The Generalized Schur Algorithm and Some Applications

Laudadio, Teresa; Mastronardi, Nicola; Dooren, Paul Van

doi:10.3390/axioms7040081

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…The generalized Schur method that is classified as a deflating subspace method is used to solve DARE. The generalized Schur algorithm is a strong algebraic tool that allows computing classical decompositions of matrices, such as the QR and LU factorizations [42] The next algorithm was used to solve DARE [43]:…”

Section: Solution To the Riccati Equation Using Matlabmentioning

confidence: 99%

Kalman Filter Estimation and Its Implementation

Ulin-Avila¹,

Ponce-Hernandez²

2021

Adaptive Filtering - Recent Advances and Practical Implementation

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In this chapter, we use the Kalman filter to estimate the future state of a system. We present the theory, design, simulation, and implementation of the Kalman filter. We use as a case example the estimation of temperature using a Resistance Temperature Detector (RTD), which has not been reported before. After a brief literature review, the theoretical analysis of a Kalman filter is presented along with that of the RTD. The dynamics of the RTD system are analytically derived and identified using Matlab. Then, the design of a time-varying Kalman filter using Matlab is presented. The solution to the Riccati equation is used to estimate the future state. Then, we implement the design using C-code for a microprocessor ATMega328. We show under what conditions the system may be simplified. In our case, we reduced the order of the system to that of a system having a 1st order response, that of an RC system, giving us satisfactory results. Furthermore, we can find two first order systems whose response defines two boundaries inside which the evolution of a second order system remains.

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Section: Solution To the Riccati Equation Using Matlabmentioning

confidence: 99%

Kalman Filter Estimation and Its Implementation

Ulin-Avila¹,

Ponce-Hernandez²

2021

Adaptive Filtering - Recent Advances and Practical Implementation

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“…In [1], the authors study the generalized Schur algorithm (GSA), which allows to compute well-known matrix decompositions, such as the QR and LU factorizations. In particular, they use the GSA to obtain new theoretical insights on the bounds of the entries of the matrix R in the QR factorization of some structured matrices, with related applications.…”

Section: Numerical Linear Algebramentioning

confidence: 99%

Advanced Numerical Methods in Applied Sciences

Brugnano

Iavernaro

2019

Axioms

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The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.

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“…This matrix is not required to have any particular structure, but we assume that A is close to a banded symmetric positive definite Toeplitz matrix T + . It is natural to use the matrix T + as a preconditioner, because linear systems of equations with a banded symmetric positive definite Toeplitz matrix can be solved rapidly and stably by exploiting the Toeplitz structure by Schur or generalized Schur algorithms described in [1,2,8,27,29] 1 , as well as by the method by Bini and Meini [4]. When the matrix A is symmetric positive definite, we can solve (5.1) by the preconditioned conjugate gradient method using T + as a preconditioner; see, e.g., [17,Algorithm 11.5.1].…”

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confidence: 99%

On the banded Toeplitz structured distance to symmetric positive semidefiniteness

Noschese

Reichel

2022

ELA

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This paper is concerned with the determination of a close real banded positive definite Toeplitz matrix in the Frobenius norm to a given square real banded matrix. While it is straightforward to determine the closest banded Toeplitz matrix to a given square matrix, the additional requirement of positive definiteness makes the problem difficult. We review available theoretical results and provide a simple approach to determine a banded positive definite Toeplitz matrix.

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The Generalized Schur Algorithm and Some Applications

Cited by 5 publications

References 23 publications

Kalman Filter Estimation and Its Implementation

Kalman Filter Estimation and Its Implementation

Advanced Numerical Methods in Applied Sciences

On the banded Toeplitz structured distance to symmetric positive semidefiniteness

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