Order parameters, convergence characteristics

In order to improve the convergence characteristics and robustness of the Gauss-Newton method, Levenberg in 1944 and later Marquardt (1963) proposed to modify the normal equations by adding a small positive number, y2, to the diagonal elements of A. Namely, at each iteration the increment in the parameter vector is obtained by solving the following equation... 

There is, of course, no more than in the closed-shell SCF procedure, any guarantee that the proposed iteration will converge but the Hamiltonian in (6.5.17) possesses a remarkable flexibility, by virtue of the arbitrary parameters it contains, and this flexibility may be exploited in order to improve the convergence characteristics of the process. Before discussing such possibilities, it is convenient to write (6.5.17) in a simpler and more general form. 

In general, no second-order correction can ensure 0 L convergence, but in the special case where b,if eff,i = fi),3f eff,3 (most notably, whenever the first and last zones have the same physical characteristics, even if their lengths differ), some terms in the general solution s second derivative with respect to L2 at 0 vanish, and then the corrected values (after elimination of Deff,3) will ensure an 0 L discrepancy between the three-zone TAP reactor and the TSTR reactor. Note that all parameters are now influenced by k ... 

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