Determination of natural frequency using Jacobi method and Jacobi-Davidson method

The first aim of this study is to determine the eigenvalues by using the Jacobi method and the Jacobi-Davidson method. The second aim is to develop the Jacobi and the Jacobi-Davidson algorithm via Mathcad. The third aim is to verify that Jacobi Davidson method is a better method compared to Jacobi method in solving eigenvalue problems. The matrix tested for both methods is real symmetrical. Different matrix size ranging from 4, 8, 16, 32, 128 and 256 was used to compute the largest eigenvalue for both methods. Through numerical experiments, we can conclude that the Jacobi-Davidson method is a better iterative method compared to Jacobi method in terms of number of iterations, execution time in seconds and error estimated. This is because the Jacobi method requires a diagonal matrix for the algorithm to terminate. In this dissertation, the classical Jacobi method was discussed. The method requires the determination of the nondiagonal element with largest modulus. In order to reduce the computational costs, the cyclic Jacobi method and threshold Jacobi method is worth to investigate for further study. In addition, different types of matrices can be tested using Jacobi-Davidson method and those matrices can be found from the Matrix Market