This chapter provides a survey of the classical algorithms for the numerical solution of algebraic Riccati and related matrix equations. Our attention is focused on algorithms based on invariant subspace techniques and on functional iterations. We describe in details the derivation of these algorithms for the CARE and briefly explain how to adapt them to the other cases. The chapter starts with the description and analysis of algorithms for solving linear matrix equations such as Sylvester, Lyapunov, and Stein equations and their generalizations. Then it continues with the study of the main classical techniques for solving algebraic Riccati equations, more specifically, methods based on Schur decomposition, Newton's iteration, iterative refinement, and the matrix sign iteration.

### 3. Classical Algorithms

#### Abstract

This chapter provides a survey of the classical algorithms for the numerical solution of algebraic Riccati and related matrix equations. Our attention is focused on algorithms based on invariant subspace techniques and on functional iterations. We describe in details the derivation of these algorithms for the CARE and briefly explain how to adapt them to the other cases. The chapter starts with the description and analysis of algorithms for solving linear matrix equations such as Sylvester, Lyapunov, and Stein equations and their generalizations. Then it continues with the study of the main classical techniques for solving algebraic Riccati equations, more specifically, methods based on Schur decomposition, Newton's iteration, iterative refinement, and the matrix sign iteration.
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Bini, DARIO ANDREA; Iannazzo, Bruno; Meini, Beatrice
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11568/1032581`
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