<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The class of ABS methods was originally developed for solving systems of linear equations in a finite number of iterations. Later it was shown that methods for solving nonlinear systems of equations, linear programming algorithms, quadratic programming, etc. are also members of the ABS class. In this paper, we show that QR factorization can also be derived from three ABS subclasses, which is an essential step in methods solving eigenvalue problems. Another possibility to use the ABS class in the eigenvalue problems is to transform a matrix to Hessenberg form which is an important initial step when solving eigenvalue problems by QR-type methods. Here we study versions of the plain QR-method which are based on the QR factorization computed by the ABS methods and compare them with the MATLAB eig () function. We also compare the ABS Hessenberg method to the MATLAB hess() function. The preliminary results show the competitiveness of 3 ABS eigenvalue solving methods with the MATLAB eig () function.</p>
</abstract>

The class of ABS methods was originally developed for solving systems of linear equations in a finite number of iterations. Later it was shown that methods for solving nonlinear systems of equations, linear programming algorithms, quadratic programming, etc. are also members of the ABS class. In this paper, we show that QR factorization can also be derived from three ABS subclasses, which is an essential step in methods solving eigenvalue problems. Another possibility to use the ABS class in the eigenvalue problems is to transform a matrix to Hessenberg form which is an important initial step when solving eigenvalue problems by QR-type methods. Here we study versions of the plain QR-method which are based on the QR factorization computed by the ABS methods and compare them with the MATLAB eig () function. We also compare the ABS Hessenberg method to the MATLAB hess() function. The preliminary results show the competitiveness of 3 ABS eigenvalue solving methods with the MATLAB eig () function.

Solution of the real and complex eigenvalue problems in the ABS class

Pure Mathematics and Applications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Solution of the real and complex eigenvalue problems in the ABS class"}

The class of ABS methods was originally developed for solving systems of linear equations in a finite number of iterations. Later it was shown that methods for...

Solution of the real and complex eigenvalue problems in the ABS class

Published Online: Nov 07, 2022

Page range: 92 - 109

Received: Jul 30, 2021

DOI: https://doi.org/10.2478/puma-2022-0028

Keywords
lienar algebra, complex eigenvalue problems, ABS methods

© 2022 József Abaffy, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Solution of the real and complex eigenvalue problems in the ABS class

Published Online: Nov 07, 2022

Page range: 92 - 109

Received: Jul 30, 2021

DOI: https://doi.org/10.2478/puma-2022-0028

Keywordslienar algebra, complex eigenvalue problems, ABS methods

© 2022 József Abaffy, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
lienar algebra, complex eigenvalue problems, ABS methods