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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Bibliographic Details
Published in:Applied numerical mathematics Vol. 220; pp. 216 - 245
Main Authors: Shirilord, Akbar, Dehghan, Mehdi
Format: Journal Article
Language:English
Published: Elsevier B.V 01.02.2026
Subjects:
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
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
ISSN:0168-9274
Online Access:Get full text
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Summary: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.
ISSN:0168-9274
DOI:10.1016/j.apnum.2025.10.012