Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X +A∗X-1A = Q

Engwerda, Jacob; Ran, André C. M.; Rijkeboer, Arie L.

doi:10.1016/0024-3795(93)90295-y

Cited by 190 publications

(120 citation statements)

References 8 publications

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“…Note that r(T) ≤ T ≤ 2r(T ) (see [9], for example). The following lemma has been proved in [3]. Proof.…”

Section: Matrix Pencilsmentioning

confidence: 98%

“…These two equations have been studied recently by several authors (see [1], [2], [3], [4], [16], [17]). For the application areas in which the equations arise, see the references given in [1] and [4].…”

Section: Introductionmentioning

confidence: 99%

“…It is proved in [3] that if (1.1) has a positive definite solution, then it has a maximal Hermitian solution X + and a minimal Hermitian solution X − . Indeed, we have 0 < X − ≤ X ≤ X + for any Hermitian solution X of (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…When the matrix A is nonsingular, the minimal positive definite solution of (1.1) can be found via the maximal solution of another equation of the same type (cf. [3,Thm. 3.3]).…”

Section: Introductionmentioning

confidence: 99%

“…3.3]). In [3], an algorithm was presented to find the minimal solution of equation (1.1) for the case where A is singular. The algorithm was based on a recursive reduction process.…”

Section: Introductionmentioning

confidence: 99%

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Iterative solution of two matrix equations

1999

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Abstract. We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations X + A * X −1 A = Q and X − A * X −1 A = Q, where Q is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation. Numerical results are reported to illustrate the convergence behaviour of various algorithms.

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“…Note that r(T) ≤ T ≤ 2r(T ) (see [9], for example). The following lemma has been proved in [3]. Proof.…”

Section: Matrix Pencilsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…When the matrix A is nonsingular, the minimal positive definite solution of (1.1) can be found via the maximal solution of another equation of the same type (cf. [3,Thm. 3.3]).…”

Section: Introductionmentioning

confidence: 99%

“…3.3]). In [3], an algorithm was presented to find the minimal solution of equation (1.1) for the case where A is singular. The algorithm was based on a recursive reduction process.…”

Section: Introductionmentioning

confidence: 99%

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Iterative solution of two matrix equations

1999

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

Poloni

2020

GAMM-Mitteilungen

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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 Q k = X 2 k of 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.

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Efficient estimation of Bayesian VARMAs with time‐varying coefficients

Chan

Eisenstat

2017

J of Applied Econometrics

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Summary Empirical work in macroeconometrics has been mostly restricted to using vector autoregressions (VARs), even though there are strong theoretical reasons to consider general vector autoregressive moving averages (VARMAs). A number of articles in the last two decades have conjectured that this is because estimation of VARMAs is perceived to be challenging and proposed various ways to simplify it. Nevertheless, VARMAs continue to be largely dominated by VARs, particularly in terms of developing useful extensions. We address these computational challenges with a Bayesian approach. Specifically, we develop a Gibbs sampler for the basic VARMA, and demonstrate how it can be extended to models with time‐varying vector moving average (VMA) coefficients and stochastic volatility. We illustrate the methodology through a macroeconomic forecasting exercise. We show that in a class of models with stochastic volatility, VARMAs produce better density forecasts than VARs, particularly for short forecast horizons.

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Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X +A∗X-1A = Q

Cited by 190 publications

References 8 publications

Iterative solution of two matrix equations

Iterative solution of two matrix equations

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

Efficient estimation of Bayesian VARMAs with time‐varying coefficients

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