Newton-Raphson technique

A significant advantage is that the constrained optimization can usually be carried out using only the first derivative of tlie energy. This avoids an explicit, and computationally expensive, calculation of the second derivative matrix, as is nomially required by Newton-Raphson techniques. 

Changes in free energy and the equilibrium constants for Reactions 1, 2, 3, and 4 are quite sensitive to temperature (Figures 2 and 3). These equilibrium constants were used to calculate the composition of the exit gas from the methanator by solving the coupled equilibrium relationships of Reactions 1 and 2 and mass conservation relationships by a Newton-Raphson technique it was assumed that carbon was not formed. Features of the computer program used were as follows (a) any pressure and temperature may be specified (b) an inert gas may be present (c) after... 

Case Run Block - This program group is used to solve the resulting program for the Program Generator. Imbedded in it are heuristics to take user input data and solve the complex models by a sophisticated Newton-Raphson technique. 

The system of non-linear equations has been solved by the multidimensional Newton-Raphson technique which involves the successive solution of a linear system of equations ... 

One method would be to use Eq. (3.62) and utilise a Newton-Raphson technique to perform a Gibbs energy minimisation with respect to the composition of either A or B. This has an advantage in that only the integral function need be calculated and it is therefore mathematically simpler. The other is to minimise the difference in potential of A and B in the two phases using the relationships... 

In this section we will discuss the specific mathematical techniques used to estimate chemical equilibria using the sequential approach, which is the foundation for all versions of the FREZCHEM model, except for versions 2 and 10 (see above). The techniques used to solve (find the roots of) the equilibrium relations can be grouped into three classes simple one-dimensional (1-D) techniques, Brents method for more complex 1-D cases, and the Newton-Raphson technique that is used for both 1-D and multidimensional cases. 

Kubicek, M. Hlavacek, V. Prochaska, F., "Global Modular Newton-Raphson Technique for Simulation of an Interconnected Plant Applied to Complex Rectification Columns" Chem. Eng. Science (1976)... 

Goldstein, R.P. and R.B. Stanfield, "Flexible Method for the Solution of Distillation Design Problems Using the Newton-Raphson Technique", I EC Process Design Development, vol 9, no 1, p 78 (June 1970)... 

Equation (14) is an implicit algebraic equation of the optimum relative diafiltration volume, Ud. It can be solved numerically by any one of a number of methods, e. g. Newton, Raphson Technique, (Lapidus, 1962). Once the value of Ud is determined, the optimum time cycles of the ultrafiltration and diafiltration stages, Tu and Td, can be calculated readily from Eqs. (12) and (5). 

In both the BP and SR methods (Secs. 4.2.5 and 4.2.7) the temperatures and the total flow rates are calculated separate of each other. An alternative is to calculate the temperatures and total flow rates together in a Newton-Raphson technique. The name 2N Newton comes from that there are two equations per stage for a total of 2 x N functions and variables per column for the Newton-Raphson. In all three of these approaches, the component flow rates are still calculated in an intermediate step. 

Oscillation in the column variables This occurs where the temperature and flow rate profiles swing widely either side of what should be the final answer, often in the Newton-Raphson-based methods. Oscillation is caused by too large a Step in the profiles from one column trial to the next. This oscillation is prevented by limiting the step or percentage change in the MESH variables to below the amount generated by the Newton-Raphson technique. 

As a final variant the SCF procedure may be solved by a Newton Raphson technique, a very important component of which comprises a partial or complete 4-index tramsformation of integrals at each cycle. As we show below, the integral transformation procedure is highly vectorisable. We feel that such a technique will perhaps prove profitable in slowly convergent close shell cases or complicated open shell cases. 

The resulting equations describing mass and heat transport are highly non-linear algebraic equations which can be solved numerically using a common procedure such as the Newton-Raphson technique. 

Goldstein, R. P. and Stanfield, R. B., Flexible Method for the Solution of Distillation Design Problems using the Newton-Raphson Technique, Ind. Eng. Chem. Process Des. Develop., 9, 78-84 (1970). 

In this method, the equations describing the system are all solved simultaneously using, for instance, the Newton-Raphson technique. Applied to phase boundary conditions, the basic relationships in Equations 2.7, 2.8, and 2.12 take a special form. Since / is either zero at the bubble point or one at the dew point. Equation... 

Equations 12.17 and 12.18 comprise C -1- 2 equations which must be solved for the C + 2 variables. The entire set of equations could be solved simultaneously using the Newton-Raphson technique. In an alternative method which simplihes the calculations, equation set 12.18 is expressed as functions of N and B ... 

The stage temperatures are determined from the calculated stage compositions by bubble point calculations. The calculations are carried out iteratively using Muller s algorithm (Wang et al., 1966). The authors have determined that convergence by this method is more reliable than the Newton-Raphson technique. 

This method (Tomich, 1970) differs from the foregoing methods mainly in that the summation statements and the energy balances (Equations 13.3 and 13.4) are solved simultaneously. The benefits of the simultaneous solution are twofold. First, distillation columns and absorbers and columns that are hybrids of both types of processes can all be solved with the same method. Second, different types of column performance specifications can be incorporated in the simultaneous solution of the equations. The method is also computationally stable and efficient because it uses Broyden s modification of the Newton-Raphson technique for solving the equations (Broyden, 1965). A brief description of the method follows ... 

In the classical Newton-Raphson technique, the Jacobian matrix is inverted every iteration in order to compute the corrections AT] and Al]. The method of Tomich, however, uses the Broyden procedure (Broyden, 1965) in subsequent iterations for updating the inverted Jacobian matrix. 

The Newton-Raphson technique is also modified in this method. A damping factor a, between zero and one, is applied to the corrections as above. The way a is calculated, however, is different from the Naphtali-Sandholm method. 

Usually, p is chosen to be a number between 4 and 10. In this way the system moves in the best direction in a restricted subspace. For this subspace the second-derivative matrix is constructed by finite differences from the stored displacement and first-derivative vectors and the new positions are determined as in the Newton-Raphson method. This method is quite efficient in terms of the required computer time, and the matrix inversion is a very small fraction of the entire calculation. The adopted basis Newton-Raphson method is a combination of the best aspects of the first derivative methods, in terms of speed and storage requirements, and the more costly full Newton-Raphson technique, in terms of introducing the most important second-de-... 

A wide variety of iterative solution procedures for solving nonlinear algebraic equations has appeared in the literature. In general, these procedures make use of equation partitioning in conjunction with equation tearing and/or linearization by Newton-Raphson techniques, which are described in detail by Myers and Seider. The equation-tearing method was applied in Section 7.4 for computing an adiabatic flash. 

To save computertime, a modified Newton-Raphson technique is often used, in which case the tangential matrix is not (always) changed after each time step [l26j,... 

Equation 7.11 represents a set of two possibly nonlinear algebraic equations in terms of yj(0) and y2(0). This algebraic problem can be solved by trial and error, using for example the Newton-Raphson technique (i pendix A). This will yield y/O) and y2(0), which will form the initial conditions for Eq. 7.9. At this point, we need to assume that the numerical (approximate) solution of Eq. 7.11 will give rise to an initial condition, which will produce a trajectory that is arbitrarily close to the one with the exact initial condition. 

The optimization is then no longer a linear problem and becomes a difficult optimization with the usual caveats such as the existence of local minima. Amat et al. used the elementary Jacobi rotation technique as an optimization technique to find the proper coefficients for the ASA s-type Gaussians. Other techniques can be applied as well, such as the Newton-Raphson technique. " Lists of s-type Gaussian coefficients and exponents may be fovmd on the Internet for different basis sets. ... 

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