We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of doubling: they construct the iterate (Formula presented.) of another naturally-arising fixed-point iteration (Xh) via a sort of repeated squaring. The equations we consider are Stein equations X − A∗ X A = Q, Lyapunov equations A∗ X + X A + Q = 0, discrete-time algebraic Riccati equations X = Q + A∗ X(I + G X)−1A, continuous-time algebraic Riccati equations Q + A∗ X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A∗Y2 = 0, and nonlinear matrix equations X + A∗ X−1A = Q. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.

### Iterative and doubling algorithms for Riccati-type matrix equations: A comparative introduction

#### Abstract

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of doubling: they construct the iterate (Formula presented.) of another naturally-arising fixed-point iteration (Xh) via a sort of repeated squaring. The equations we consider are Stein equations X − A∗ X A = Q, Lyapunov equations A∗ X + X A + Q = 0, discrete-time algebraic Riccati equations X = Q + A∗ X(I + G X)−1A, continuous-time algebraic Riccati equations Q + A∗ X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A∗Y2 = 0, and nonlinear matrix equations X + A∗ X−1A = Q. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.
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2020
Poloni, F.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11568/1055062`
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