#### Volume 54, pp. 68-88, 2021.

## On the solution of the nonsymmetric T-Riccati equation

Peter Benner and Davide Palitta

### Abstract

The nonsymmetric T-Riccati equation is a quadratic matrix equation where the linear part corresponds to the so-called T-Sylvester or T-Lyapunov operator that has previously been studied in the literature. It has applications in macroeconomics and policy dynamics. So far, it presents an unexplored problem in numerical analysis, and both theoretical results and computational methods are lacking in the literature. In this paper we provide some sufficient conditions for the existence and uniqueness of a nonnegative minimal solution, namely the solution with component-wise minimal entries. Moreover, the efficient computation of such a solution is analyzed. Both the small-scale and large-scale settings are addressed, and Newton-Kleinman-like methods are derived. The convergence of these procedures to the minimal solution is proven, and several numerical results illustrate the computational efficiency of the proposed methods.

Full Text (PDF) [602 KB], BibTeX

### Key words

T-Riccati equation, M-matrices, minimal nonnegative solution, Newton-Kleinman method

### AMS subject classifications

65F30, 15A24, 49M15, 39B42, 40C05

### Links to the cited ETNA articles

[17] | Vol. 33 (2008-2009), pp. 53-62 M. Heyouni and K. Jbilou: An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation |

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