A constructive method for the solution of the stability problem

Howland, James Lucien; Senez, John Albert

doi:10.1007/bf02162400

Cited by 12 publications

(3 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…[4] give a similar treatment of the block Arnoldi process; however, their algorithm computes an orthogonal basis of the entire controllable subspace (i.e., the Krylov space K(A, B)), the construction of Am is carried out using (20) rather than by the Arnoldi process and finally, they do not consider (19). In our algorithm, we use their implementation of (16) for the updates given in (17). This ensures that (19) (14) is then given by (21)…”

mentioning

confidence: 99%

Krylov Subspace Methods for Solving Large Lyapunov Equations

Jaimoukha¹,

Kasenally²

1994

SIAM J. Numer. Anal.

266

192

View full text Add to dashboard Cite

Abstract. This paper considers several methods for calculating low-rank approximate solutions to large-scale Lyapunov equations of the form AP+PA +BB 0. The interest in this problem stems from model reduction where the task is to approximate high-dimensional models by ones of lower order. The two recently developed Krylov subspace methods exploited in this paper are the Arnoldi method [Saad, Math. Comput., 37 (1981)

show abstract

mentioning

confidence: 99%

Krylov Subspace Methods for Solving Large Lyapunov Equations

Jaimoukha¹,

Kasenally²

1994

SIAM J. Numer. Anal.

266

192

View full text Add to dashboard Cite

show abstract

“…10 for n = 4). Thus for n > 2 we generalize the idea used in [7] for special matrices Z, W . We choose some block of entries of X 11 , calculate the remaining entries and then we obtain the appropriate values of chosen entries as a solution to some linear system, see [18] for details.…”

Section: Solution To the Lyapunov Equationmentioning

confidence: 99%

Application of the infinitely many times repeated BNS update and conjugate directions to limited-memory optimization methods

Vlček¹,

Lukšan²

2019

Programs and Algorithms of Numerical Mathematics 19

View full text Add to dashboard Cite

show abstract

“…Howland und Senez [6] haben ein Verfahren entwickelt, bei dem die Stabilitiitsfrage durch direkte L6sung dieser Liapunovgleichung ftir reelle Matrizen gel/Sst wird; diese Methode wird yon Meyer-Spasche ( [7], [9]) auf komplexe Matrizen erweitert. Howland und Senez [6] haben ein Verfahren entwickelt, bei dem die Stabilitiitsfrage durch direkte L6sung dieser Liapunovgleichung ftir reelle Matrizen gel/Sst wird; diese Methode wird yon Meyer-Spasche ( [7], [9]) auf komplexe Matrizen erweitert.…”

Section: Fsbersichtunclassified

Criteria for the Liapunov stability and their numerical application

Elsaesser

Niethammer

1977

Computing

View full text Add to dashboard Cite

Zusammenfassung --AbstractKriterien for die Liapunovstabilit/it und ihre numerische Anwendung. Drei verschiedene Methoden zur numerischen L6sung des Stabilit/itsproblems, d. h. der Frage, ob die Eigenwerte einer reellen oder komplexen Matrix A in der linken Halbebene liegen, werden verglichen: Die Bereehnung des charakteristischen Polynoms yon A mit der Anwendung des Routh-Kriteriums, die direkte L6sung der Liapunovgleichung und die Transformation yon A auf die sogenannte Schwarzsche Form; aus der letzteren wird ein Routh-Schema fiir komplexe Polynome hergeleitet. Ferner l~iBt sich dutch eine geeignete M6bius-Transformation eine Methode zur EinschlieBung von Eigenwerten komplexer Matrizen angeben, die bei einer Matrix aus einem 6konometrischen Modell erprobt wird. Criteria for the Liapunov Stability and Their Numerical Application.There are considered three different approaches for solving numerically the stability problem, i.e. the question whether the eigenvalues of a matrix A have negative real parts: Calculating the characteristic polynomial and applying the Routh scheme, direct solution of the Liapunov equation and transforming A to the so-called Schwarz form. From the latter a Routh scheme for complex polynomials is derived. Further, by an appropriate M6bius transformation, an inclusion method for the eigenvalues of a complex matrix is proposed and applied to a matrix resulting from an econometric model.

show abstract

A constructive method for the solution of the stability problem

Cited by 12 publications

References 4 publications

Krylov Subspace Methods for Solving Large Lyapunov Equations

Krylov Subspace Methods for Solving Large Lyapunov Equations

Application of the infinitely many times repeated BNS update and conjugate directions to limited-memory optimization methods

Criteria for the Liapunov stability and their numerical application

Contact Info

Product

Resources

About