Newton-Raphson method convergence

If a function has more than one real root, the particular root to which the Newton-Raphson method converges will depend on the initial estimate chosen. Thus, to obtain a particular root, some guidance must be provided by the user. 

Applications of approximate formulations of the Newton-Raphson method such as the ones proposed by Sujata22 and Holland17 may fail to converge for some examples. For example, the 2N Newton-Raphson method converged for the test problem called Example 3 by Boyum4 while the approximate methods of Sujata and Holland failed. 

This equation must be solved for y The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL(Ref. 224). 

The advantage of this approach is that it is easier to program than a full Newton-Raphson method. If the transport coefficients do not vary radically, then the method converges. If the method does not converge, then it maybe necessary to use the full Newton-Raphson method. 

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 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... 

While it is technically erroneous to claim that the linearization method does not require any initialization (J2), it is true that the initialization procedure used appear to be quite effective. A more comprehensive discussion of initialization procedure will be given in Section III,A,5. With this initialization procedure, the linearization method appears to converge very rapidly, usually in less than 10 iterations for formulations A and B. Since the evaluation of f(x) and its partial derivatives is not required, the method is also simpler and easier to implement than the Newton-Raphson method. 

Finally, for formulation D the flows in the tree branches can be computed sequentially assuming zero chord flows. This initialization procedure was used by Epp and Fowler (E2) who claimed that it led to fast convergence using the Newton-Raphson Method. 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

Algorithmic Details for NLP Methods All the above NLP methods incorporate concepts from the Newton-Raphson method for equation solving. Essential features of these methods are that they rovide (1) accurate derivative information to solve for the KKT con-itions, (2) stabilization strategies to promote convergence of the Newton-like method from poor starting points, and (3) regularization of the Jacobian matrix in Newton s method (the so-called KKT matrix) if it becomes singular or ill-conditioned. 

The Newton-Raphson method consists in solving simultaneously the conservation and mass action equations. Because of its simplicity and rather fast convergence, it is well-fitted to sets of non-linear equations in several unknowns, as described in Chapter 3. 

The Muller method converges more quickly than the Newton-Raphson method when the functions have more curvature. However, it is more complex to program and more susceptible to numerical divergence problems. 

The convergence properties are similar to those of the Newton-Raphson method, usually with more iterations but less equivalent function evaluations. 

There are a few points with respect to this procedure that merit discussion. First, there is the Hessian matrix. With elements, where n is the number of coordinates in the molecular geometry vector, it can grow somewhat expensive to construct this matrix at every step even for functions, like those used in most force fields, that have fairly simple analytical expressions for their second derivatives. Moreover, the matrix must be inverted at every step, and matrix inversion formally scales as where n is the dimensionality of the matrix. Thus, for purposes of efficiency (or in cases where analytic second derivatives are simply not available) approximate Hessian matrices are often used in the optimization process - after aU, the truncation of the Taylor expansion renders the Newton-Raphson method intrinsically approximate. As an optimization progresses, second derivatives can be estimated reasonably well from finite differences in the analytic first derivatives over the last few steps. For the first step, however, this is not an option, and one typically either accepts the cost of computing an initial Hessian analytically for the level of theory in use, or one employs a Hessian obtained at a less expensive level of theory, when such levels are available (which is typically not the case for force fields). To speed up slowly convergent optimizations, it is often helpful to compute an analytic Hessian every few steps and replace the approximate one in use up to that point. For really tricky cases (e.g., where the PES is fairly flat in many directions) one is occasionally forced to compute an analytic Hessian for every step. 

SC (simultaneous correction) method. The MESH equations are reduced to a set of N(2C +1) nonlinear equations in the mass flow rates of liquid components ltJ and vapor components and the temperatures 2J. The enthalpies and equilibrium constants Kg are determined by the primary variables lijt vtj, and Tf. The nonlinear equations are solved by the Newton-Raphson method. A convergence criterion is made up of deviations from material, equilibrium, and enthalpy balances simultaneously, and corrections for the next iterations are made automatically. The method is applicable to distillation, absorption and stripping in single and multiple columns. The calculation flowsketch is in Figure 13.19. A brief description of the method also will be given. The availability of computer programs in the open literature was cited earlier in this section. 

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. 

The Newton-Raphson methods of energy minimization (Berkert and Allinger, 1982) utilize the curvature of the strain energy surface to locate minima. The computations are considerably more complex than the first-derivative methods, but they utilize the available information more fully and therefore converge more quickly. These methods involve setting up a system of simultaneous equations of size (3N — 6) (3N — 6) and solving for the atomic positions that are the solution of the system. Large matrices must be inverted as part of this approach. 

He showed that positive dX give d that reduce Gibbs free energy. This method is analogous to that of steepest descent, a first-order method for minimization of Gibbs free energy. Ma and Ship-man (11) used Naphtali s method to estimate compositions at equilibrium and the Newton-Raphson method to achieve convergence. 

In addition to the solution criteria in Sec. 4.2.3, the Newton-Raphson method requires an additional check on convergence, This check, termed the norm or the square root of the sum. of the squares, tests that all of the functions are driven nearly to zero ... 

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