Fast Givens Rotation Approach to Second Order Sequential Best Rotation Algorithms

Abstract

The second order sequential best rotation (SBR2) algorithm is a popular algorithm to decompose a parahermitian matrix into approximated polynomial eigenvalues and eigenvectors. The work horse behind SBR2 is a Givens rotation interspersed by delay operations. In this paper, we investigate and analyse the application of a fast Givens rotation in order to reduce the computation complexity of SBR2. The proposed algorithm inherits the SBR2's proven convergence to a diagonalised and spectrally majorised solution for the polynomial eigenvalues. We provide some analysis and examples for the execution speed of this fast Givens-based SBR2 compared to a standard SBR2 implementation.

Extension of Power Method to Para-Hermitian Matrices: Polynomial Power Method

“…The proposed method finds potential applications in areas such as data compaction [18], single broadband source to sensors transfer and source spectral density estimation [29], broadband transient signal detection [30]- [32], and the calculations of an approximate PEVD through deflation via subsequent extractions and eliminations of the principal eigenpairs. The approach can also be generalised to calculating a polynomial singular value decomposition [33].…”

Space-Time Covariance Matrix Estimation: Loss of Algebraic Multiplicities of Eigenvalues

“…In terms of applications, the effect can be both beneficial or detrimental, and favours a revival of algorithms that target spectrally majorised eigenvalues for polynomial matrix factorisations, which are supported by substantial algorithmic developments and implementations [3], [12]. Alternatively, analytic eigenvalue and eigenvector extraction algorithms [35]- [37], [39] can also yield such solutions with guaranteed spectral majorisation where current time domain methods may fail due to a large dynamic range in the eigenvalues.…”

Support Estimation of Analytic Eigenvectors of Parahermitian Matrices

“…Therefore, the correct time domain support estimation directly impacts on the complexity of any subsequent applications [1]- [14], [44], [45], and complements the significant efforts that have been dedicated to the efficient implementation of polynomial matrix factorisations, see e.g. [46]- [48].…”