Optimizing Matrix Operations: Unveiling the Efficiency of Sparse Representations below a Third Density

This paper investigates the critical decision of choosing between dense and sparse matrix representations in computational environments. Drawing upon a clearly defined criterion, we demonstrate that employing a sparse matrix representation is advantageous for memory and time efficiency when the number of non-zero elements is less than or equal to approximately one-third of the total matrix elements. A comprehensive comparative analysis is provided, detailing the time and space complexities for various fundamental matrix operations under both dense and sparse representations, thereby quantitatively justifying the proposed approach. This work underscores the significant benefits of intelligent matrix representation in addressing the escalating demands of large-scale scientific and data-intensive computations.