The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices
The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices ar...
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2023-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/9/831 |
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| author | Mutti-Ur Rehman Jehad Alzabut Nahid Fatima Tulkin H. Rasulov |
| author_facet | Mutti-Ur Rehman Jehad Alzabut Nahid Fatima Tulkin H. Rasulov |
| author_sort | Mutti-Ur Rehman |
| collection | DOAJ |
| description | The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new results for the numerical approximation of the largest singular values corresponding to Bernstein–Vandermonde, Bernstein–Bezoutian, Cauchy—polynomial-Vandermonde and quasi-rational Bernstein–Vandermonde structured matrices. The numerical approximation for the reciprocal of the largest singular values returns the structured singular values. The new results for the numerical approximation of bounds from below for structured singular values are accomplished by computing the largest singular values of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices. Furthermore, we present the spectral properties of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and structured quasi-rational Bernstein–Vandermonde matrices by computing the eigenvalues, singular values, structured singular values and its lower and upper bounds and condition numbers. |
| first_indexed | 2024-03-10T23:02:15Z |
| format | Article |
| id | doaj.art-3cdbbe522b3f44a9b7487ec0c5d0b976 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T23:02:15Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-3cdbbe522b3f44a9b7487ec0c5d0b9762023-11-19T09:32:19ZengMDPI AGAxioms2075-16802023-08-0112983110.3390/axioms12090831The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde MatricesMutti-Ur Rehman0Jehad Alzabut1Nahid Fatima2Tulkin H. Rasulov3Department of Mathematical Analysis, Bukhara State University, Bukhara 200100, UzbekistanDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematical Analysis, Bukhara State University, Bukhara 200100, UzbekistanThe numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new results for the numerical approximation of the largest singular values corresponding to Bernstein–Vandermonde, Bernstein–Bezoutian, Cauchy—polynomial-Vandermonde and quasi-rational Bernstein–Vandermonde structured matrices. The numerical approximation for the reciprocal of the largest singular values returns the structured singular values. The new results for the numerical approximation of bounds from below for structured singular values are accomplished by computing the largest singular values of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices. Furthermore, we present the spectral properties of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and structured quasi-rational Bernstein–Vandermonde matrices by computing the eigenvalues, singular values, structured singular values and its lower and upper bounds and condition numbers.https://www.mdpi.com/2075-1680/12/9/831<i>μ</i>-valuessingular valueseigenvaluesstructured matrices |
| spellingShingle | Mutti-Ur Rehman Jehad Alzabut Nahid Fatima Tulkin H. Rasulov The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices Axioms <i>μ</i>-values singular values eigenvalues structured matrices |
| title | The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices |
| title_full | The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices |
| title_fullStr | The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices |
| title_full_unstemmed | The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices |
| title_short | The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices |
| title_sort | stability analysis of linear systems with cauchy polynomial vandermonde matrices |
| topic | <i>μ</i>-values singular values eigenvalues structured matrices |
| url | https://www.mdpi.com/2075-1680/12/9/831 |
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