We propose a new matrix class named GSDD \(_k^*\) matrices, which forms a subclass of nonsingular H-matrices. This generalization extends classical strictly diagonally dominant (SDD) matrices by permitting controlled relaxation of diagonal dominance conditions through an iterative parameter k. Our investigation focuses on vital properties of the GSDD \(_k^*\) class and the relationships among GSDD \(_k^*\) matrix families for varying positive integers k. To enhance practical applicability, we propose a construction method that combines a positive diagonal matrix X with a given GSDD \(_k^*\) matrix A to transform the product AX into an SDD matrix. By leveraging existing results and techniques associated with S-SDD matrices, we address the challenge of handling parameters in practical applications. As a significant contribution, we establish a parameter-free upper bound for the infinite norm of the inverse of the GSDD \(_k^*\) matrix and derive a parameter-free error bound for the linear complementarity problem. Moreover, we formulate an efficient algorithmic framework to tackle the aforementioned issues. Subsequently, we show the effectiveness and practical utility of our proposed algorithms through numerical examples and experimental results.
