The interval Lyapunov matrix equation: analytical results and an efficient numerical technique for outer estimation of the united solution set. (English) Zbl 1255.15002

Hashemi, Behnam; Dehghan, Mehdi

Math. Comput. Modelling 55, No. 3-4, 622-633 (2012).

Summary: This note tries to propose an efficient method for obtaining outer estimations for the so-called united solution set of the interval Lyapunov matrix equation \(\mathbf AX+X\mathbf A^T=\mathbf F\), where \(\mathbf A\) and \(\mathbf F\) are known real interval matrices while \(X\) is the unknown matrix; all of dimension \(n \times n\). We first explore the equation in the more general setting of AE-solution sets, and show that only a small part of Shary’s results on the AE-solution sets of interval linear systems can be generalized to the interval Lyapunov matrix equation. Then, we propose our modification of Krawczyk operator which enables us to reduce the computational complexity of obtaining an outer estimation for the united solution set to cubic, provided that the midpoint of \(\mathbf A\) is diagonalizable.

MSC:

15A06	Linear equations (linear algebraic aspects)
65F30	Other matrix algorithms (MSC2010)
65G40	General methods in interval analysis

Keywords:

Lyapunov matrix equation; interval analysis; united solution set; AE-solution sets

Software:

Algorithm 432; ParLinSys; INTLAB

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