On the computation of probabilities and eigenvalues for random and non-random matrices

[摘要] (cont.) An expression for the expected number of real eigenvalues En=o kpn,k is obtained in paper [2]. Results relating the rational and irrational parts of the summations n =o kpn,ki, En k=0 (Pk n,k and En= - n,k 0 k)Pn,k are conjectured. Three eigenvalue algorithms, the block Davidson, the block KrylovSchur and the Locally optimal Block Pre-conditioned Conjugate Gradient Method (LOBPCG) are analyzed and their performance on different types of matrices are studied. The performance of the algorithms as a function of the parameters , block size, number of blocks and the type of preconditioner is also examined in this thesis. The block Krylov Schur Algorithm for the matrices which are used for the experiments have proved to much superior to the others in terms of computation time. Also its been more efficient in finding eigenvalues for matrices representing grids with Neumann boundary conditions which have at least one zero eigenvalue. There exists one optimal combination of block size and number of blocks at which the time for eigenvalue computation is minimum. These parameters have different effects for different cases. The block Davidson algorithm has also been incorporated with the locking mechanism and this implementation is found to be much superior to its counterpart without the locking mechanism for matrices which have at least one zero eigenvalue.

[发布机构] Massachusetts Institute of Technology