Scalar correction method for finding least-squares solutions on Hilbert spaces and its applications

We use the idea of two-point stepsize gradient methods, developed to solve unconstrained minimization problems on RnRn, for computing least-squares solutions of a given linear operator equation on Hilbert spaces. Among them we especially pay attention to corresponding modification of the scalar correction method. An application of this approach is presented related to computation of {1, 3} inverses and the Moore–Penrose inverse of a given complex matrix. Convergence properties of the general gradient iterative scheme for computation of various pseudoinverses are investigated. The efficiency of the presented algorithm is theoretically verified and approved by selected test matrices.