Study on the Frequency Analysis Using a Modified Iterative Matrix Transformation Algorithm

In numerical linear algebra, the Jacobi algorithm is an iterative method for the calculation of the frequencies of a matrix obtained from a structure. The method involves iteratively applying a series of orthogonal matrices and, thus, similarity transformations to the original stiffness matrix until it is transformed into a diagonal matrix. The frequencies of eigenvalues are aligned along the diagonal of the matrix. The Jacobi method is known for its simplicity and ease of implementation, but its convergence rate is slow for matrices obtained from large complex structures.

As such, in this paper, a more efficient modified iteration algorithm was studied so that the presented method can be used in practice to evaluate the eigenvalues accordingly.

The eigenvectors of the matrix can be obtained as the transformation matrices, such as constructed orthogonal matrices applied during the Jacobi iterations.

The presented method proved to be computationally efficient and accurate for frequency analysis, particularly for large datasets. It is also less sensitive to conditions of the original starting matrix than some other eigenvalue algorithms. However, it can be more complex to implement than the Jacobi method and requires more matrices to apply before proceeding to iterative transformation to ensure convergence.