Search

Recently a selection of sequential matrix diagonalisation (SMD) algorithms have been introduced which approximate polynomial eigenvalue decomposition of parahermitian matrices. These variants differ only in the search methods that are used to bring energy onto the zero-lag. Here we analyse the search methods in terms of their computational complexities for different sizes of parahermitian matrices which are verified through simulated execution times. Another important factor for these search methods is their ability to transfer energy. Simulations show that the more computationally complex search methods transfer a greater proportion of the off-diagonal energy onto the zero-lag over a selected range of parahermitian matrix sizes. Despite their higher cost per iteration experiments indicate that the more complex search algorithms still converge faster in real time.

This paper introduces a causality constrained sequential matrix diagonalisation (SMD) algorithm, which generates a causal paraunitary transformation that aprroximately diagonalises and spectrally majorises a parahermitian matrix, and can be used to determine a polynomial eigenvalue decomposition. This algorithm builds on a multiple shift technique which speeds up diagonalisation by diagonalisation per iteration step based on a particular search space, which is contrained to permit a maximum number of causal time shifts. The results presented in this paper show the performance in comparison to existing algorithms, in particular an unconstrained multiple shift SMD algorithm, from which our proposed method derives.

A recent class of sequential matrix diagonalisation (SMD) algorithms have been demonstrated to provide a fast converging solution to iteratively approximating the polynomial eigenvalue decomposition of a parahermitian matrix. However, the calculation of an EVD, and the application of a full unitary matrix to every time lag of the parahermitian matrix in the SMD algorithm results in a high numerical cost. In this paper, we replace the EVD with a limited number of Givens rotations forming a cyclic-by-row Jacobi sweep. Simulations indicate that a considerable reduction in computational complexity compared to SMD can be achieved with a negligible sacrifice in diagonalisation performance, such that the benefits in applying the SMD are maintained.