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1THE RITZ-LANCZOS ALGORITHM IN STRUCTURAL DYNAMIC ANALYSIS
Published in Franklin Y. Cheng, FU Zizhi, Computational Mechanics in Structural Engineering, 2003
The eigenvalue problem of very large structures is always an important and basic problem in the field of dynamic analysis. In 1950 Lanczos[2] proposed an algorithm of three terms recurrence formula to produce a set of orthogonal vectors, and then to transform the original matrix to a tridiagonal matrix. As there are many efficient solvers for the eigenvalue problem of a tridiagonal matrix such as the QL method, the Lanczos algorithm seems to be the fastest. Unfortunately it has a fatal defect, i.e. numerical instability caused by the loss of orthogonality, which has been thoroughly studied by Paige[3] in his thesis, and solved by other researchers[4, 5] with various reorthogonalization methods. The following problem now remains: the Lanczos method produced a set of orthogonalized vectors, in which the first several vectors are good approximations to the eigenvectors. When the tolerance of eigenvalue error is prescribed and the number 1 of Lanczos vectors is required, the number of Lanczos vectors j is uncertain. It has been suggested[6] that j = 21. Apparently this value does not relate to the prescribed tolerance and in most cases it is not sufficient. We propose the following Ritz- Lanczos (R-L) algorithm, slightly different from [1] but more efficient.
Asymmetric compressive stability of rotating annular plates
Published in European Journal of Computational Mechanics, 2019
H. Bagheri, Y. Kiani, M. R. Eslami
In Equation (24) is the elastic stiffness matrix, is the geometrical stiffness matrix due to uniform compression, and is the geometrical stiffness matrix due to rotation. The above system should be solved as an eigenvalue problem to obtain the critical states of the plate and the associated buckled shapes. In this study a Matlab code is developed to obtain the eigenvalues and eigenvectors. In this code, the eigenvalues and eigenvectors of the system of equations are obtained based on the Lanczos algorithm. The process to obtain those parameters is as follows: For each circumferential mode number starting from zero, the eigenvalue problem is solved and the minimum eigenvalue which is is obtained. Searching among all of these minimums which are extracted for different number of nodal diameters and choosing the minimum one, one may reach to the critical buckling parameter of the plate and the associated mode number .
Construction of H-Symmetric pentadiagonal matrices by three spectra
Published in Applied Mathematics in Science and Engineering, 2022
In this section, we consider an inverse eigenvalues problem for pentadiagonal H-Symmetric matrix P using three spectra consisting of complex eigenvalues. That is we want to construct pentadiagonal matrix P such that may have some complex numbers. Indeed, we construct a pentadiagonal matrix P by prescribed eigendata such that are real and distinct, but has some complex eigenvalues. Since P has complex eigenvalues, thus the corresponding eigenvectors do not make a H-Orthonormal matrix. Therefore, we can not use the Lanczos algorithm to construct matrix P. Using (11) we compute , where On the other hand, the eigenvectors of are -Orthonormal, thus simple calculation shows that From (14), we find Multiplying both sides of (16) by and taking sum in i and considering the condition , we obtain therefore, we find Now using the first and the second components of , i.e. and the spectrum we may use the Lanczos algorithm to construct . The entries and are found by (15) and (17). Using trace formula we find as follows: Therefore, P is constructed completely.