Showing papers on "Gâteaux derivative published in 1967"
TL;DR: In this paper, the convergence of the conjugate gradient method for finite systems of nonlinear algebraic equations has been discussed, and numerical examples are given comparing different methods of solution and describing certain difficulties that arise with CG methods.
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:
51 citations