Interpolation Newton method

Determine the relative rates of convergence for (1) Newton s method, (2) a finite difference Newton method, (3) quasi-Newton method, (4) quadratic interpolation, and (5) cubic interpolation, in minimizing the following functions ... 

At step k, this results in a rank k update to the Hessian, as opposed to the rank 1 and 2 formulas used in the quasi-Newton methods, leading to improved convergence properties. The line search is avoided by using a constrained quartic polynomial to estimate the position of the minimum. The energy and gradient at the line search minimum are obtained by interpolation rather than by recalculation. 

As expected, both steps are opposite the gradient for small t. Thus, the level-shifted Newton method and quadratic steepest descent differ only in the interpolation between small steps (which are opposite the gradient) and large steps (which are equivalent to the Newton step). 

Efficient single-variable (or one-dimensional) optimization methods include Newton and quasi-Newton methods and polynomial approximation (Edgar et al., 2001). The second category includes quadratic interpolation, which utilizes three points in the interval of uncertainty to fit a quadratic polynomial to f x) over this interval. Let and Xc denote three values of x in... 

Example 3.1 Gregory-Newton Method for Interpolation of Equally Spaced Data. 

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

Pseudo-Newton-Raphson methods have traditionally been the preferred algorithms with ab initio wave function. The interpolation methods tend to have a somewhat poor convergence characteristic, requiring many function and gradient evaluations, and have consequently primarily been used in connection with semi-empirical and force field methods. 

The four relations cited in Step 22 are solved simultaneously by trial to find the temperature of the gas. Usually it is in the range 1500-1800°F. The Newton-Raphson method is used in the program of Table 8.18. Alternately, the result can be obtained by interpolation of a series of hand calculations... 

THE PROGRAM USES THE NEWTON-GREGORY FORWARD AND BACKWARD INTERPOLATIONS AND STIRLING S CENTRAL DIFFERENCE METHOD. 

Newton-Raphson methods can be combined with extrapolation procedures, and the best known of these is perhaps the Geometry Direct Inversion in the Iterative Subspace (GDIIS), which is directly analogous to the DIIS for electronic wave functions described in Section 3.8.1. In the GDIIS method, the NR step is not taken from the last geometry but from an interpolated point with a corresponding interpolated gradient based on the previously calculated points on the surface. 

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