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Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction
Author(s) -
Poloni Federico
Publication year - 2020
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.202000018
Subject(s) - mathematics , algebraic riccati equation , riccati equation , matrix (chemical analysis) , linear quadratic regulator , algebraic equation , lyapunov function , type (biology) , nonlinear system , optimal control , mathematical analysis , differential equation , mathematical optimization , ecology , materials science , composite material , biology , physics , quantum mechanics
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 iterateQ k = X2 kof another naturally‐arising fixed‐point iteration ( X h ) 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 ) −1 A , continuous‐time algebraic Riccati equations Q + A ∗ X + X A − X G X = 0 , palindromic quadratic matrix equations A + Q Y + A ∗ Y 2 = 0 , and nonlinear matrix equations X + A ∗ X −1 A = 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.