Configuration Interaction Methods and Large-scale Matrix Diagonalization

Abstract

This thesis presents investigations made with matrices representing quantum mechan- ical many-body systems in order to provide recommendations for a future eigensolver. The matrices are generated using the No-Core Shell Model for bosons (NCSMb) code. Finding the eigenvalues is of interest since these correspond to the energy spectrum of the system of particles, where examples of such systems include nucleons in a nuclei or trapped atomic gases at low temperatures. The eigensolver will be based on the Lanczos algorithm, a method well suited to reduce the complexity of the eigenvalue-solving of large, sparse matrices. Investigations are performed concerning the Lanczos method and of how to adapt it in an appropriate way in terms of choice of starting vectors, methods of orthogonalization and of how to accelerate convergence. Also, the possibility of Block Lanczos is examined. Guidelines are given in order to make a future eigensolver as effective as possible. Furthermore, various tests are made investigating the properties of the matrices generated by NCSMb such as the dimension of the matrix for different model spaces, the number of nonzero elements and the distribution of the matrix elements among others.