Iterative Matrix Techniques Based on Averages
Matrices have an important role in modern engineering problems like artificial intelligence, biomedicine, machine learning, etc. The present paper proposes new algorithms to solve linear problems involving finite matrices as well as operators in infinite dimensions. It is well known that the power m...
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MDPI AG
2025-07-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/18/7/439 |
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| author | María A. Navascués |
| author_facet | María A. Navascués |
| author_sort | María A. Navascués |
| collection | DOAJ |
| description | Matrices have an important role in modern engineering problems like artificial intelligence, biomedicine, machine learning, etc. The present paper proposes new algorithms to solve linear problems involving finite matrices as well as operators in infinite dimensions. It is well known that the power method to find an eigenvalue and an eigenvector of a matrix requires the existence of a dominant eigenvalue. This article proposes an iterative method to find eigenvalues of matrices without a dominant eigenvalue. This algorithm is based on a procedure involving averages of the mapping and the independent variable. The second contribution is the computation of an eigenvector associated with a known eigenvalue of linear operators or matrices. Then, a novel numerical method for solving a linear system of equations is studied. The algorithm is especially suitable for cases where the iteration matrix has a norm equal to one or the standard iterative method based on fixed point approximation converges very slowly. These procedures are applied to the resolution of Fredholm integral equations of the first kind with an arbitrary kernel by means of orthogonal polynomials, and in a particular case where the kernel is separable. Regarding the latter case, this paper studies the properties of the associated Fredholm operator. |
| format | Article |
| id | doaj-art-d7ad1a2dd94e442da5c02eae41c9cb7e |
| institution | DOAJ |
| issn | 1999-4893 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj-art-d7ad1a2dd94e442da5c02eae41c9cb7e2025-08-20T02:48:19ZengMDPI AGAlgorithms1999-48932025-07-0118743910.3390/a18070439Iterative Matrix Techniques Based on AveragesMaría A. Navascués0Department of Applied Mathematics, Universidad de Zaragoza, 50018 Zaragoza, SpainMatrices have an important role in modern engineering problems like artificial intelligence, biomedicine, machine learning, etc. The present paper proposes new algorithms to solve linear problems involving finite matrices as well as operators in infinite dimensions. It is well known that the power method to find an eigenvalue and an eigenvector of a matrix requires the existence of a dominant eigenvalue. This article proposes an iterative method to find eigenvalues of matrices without a dominant eigenvalue. This algorithm is based on a procedure involving averages of the mapping and the independent variable. The second contribution is the computation of an eigenvector associated with a known eigenvalue of linear operators or matrices. Then, a novel numerical method for solving a linear system of equations is studied. The algorithm is especially suitable for cases where the iteration matrix has a norm equal to one or the standard iterative method based on fixed point approximation converges very slowly. These procedures are applied to the resolution of Fredholm integral equations of the first kind with an arbitrary kernel by means of orthogonal polynomials, and in a particular case where the kernel is separable. Regarding the latter case, this paper studies the properties of the associated Fredholm operator.https://www.mdpi.com/1999-4893/18/7/439power methodeigenvalueslinear systemsiterative methodsFredholm integral equationsspectrum |
| spellingShingle | María A. Navascués Iterative Matrix Techniques Based on Averages Algorithms power method eigenvalues linear systems iterative methods Fredholm integral equations spectrum |
| title | Iterative Matrix Techniques Based on Averages |
| title_full | Iterative Matrix Techniques Based on Averages |
| title_fullStr | Iterative Matrix Techniques Based on Averages |
| title_full_unstemmed | Iterative Matrix Techniques Based on Averages |
| title_short | Iterative Matrix Techniques Based on Averages |
| title_sort | iterative matrix techniques based on averages |
| topic | power method eigenvalues linear systems iterative methods Fredholm integral equations spectrum |
| url | https://www.mdpi.com/1999-4893/18/7/439 |
| work_keys_str_mv | AT mariaanavascues iterativematrixtechniquesbasedonaverages |