Direct Sequential Generation of Optimal Eigenvectors of the Symmetric Kernel of the Discrete Hankel Transform

Generating orthonormal Laguerre-Gaussian-power-like eigenvectors of the symmetric kernel matrix T of the discrete Hankel transform (DHT) is the cornerstone in developing the discrete fractional Hankel transform (DFRHT). The direct sequential generation of the columns of each partition of the modal matrix of matrix T pertaining to one distinct eigenvalue is considered. The focus of the research is the assessment of the direct sequential algorithms: the Gram-Schmidt algorithm (GSA), modified Gram-Schmidt algorithm (MGSA), sequential orthogonal procrustes algorithm (SOPA) and direct sequential evaluation by constrained optimization algorithm (DSEOA). Although the four algorithms are theoretically equivalent, they are quite numerically distinct. The difference in performance becomes more pronounced as the order of the square matrix T becomes larger. The assessment shows that the MGSA outperforms both the GSA and the SOPA and that the DSEOA is by far the best in terms of computation accuracy.