Newton-Raphson methods extrapolations

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. 

MATLAB in the Optimization Toolbox has the routine f solve. This routine implements an advanced version of the Newton-Raphson method as described above. However, rather than just a linear extrapolation, it implements a mixed quadratic and cubic extrapolation procedure. To learn more about this function, type... 

A numerical technique that sometimes is useful is the Newton-Raphson root finding method (Figure 1.3). The basic idea is that if the slope of a line/curve y=f(x) is the first derivative dy/dx), maybe one can extrapolate the slope back to where the function crossed y = 0. In order to use this idea, you have to rearrange your equation so that y(x) = 0. Then the key equation is... 

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