The solution of linear systems by Richardson's method

where the problem to be solved is Ax = d and r k = d Ax k. The mechanization of this method suffers from three deficiencies: a) It requires, at the outset, the generation of the ~k'S, k = 0, i, °.. N where N is sufficiently large so as to insure convergence upon the calculation of XN+ I. b) It is an unstable numerical process in that a loss of significant figures prevents convergence when N gets moderately large. c) It requires bounds on the smallest and largest eigenvalue of the matrix A. A modification of this method has been suggested by Do Young and is also described in 0]where a three term recurrence relation supplants equation (1) so as to obtain: