Advances in Chemical Physics, Vol.1 (Interscience 1958) by I., Editor Prigogine

By I., Editor Prigogine

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Test xk for convergence. 2. Compute a descent direction pk so that where q k controls the accuracy of the solution and some symmetric matrix ITkmay approximate Hk. 3. Compute a step length A so that for X k + l = xk + hpk, with 0 < OL < @ < 1. 4. Set x k + ] = xk + Xpk. 38 Optimization Methods in Computational Chemistry Newton variants are constructed by combining various strategies for the above. These involve procedures for formulating Hkor Hk,dealing with structures of indefinite Hessians, and solving for the modified Newton search direction.

For simplicity, all difference parameters {hi}may be taken as a fixed value h. As the difference formulation [38] will not necessarily produce a symmetric matrix f i k , whose columns are the vectors &,}, i = 1, . . , n, a symmetrizing procedure can be used to construct the matrix 6 - Hk = i(fik + a). WI With exact arithmetic, discrete Newton methods converge quadratically if each hi goes to zero as llgll does;6 however, the roundoff error limits the smallest feasible size of difference interval practice and, hence, the accuracy (a combination of roundoff and truncation errors) that can be obtained.

More generally, the shape of the elliptical contours near local minima depends on the eigenvalues { X i } and eigenvectors (v,} of the Hessian in that neighborhood. The axes of the elliptical contours point in the direction of the orthogonal eigenvectors, and the length of each axis corresponding to the ith eigenvector is proportional to l/Ai. Thus for f ( x )defined in Eq. In Figure 5d, for example, the axes of the ellipses are along the basis vectors, the eigenvectors corresponding to the diagonal matrix.

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