The Newton Algorithm

Thus, the Newton algorithm for the nonlinear regularized least-squares inversion can be expressed by the formula... 

Evaluation of (5.1-37) requires several iterations. Helpful is, for instanee, the Newton Algorithm, see Stiehlmair and Fair (1998). 

We discuss thoroughly in Chapter 4 the feasibility of time independent approaches for computing the ABC Green s function. We eventually come full circle, in the end developing a technique called the Newton algorithm similar in spirit to the PSG, but vastly improved in efficiency. We test the Newton algorithm on the calculation of initial state selected reaction probabilities for the three dimensional D-I-H2 reaction, and find both rapid convergence and strict accuracy control. 

The Newton algorithm wa.s applied to calculating initial state selected reaction probabilities for three dimensional D-fH2(u,i) — DH-I-H with zero total angular momentum. We found that the probabilities with initial j = 1 were the largest, and attributed this effect to a small amount of orbital angular momentum helping to focus the system into more reactive geometries. 

To speculate on the requirements for larger systems, we recall that the operation count for the Newton algorithm is NiterinF -f- l)iV, where... 

In the next Chapter, we apply the Newton algorithm to the calculation of numerically exact cross sections and rate constants for the vibrationally excited reaction D-fH2(v = l,i) — DH-fH. We show that the ABC Newton method is the most direct route to date for accurate reaction cross sections. 

A brief summary of the Newton algorithm appeared previously in the following article ... 

We now discuss the formalism used in the present Chapter to obtain the initial state selected rate constant for an atom-diatom reaction. We briefly review the general rate constant formulae and the ABC method of obtaining reaction probabilities. The Newton algorithm for the ABC Green s function was thoroughly discussed in the previous Chapter. 

The basic idea of the Newton algorithm is this given an initial guess, call it x to a root of/(x) = 0, a refined guess, x, is computed based on the x-intercept of the line tangent tof x) at xK That is, consider the equation of the line tangent to fix) at x (this is just the Taylor series expansion of the function ignoring all but linear terms) ... 

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