DEVELOPMENT OF NEW ITERATIVE TECHNIQUES AND EXTRAPOLAnON METHODS FOR CONSISTENT OR INCONSISTENT SINGULAR LINEAR SYSTEMS

Consider the linear system of equations Bx=f , where B is an N xN singular matrix. In an earlier work [5] by the author it was shown that iterative techniques coupled with standard vector extrapolation methods can be used to obtain (approximations to) a solution of this sys­ tem when it is consistent. In the present work we expand on the approach of [5] to treat the case in which this system is in general inconsistent. We develop a family of new iterative techniques and vector extrapolation methods that enable us to obtain (approximations to) a unique vector that lies in the subspace spanned by those eigenvectors and principal vectors of B that belong to its nonzero eigenvalues. This vector turns out to be a solution in case B is diagonalizable and the system is consistent. When B is a nonnal matrix, this vector is the generalized inverse solution whether the system is consistent or not. We show that this vector can be constructed from a finite number of iterations, this number being at most N+2. We also provide detailed convergence analyses of the new iterative techniques and vector extra­ ~lation methods and give their precise rates ofconvergence. -