Steepest Decsent for Generalised and Regularised Solution of Linear Operator Equations.

Let H1 H2 be Hilbert spaces, T a bounded linear operator on H1 into H2 such that, the range of T, R (T) is closed. Let T* denote the adjoint of T. In this paper, we review the convergence of the method of the steepest descent to a solution of the equation T*Tx= T*b,b =H2, for any initial approximation x0=H1. The method converges to the unique minimum norm, or generalised, solution if, and only if, x0 is in the range of T*. Further, we establish the convergence of the method steepest descent to the unique regularised solution (T*T+ui) 1Tb,b=H2 if x0 is in the range of T*