Multiple shift maximum element sequential matrix diagonalisation for parahermitian matrices

Abstract

Abstract-A polynomial eigenvalue decomposition of parahermitian matrices can be calculated approximately using iterative approaches such as the sequential matrix diagonalisation (SMD) algorithm. In this paper, we present an improved SMD algorithm which, compared to existing SMD approaches, eliminates more off-diagonal energy per step. This leads to faster convergence while incurring only a marginal increase in complexity. We motivate the approach, prove its convergence, and demonstrate some results that underline the algorithm's performance.

Impact of source model matrix conditioning on PEVD algorithms

“…The current most popular PEVD algorithms [4,[8][9][10] have the goal of diagonalising a parahermitian matrix R(z) starting from an initial approximation S (0) (z). The ith iteration of all algorithms consists of three common steps operating on S (i−1) (z), which vary with implementation.…”

“…A number of PEVD algorithms have been introduced [4,[6][7][8][9][10], and offer various performance characteristics. The algorithms in [4,6,10] have been demonstrated on parahermitian matrices R(z) ∈ C M×M derived from random A(z) ∈ C M×K as R(z) = A(z)Ã(z).…”

“…The algorithms in [4,6,10] have been demonstrated on parahermitian matrices R(z) ∈ C M×M derived from random A(z) ∈ C M×K as R(z) = A(z)Ã(z). For K < M , R(z) is guaranteed to be rank deficient, but when K ≥ M it is possible for R(z) to have full rank.…”

“…In [9], a source model convolutively mixes spectrally majorised sources by means of an arbitrary paraunitary matrix, such that the ground truth PEVD with finite order factors and equality in (1) is guaranteed. Since these publications [4,[6][7][8][9][10] use differently conditioned problems, a direct comparison between algorithms proposed in individual papers is not always straightforward.…”

“…An ensemble of randomised parahermitian matrices with different dynamic ranges and types of majorisation are factorised by a number of PEVD algorithms belonging to the second order sequential best rotation (SBR2 family, [4,8] and the sequential matrix diagonalisation (SMD family, [9,10]). …”