Low-rank updates and divide-and-conquer methods for quadratic matrix equations

In this work, we consider two types of large-scale quadratic matrix equations: continuous-time algebraic Riccati equations, which play a central role in optimal and robust control, and unilateral quadratic matrix equations, which arise from stochastic processes on 2D lattices and vibrating systems....

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Bibliographic Details
Published in:Numerical algorithms Vol. 84; no. 2; pp. 717 - 741
Main Authors: Kressner, Daniel, Kürschner, Patrick, Massei, Stefano
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2020
Springer Nature B.V
Subjects:
Algebra
Algorithms
Approximation
Computer Science
Eigenvalues
Iterative methods
Methods
Numeric Computing
Numerical Analysis
Original Paper
Quadratic equations
Riccati equation
Robust control
Stochastic processes
Theory of Computation
Hierarchical matrices
Divide-and-conquer
Riccati equation
Low-rank update
Unilateral quadratic matrix equation
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:In this work, we consider two types of large-scale quadratic matrix equations: continuous-time algebraic Riccati equations, which play a central role in optimal and robust control, and unilateral quadratic matrix equations, which arise from stochastic processes on 2D lattices and vibrating systems. We propose a simple and fast way to update the solution to such matrix equations under low-rank modifications of the coefficients. Based on this procedure, we develop a divide-and-conquer method for quadratic matrix equations with coefficients that feature a specific type of hierarchical low-rank structure, which includes banded matrices. This generalizes earlier work on linear matrix equations. Numerical experiments indicate the advantages of our newly proposed method versus iterative schemes combined with hierarchical low-rank arithmetic.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-019-00776-w