Tree Quasi-Separable matrices

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Tree Quasi-Separable matrices

A simultaneous generalization of sequentially and hierarchically semiseparable representations

Journal Article (2025)

Author(s)

N. Govindarajan (Katholieke Universiteit Leuven, TU Delft - Team Michel Verhaegen)

Shivkumar Chandrasekaran (University of California–Santa Barbara)

P. Dewilde (Technische Universität München, TU Delft - Signal Processing Systems)

Research Group

Signal Processing Systems

DOI related publication

https://doi.org/10.1137/24M1682348

GIRS HSS matrices Quasi-separable matrices Rank-structured matrices SSS matrices TQS matrices

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https://resolver.tudelft.nl/uuid:9960c555-5634-4282-b355-5007fbf80d02

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Publication Year

2025

Language

English

Research Group

Signal Processing Systems

Bibliographical Note

Green Open Access added to TU Delft Institutional Repository as part of the Taverne amendment. More information about this copyright law amendment can be found at https://www.openaccess.nl. Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. @en

Issue number

2

Volume number

46

Pages (from-to)

1562-1586

Reuse Rights

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Abstract

We present a unification and generalization of what is known in the literature as sequentially and hierarchically semiseparable (SSS and HSS) representations for matrices. These so-called tree quasi-separable (TQS) matrices contain sparse matrices with tree-structured adjacency graphs as an important subcase. TQS matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction that the size of the representation is dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers. A sketch of the associated algorithms is provided.

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