Newton-Raphson generalization

Generalizing the Newton-Raphson method of optimization (Chapter 1) to a surface in many dimensions, the function to be optimized is expanded about the many-dimensional position vector of a point xq... 

By iteration, the general expression for the Newton Raphson method may be written (if f can be evaluated and is continuous near the root) ... 

The Newton-Raphson method has been applied to pipeline network problems since 1954 (Wl). Its performance has been generally very good, although convergence difficulties have been reported (S2), when starting from inappropriate initial guesses. In some cases large oscillations around... 

For the special case for which n = 2, it can be shown that the linearization method defined above becomes identical to the Newton-Raphson method. The result may be generalized to apply to any homogeneous function of degree n. 

The Newton-Raphson algorithm is further developed into a fairly generally applicable tool for the solving of sets of non-linear equations. 

The dispersion coefiicients can now be foimd from the Ni/q by a nonlinear calculation procedure, such as the Newton-Raphson method, utilizing the expressions for Nt, Eqs. (56) or (57). In the general case, values of Pli, Psit Pl2, Pr2, and the physical dimensions of the apparatus are substituted into Eq. (56) or (57), and then Pl and Pr can be found from the simultaneous (nonlinear) solution of the expressions for Ni and N2. The variances of the dispersion coefficients could also be found from the variances of the Ki by standard statistical methods. 

This process is easily generalized to systems with more variables and hence higher-dimension phase spaces. In general we specify some (N — l)-dimen-sional plane in the N-dimensional phase space and perform a similar single-cycle integration. We then have N — 1 differences, which form a vector Ax which is a function of the N — 1 initial concentration xq. We wish to make all the components of Ax zero, and the appropriate Newton Raphson form is then... 

Thus the Hessian will become singular if we include rotations between the active orbitals. Redundant parameters must not be included in the Newton-Raphson procedure.They are trivial to exclude for the examples given above, but in more general cases a redundant variable may occur as a linear combination of S and T and it might be difficult to exclude them. One of the advantages of the CASSCF method is that all parameters except those given above are non-redundant. 

Computationally the super-CI method is more complicated to work with than the Newton-Raphson approach. The major reason is that the matrix d is more complicated than the Hessian matrix c. Some of the matrix elements of d will contain up to fourth order density matrix elements for a general MCSCF wave function. In the CASSCF case only third order term remain, since rotations between the active orbitals can be excluded. Besides, if an unfolded procedure is used, where the Cl problem is solved to convergence in each iteration, the highest order terms cancel out. In this case up to third order density matrix elements will be present in the matrix elements of d in the general case. Thus super-CI does not represent any simplification compared to the Newton-Raphson method. 

In the block-diagonal Newton-Raphson minimization, the generally small size of the off-diagonal terms is exploited and the matrix describing the curvature is reduced to N 3 3 matrices, i.e., to 9N elements (Fig. 3.8). Due to the approximations... 

Note that, by using the quasi-linearization method, the solution of a non-linear problem can be reduced to solution of a succession of linear problems. The method is a further development of the Newton-Raphson method (Dulnev and Ushakovskaya, 1988) and its generalized version. 

The most widely used methods fall into two general categories (1) steepest descent and related methods such as conjugate gradient, which use first derivatives, and (2) Newton-Raphson procedures, which additionally use second derivatives. 

CASVB has been shown to be applicable to electronically excited states as well [18], by the adoption of a clever generalization [21] of the second-order stabilized Newton-Raphson optimization procedure [22] used by it, SC [4], and OBS-GMCSC [1][2]. It turns out that essentially the same procedure works for OBS-GMCSC as well [23], allowing it to exploit basis-set optimization where it can be especially advantageous, i.e. for excited states. 

Sparse Matrix Methods. In order to get around the limitations of the sequential modular architecture for use in design and optimization, alternate approaches to solving flowsheeting problems have been investigated. Attempts to solve all or many of the nonlinear equations simultaneously has led to considerable interest in sparse matrix methods generally as a result of using the Newton-Raphson method or Broyden s method (22, 23, 24 ). ... 

Despite these potential difficulties, efforts to attack this problem have been undertaken and some progress has been made. The nonlinear equations are generally attacked by methods (e.g. Newton-Raphson) which require periodic solution of linear equations. 

Class II Methods. The methods of Class II are those that use the simultaneous Newton-Raphson approach, in which all the equations are linearized by a first order Taylor series expansion about some estimate of the primitive variables. In its most general form, this expansion includes terms arising from the dependence of the thermo-physical property models on the primitive variables. The resulting system of linear equations is solved for a set of iteration variable corrections, which are then applied to obtain a new estimate. This procedure is repeated until the magnitudes of the corrections are sufficiently small. 

The integrated expression is not easily solved for C hence, unlike the case of first-order kinetics, no attempt is made to write a general expression for the accumulated residues. Instead, the equation was solved numerically for a range of values for the constants, Vm and Km, using the Newton-Raphson method for numerical approximation. This was programmed for a computer easily, although log table and slide rule or calculator will do the same job but in more time. 

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