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- W2385763106 abstract "By employing and modifying the well-known method of . B. KmTopmwi in de- aling with Newton's method for functional equations,the author has established in this paper a few convergence theorems of the gradient method concerning non-li- near minimization problems. The successive approximation method of gradient for the equation G(X)=0, X=(x_1,…, x_n)is known to be The following results are typical(cf.(1948)): Theorem 1.Let G(X)eC~2 and let G(X) satisfy the following conditions: 1) G(X_o)=η_o 2) in the sphere(S):‖X-X_o‖≤H(Bη_o)~we have where Then there exists a solution X~* of G(X)=0 in the sphere(S) which can be approximated by Xn of the gradient method with the following error estimation Theorem 2.Suppose that 1) ‖((dG)/(dX))_(x_o)‖≥C_o=1/B_o. 2) |G(X_o)|/‖((dG)/(dX))_(x_o)‖≤η_o. 3) in the sphere(C):‖X-X_o‖≤N(h_o)η_o={[1-(1-2h_o)~(1/2)]/h_o}η_o,the function G(X) has bounded second derivative: ‖(d~2G)/(dX~2)‖≤K, where (h_o=B_oη_oK≤(1/2). Then there is a solution X~* of G(X)=0 in (C) which can be approximated by X_n of the gradient method with the error estimation: The author has also considered certain modified forms of the gradient method, of which certain practical advantage,when compared with resuit,has been mentioned.Finally,a necessary and sufficient condition for equivalence betwe- en the Newton method and the gradient method has been obtained. The paper is closed with a numerical illustration." @default.
- W2385763106 created "2016-06-24" @default.
- W2385763106 creator A5044229427 @default.
- W2385763106 date "1955-01-01" @default.
- W2385763106 modified "2023-09-25" @default.
- W2385763106 title "ON THE CONVERGENCE OF THE GRADIENT METHOD FOR NON-LINEAR MINIMIZATION PROBLEMS" @default.
- W2385763106 hasPublicationYear "1955" @default.
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