Analytic theory of differential equations; the proceedings by P. F. Hsieh, A. W. J. Stoddart

By P. F. Hsieh, A. W. J. Stoddart

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Extra info for Analytic theory of differential equations; the proceedings of the conference at Western Michigan University, Kalamazoo, from 30 April to 2 May 1970

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6. Let /e C2(RN) and suppose that gT(x)d = 0 and dTH(x)d < 0 for some x and d. Then d is a descent direction for / at x. 2). 7. Let /e Cl(RN). Then among all search directions d at some point x, that direction in which / descends most rapidly in a neighborhood of x isd = —g(x). PROOF: We want to minimize the directional derivative of / at x over all search directions. 3) that this is the same problem as that of minimizing gT(x)y for all y such that |y|| = 1. 2 THE METHOD OF STEEPEST DESCENT 11 /(x) is minimized on the line x = xk + rdk, — oo < T < oo for T = ik.

The conjugate gradient method has the property that x m = x for some m < N, where N is the order of H. PROOF: Suppose the contrary is true. 39) implies that these N + 1 iV-dimensional nonzero vectors are mutually orthogonal. Since this is impossible, the theorem is proved. 15), the conjugate gradient method has the property of finite termination. 18). There are two remarks to be made concerning the finite termination property in the context of practical computation. The first is that rounding errors prevent our obtaining x exactly and thus permit the iterations to continue for k > m.

If H is positive definite, then all of its eigenvalues are positive and the range of / is [c, oo). Clearly, z = 0 is the strong global minimizer of /. If H is negative definite, then the range of / is (— oo, c] and z = 0 is the strong global maximizer. If H has both positive and negative eigenvalues, then the range of / is (— oo, oo). / is still stationary at z = 0 but possesses no minimizer or maximizer there. We consider again the case in which H is positive definite. For k > c the level surface Lk is the ellipsoid where £ = 2(k — t) > 0, as sketched in Fig.

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