Solving Rectangular Nonlinear System of Equation by Filtered Tikhonov Regularization

Abstract

The paper presents methods for solving a large-scale rectangular system of equations based on Tikhonov regularization procedures wherein, incorporated, the Singular Value Decomposition (SVD) as basis of numerical computation. We obtain the regularization parameter by adopting the Penrose-pseudo-inverse process with a view to diminishing occurrence of huge condition number appearing in the left-hand side of the equation for meaningful solution. We obtain the rank of a rectangular matrix as well as approximation of Low rank matrix, a very important tool in image reconstruction from the noisy data.  It is demonstrated that the symmetric matrix coming from the normal equation is reduced to a tridiagonal matrix using the Givens- QR - transformation process wherein, the norm of the inverse tridiagonal matrix may be obtained in an economical way