A unified preconditioned minimal residual (PMR) algorithm for matrix problems: Linear systems, multiple right-hand sides linear systems, least squares problems, inversion and pseudo-inversion with application to color image encryption
In this article, we introduce a preconditioned minimal residual (PMR) algorithm designed to address a wide range of matrix equations and linear systems. We illustrate the efficacy of this algorithm through several numerical examples, including the solution of matrix equations. Notably, we tackle var...
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| Vydáno v: | Applied numerical mathematics Ročník 220; s. 216 - 245 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
01.02.2026
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| Témata: | |
| ISSN: | 0168-9274 |
| On-line přístup: | Získat plný text |
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| Abstract | In this article, we introduce a preconditioned minimal residual (PMR) algorithm designed to address a wide range of matrix equations and linear systems. We illustrate the efficacy of this algorithm through several numerical examples, including the solution of matrix equations. Notably, we tackle various significant problems such as the minimization of Frobenius norms, least squares optimization, and the computation of the Moore-Penrose pseudo-inverse. Convergence analysis shows that it converges without any constraints and for any initial guess, although this algorithm is more efficient when the matrices are sparse. To validate the effectiveness of our proposed iterative algorithm, we offer various numerical examples by large matrices. As an application of the matrix equation, we explore a method for encrypting and decrypting color images. |
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| AbstractList | In this article, we introduce a preconditioned minimal residual (PMR) algorithm designed to address a wide range of matrix equations and linear systems. We illustrate the efficacy of this algorithm through several numerical examples, including the solution of matrix equations. Notably, we tackle various significant problems such as the minimization of Frobenius norms, least squares optimization, and the computation of the Moore-Penrose pseudo-inverse. Convergence analysis shows that it converges without any constraints and for any initial guess, although this algorithm is more efficient when the matrices are sparse. To validate the effectiveness of our proposed iterative algorithm, we offer various numerical examples by large matrices. As an application of the matrix equation, we explore a method for encrypting and decrypting color images. |
| Author | Shirilord, Akbar Dehghan, Mehdi |
| Author_xml | – sequence: 1 givenname: Akbar orcidid: 0000-0001-5027-581X surname: Shirilord fullname: Shirilord, Akbar email: akbar.shirilord@aut.ac.ir – sequence: 2 givenname: Mehdi orcidid: 0000-0002-2573-9755 surname: Dehghan fullname: Dehghan, Mehdi email: mdehghan@aut.ac.ir, mdehghan.aut@gmail.com |
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| Keywords | 65F50 Multiple right-hand sides systems Heat equation 65F10 Matrix inversion 65Fxx 65F20 Optimal stepwise gradient algorithm Least squares solution Preconditioned minimal residual (PMR) algorithm Moore–Penrose pseudo-inverse 15A09 |
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| SubjectTerms | Heat equation Least squares solution Matrix inversion Moore–Penrose pseudo-inverse Multiple right-hand sides systems Optimal stepwise gradient algorithm Preconditioned minimal residual (PMR) algorithm |
| Title | A unified preconditioned minimal residual (PMR) algorithm for matrix problems: Linear systems, multiple right-hand sides linear systems, least squares problems, inversion and pseudo-inversion with application to color image encryption |
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