An algorithm for the solution of ill-posed linear systems arising from the discretization of Fredholm integral equation of the first kind

An algorithm for obtaining approximate solutions of ill-posed systems of linear equations arising from the discretization of Fredholm integral equation of the first kind is described. The ill-posed system is first replaced by an equivalent consistent system of linear equations. The method calculates the minimum length least squares solution of the consistent system. Starting from rank = 1 of the consistent system, the rank is increased by one in succession and a new solution is calculated. This is repeated until a certain simple criterion is satisfied. Linear programming techniques are used for which successive solutions are the basic solutions in the successive simplex tableaux. The algorithm is numerically stable. Numerical results show that this method compares favorably with other direct methods.