coming from the solution of a linear system of equations that is determined trivially by differentiati ng the nonlinear equations with respect to a structural parameter. The technique can be extended easily to derivatives of any order and always requires solutions of the same linear system with different right-hand sides. The coefficient matrix of this linear system always maintains the structure of the matrix of the eigenproblem and three methods of solution are presented to take full advantage from its symmetry and sparseness, if present. Application to flutter and divergence problems is emphasized, as the basic idea of the method unifies the determination of a solution and of its derivatives in one single approach.

Calculation of Eigenvalue and Eigenvector Derivatives for Algebraic Flutter and Divergence Eigenproblems

Bathe's algorithm of subspace iteration for the solution of the eigenvalue problem with symmetric matrices is improved by incorporating an acceleration technique using Chebyshev polynomials. This method of acceleration is particularly effective for this kind of iteration. The rate of convergence of the iteration scheme presented is considerably improved when compared with the original one, and satisfactory rates of convergence can be obtained for a wider range of eigenvalues.

Subspace iteration accelrated by using Chebyshev polynomials for eigenvalue problems with symmetric matrices

Discussion of papers By O. C. Zienkiewicz, R. L. Taylor and J. M. Too and S. F. Pawsey and R. W. Clough

The subspace iteration method is widely used for the computation of a few smallest eigenvalues and the corresponding eigenvectors of large eigenproblems. Under certain conditions, this method exhibits slow convergence. For improving the convergence of the method, a few acceleration techniques are already available in the literature. In this paper, yet another method is presented. The method has been applied for the computation of a few smallest eigenpairs of some typical eigenprobiems. The results indicate that the acceleration method is efficient for large eigenproblems. Simplicity, ease of computer implementation, absence of parameters whose values are to be chosen based on experience and effectiveness for large problems are the attractive features of the proposed method.

B.R. Corar

Papers

Implementation of simultaneous iteration for vibration analysis