Convergence of the conjugate gradient method with computationally convenient modifications

Abstract

The conjugate gradient (CG) method has been well analyzed [6] as a method of solving linear operator equations; its practical aspects in this regard have also been thoroughly reported for systems of linear algebraic equations in an excellent s tudy b y ENGELI, et al. [2]. For nonlinear equations, it is possible to implement the CG idea in several ways [3, 4, 7a, 7b, 8]; earlier papers [la, 1hi described the theoretical aspects of one particular implementation, while the present report continues with a discussion of the effect on the convergence of introducing certain computat ional ly convenient alterations. Numerical examples are given comparing different methods of solution and describing certain difficulties tha t m a y arise with CG methods for finite systems of (nonlinear) algebraic equations. Let 1 J(x) be a norm continuous operator in x from a real separable Hilbert space ~ with inner product ( . , • ) into ~ ; for each x in ~ let there exist the Frechet derivative J~ with range ~ and Gateaux derivative J~'. We assume tha t IIJ~lt and fIJ~'ll are uniformly bounded and tha t J~ is a self-adjoint uniformly positive definite operator, i.e. O < a I < ~ J ~ A I , lI/2'fl T~e CG iteration to solve J(x) = 0 is as follows: