Generalized Polynomial Power Method

Abstract

The polynomial power method repeatedly multiplies a polynomial vector by a para-Hermitian matrix containing spectrally majorised eigenvalue to estimate the dominant eigenvector corresponding the dominant eigenvalue. To limit the order of the resulting vector, truncation is performed in each iteration. This paper extends the polynomial power method from para-Hermitian matrices to a general polynomial matrix for determining its dominant left-and right-singular vectors and the corresponding singular value. The proposed extension assumes that the dominant singular is positive on the unit circle. The resulting algorithm is compared with a state-of-the-art PSVD algorithm and provides better accuracy with reduced computation time and lower approximation orders for the decomposition.

Low-Rank Para-Hermitian Matrix EVD via Polynomial Power Method with Deflation

“…Over an ensemble of low-rank para-Hermitian matrices, the proposed method has outperformed state-of-the-art algorithms in terms of accuracy, speed, and implementation complexity. The algorithm can similarly be extended to compute the PSVD of low-rank general polynomial matrices based on a generalized polynomial power method [17] as an alternative to a full PSVD in [36]. The proposed technique can also be directly applied to a number of low-rank applications where the number of channels can substantially exceed the number of sources [37][38][39] or in problems that are rank one [40].…”

“…Due to the existence of the analytic EVD for para-Hermitian polynomial matrices [13][14][15], and the fact that analytic functions can be arbitarily closely approximated by polynomials of sufficiently high order, the deflation concept appears viable for polynomial matrices. Therefore, the recently proposed polynomial equivalent of the power method for para-Hermitian polynomial matrices [16], and its extension to general polynomial matrices [17], motivate the extension of Hotelling's deflation to polynomial matrices. With this extension, the polynomial power method can be utilized for a partial or full PEVD of a para-Hermitian polynomial matrix.…”