Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A0, the eigenpairs of A(t) ≡ (1 − t)A0 + tA1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix: • bifurcation, • ill conditioning due to nonorthogonal eigenvectors, • jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented

The application of parallel computer technology to the dynamic analysis of suspension bridges

This research is concerned with the application of distributed computer technology to the solution of non-linear structural dynamic problems, in particular the onset of aerodynamic instabilities in long span suspension bridge structures, such as flutter which is a catastrophic aeroelastic phenomena. The thesis is set out in two distinct parts:- Part I, presents the theoretical background of the main forms of aerodynamic instabilities, presenting in detail the main solution techniques used to solve the flutter problem. The previously written analysis package ANSUSP is presented which has been specifically developed to predict numerically the onset of flutter instability. The various solution techniques which were employed to predict the onset of flutter for the Severn Bridge are discussed. All the results presented in Part I were obtained using a 486DX2 66MHz serial personal computer. Part II, examines the main solution techniques in detail and goes on to apply them to a large distributed supercomputer, which allows the solution of the problem to be achieved considerably faster than is possible using the serial computer system. The solutions presented in Part II are represented as Performance Indices (PI) which quote the ratio of time to performing a specific calculation using a serial algorithm compared to a parallel algorithm running on the same computer system