@article{Tansri2023, 
author = {Kanjanaporn Tansri and Pattrawut Chansangiam},
title = {Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {5},
pages = {11781-11798},
keywords = {gradient-descent, iterative method, least-squares solution, weighted norm, convergence analysis},
url = {https://www.sciopen.com/article/10.3934/math.2023596},
doi = {10.3934/math.2023596},
abstract = {Consider a linear system    A  x  =  b where the coefficient matrix    A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices    W. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when    W  =  I. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time.}
}