scholarly journals Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems

1991 ◽  
Vol 32 (4) ◽  
pp. 437-456 ◽  
Author(s):  
D. C. Dzeng ◽  
W. W. Lin
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Positive Semidefinite ◽  
Symmetric Matrices ◽  
Homotopy Continuation ◽  
Smooth Curves ◽  
Homotopy Continuation Method ◽  
Symmetric Eigenvalue Problems

AbstractWe consider a generalised symmetric eigenvalue problem Ax = λMx, where A and M are real n by n symmetric matrices such that M is positive semidefinite. The purpose of this paper is to develop an algorithm based on the homotopy methods in [9, 11] to compute all eigenpairs, or a specified number of eigenvalues, in any part of the spectrum of the eigenvalue problem Ax = λMx. We obtain a special Kronecker structure of the pencil A − λM, and give an algorithm to compute the number of eigenvalues in a prescribed interval. With this information, we can locate the lost eigenpair by using the homotopy algorithm when multiple arrivals occur. The homotopy maintains the structures of the matrices A and M (if any), and the homotopy curves are n disjoint smooth curves. This method can be used to find all/some isolated eigenpairs for large sparse A and M on SIMD machines.

scholarly journals New collocation path-following approach for the optimal shape parameter using Kernel method

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Zouhair Saffah ◽  
Abdelaziz Timesli ◽  
Hassane Lahmam ◽  
Abderrahim Azouani ◽  
Mohamed Amdi
Keyword(s):  
Shape Parameter ◽  
Kernel Method ◽  
Continuation Method ◽  
Path Following ◽  
High Order ◽  
Homotopy Continuation ◽  
Homotopy Continuation Method ◽  
Optimal Value ◽  
Rbf Kernel ◽  
Order Algorithm

AbstractThe goal of this work is to develop a numerical method combining Radial Basic Functions (RBF) kernel and a high order algorithm based on Taylor series and homotopy continuation method. The local RBF approximation applied in strong form allows us to overcome the difficulties of numerical integration and to treat problems of large deformations. Furthermore, the high order algorithm enables to transform the nonlinear problem to a set of linear problems. Determining the optimal value of the shape parameter in RBF kernel is still an outstanding research topic. This optimal value depends on density and distribution of points and the considered problem for e.g. boundary value problems, integral equations, delay-differential equations etc. These have been extensively attempts in literature which end up choosing this optimal value by tests and error or some other ad-hoc means. Our contribution in this paper is to suggest a new strategy using radial basis functions kernel with an automatic reasonable choice of the shape parameter in the nonlinear case which depends on the accuracy and stability of the results. The computational experiments tested on some examples in structural analysis are performed and the comparison with respect to the state of art algorithms from the literature is given.

scholarly journals High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem

2020 ◽  
Vol 96 ◽  
pp. 102639
Author(s):  
Carolin Penke ◽  
Andreas Marek ◽  
Christian Vorwerk ◽  
Claudia Draxl ◽  
Peter Benner
Keyword(s):  
Eigenvalue Problem ◽  
High Performance ◽  
Eigenvalue Problems ◽  
Symmetric Eigenvalue Problems
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