A fast method to estimate the Moore-Penrose inverse for well-determined numerical rank matrices based on the Tikhonov regularization

‎This paper introduces a novel approach for estimating the Moore-Penrose inverse‎. ‎The method proposed relies on Tikhonov regularization‎, ‎which requires the computation of all positive singular values of an \(m \times n\) matrix‎. ‎Additionally‎, ‎we present a highly efficient and accurate procedure for estimating these singular values‎. ‎This procedure assumes the well-determined numerical rank of matrices \(A^*A\) (if \(m \geq n\)) and \(AA^*\) (if \(m \leq n\))‎. ‎Furthermore‎, ‎we demonstrate the application of our proposed method in solving linear discrete well-posed problems‎. ‎The paper concludes with numerical simulations to illustrate the advantages of our novel approach‎. ‎Notably‎, ‎we compare the execution time associated with our technique to that of some relevant methods in the existing literature‎, ‎demonstrating that our method outperforms others in terms of computational efficiency‎. ‎To further substantiate our findings‎, ‎we conduct computational experiments to measure execution time and speedup‎. ‎The results affirm the efficiency of our proposed method‎, ‎showcasing reduced execution times compared to other methods‎. ‎This contributes to establishing our approach's practical viability and effectiveness in diverse applications‎.