Core Concepts
A new matrix algebra approach is proposed to represent and efficiently compute with quasi-Toeplitz matrices, providing substantial computational advantages over existing methods.
Abstract
The paper analyzes the structural and computational properties of the linear space Pα spanned by the powers of a semi-infinite tridiagonal matrix Aα. It is shown that any matrix in Pα can be represented as the sum of a symmetric Toeplitz matrix and a Hankel matrix, where the Hankel component depends on the parameter α.
Key highlights:
- A basis {P0, P1, P2, ...} of Pα is determined, where each Pn is the sum of a symmetric Toeplitz matrix and a Hankel matrix.
- It is proven that Pα is a matrix algebra, and conditions are provided for Pα to be a Banach algebra.
- For quasi-Toeplitz (QT) matrices with symmetric Toeplitz part, representing them as Pα(a) + KA, where Pα(a) is in the Pα algebra and KA is compact, is shown to be more efficient than the standard representation A = T(a) + EA.
- Experimental results demonstrate substantial acceleration in solving matrix equations and computing matrix square roots for QT matrices using the new representation compared to existing methods.
Stats
The 2-norm and infinity norm of matrices in Pα are bounded if |α| ≤ 1, or if the generating Laurent series is analytic in an annulus around the unit circle.
The computational cost of matrix arithmetic operations in Pα is O((m+n)log(m+n)), where m and n are the degrees of the involved Laurent polynomials.
Quotes
"Representing a QT matrix A = T(a) + EA, where T(a) is symmetric, in the form Pα(a) + KA, for KA compact, is more convenient from the theoretical and the computational point of view."
"From the computational point of view, we compared the representation of QT matrices given in the CQT-Toolbox, designed in [6], with the new representation based on the algebra Pα. As test problems, we considered the solution of a matrix equation with QT coefficients performed by means of fixed point iterations, and the computation of the square root of a real symmetric QT matrix. In both cases, with the new representation we obtain substantial acceleration in the CPU time with respect to the standard representation given in [6]."