  This article aims to present the $AT$ algorithm, a novel two-step iterative
approach for approximating fixed points of weak contractions within complete
normed linear spaces. The article demonstrates the convergence of $AT$
algorithm towards fixed points of weak contractions. Notably, it establishes
the algorithm's strong convergence properties, highlighting its faster
convergence compared to established iterative methods such as $S$, normal-$S$,
Varat, Mann, Ishikawa, $F^{*} $, and Picard algorithms. Additionally, the study
explores the $AT$ algorithm's almost stable behavior for weak contractions.
Emphasizing practical applicability, the paper offers data-dependent results
through the $AT$ algorithm and substantiates findings with illustrative
numerical examples
