On Approximating Matrix Inverse

In some real-world applications such as 3D prints, computing the inverse of a linear transformation is of interest. Also, to solve some nonlinear matrix equations, a sequence converging to the inverse matrix is needed. In this paper, we present an effective algorithm for approximating the inverse matrix. An optimization problem is introduced for finding the inverse matrix. The steepest-descent method in conjunction with Barzilai-Borwein step length (SDBBI) is suggested to solve the optimization problem. We show the global R-linear convergence of the algorithm. We also suggest using conjugate gradient method instead of steepest-descent (CGBBI) to provide linearly independent steps. To accelerate the convergence, we then suggest using Newton method in final iterations which results in the hybrid algorithm HBBNI. We implement three algorithms, SDBBI, CGBBI and HBBNI, using MATLAB 2020 on a windows 11 computer with 16GB RAM and compare our proposed algorithms with well-known matrix decompositions, the LU factorization and the SVD. Numerical results on medium size problems confirm the efficiency of CGBBI algorithm in approximating the inverse matrix faster and more accurate than existing methods while for larger matrices, HBBNI performs the best.