The Newton-Raphson method applied to solutions

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. 

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In practice, of course, the surface is only quadratic to a first approximation and so a number of steps will be required, at each of which the Hessian matrix must be calculated and inverted. The Hessian matrix of second derivatives must be positive definite in a Newton-Raphson minimisation. A positive definite matrix is one for which all the eigenvalues are positive. When the Hessian matrix is not positive definite then the Newton-Raphson method moves to points (e.g. saddle points) where the energy increases. In addition, far from a mimmum the harmonic approximation is not appropriate and the minimisation can become unstable. One solution to this problem is to use a more robust method to get near to the minimum (i.e. where the Hessian is positive definite) before applying the Newton-Raphson method. 

In this method the equations desalbing 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 spedal form. Since y/ is either zero at the bubble point or one at the dew point. Equation 2.7 reduces to the trivial relationship Z = Xj or Z = Yj. Equation 2.8 need not be involved in the simultaneous solution of the system equations. Once the phase boundary conditions are determined, this equation may be solved independently to calculate the heat duty required to bring the feed to its bubble point or dew point. Thus, the only equation to be solved for bubble or dew point calculations is Equations 2.12. 

The problem of finding stationary points of a potential energy surface is an old one, and numerous methods have been developed to solve it. The most obvious method is applying the Newton-Raphson method to the equation W = 0. The Newton-Raphson method tends to yield a solution whenever... 

An alternative to the above exploration will be to treat Eq. (1.43) as another nonlinear equation in Lx and to apply the Newton-Raphson method for its solution ... 

After the last column trial of a given system trial has been performed, the capital 0 method for systems is applied to find a new set of product-component-flow rates which satisfy all of the system component-material balances simultaneously while being in agreement with the specified values of the terminal flow rates W and (k = 1, 2, 3, 4). The solution set of pis is determined by finding the set of positive 0 s that make g = g = 0, (k = 1, 2, 3, 4), simultaneously. This set of 0 s may be found by use of the Newton-Raphson method the equations for this method are represented by... 

The Newton-Raphson method is formulated first for an absorber in which one chemical reaction occurs per plate. Then, the method is modified as required to describe distillation columns in which chemical reactions occur. Although the resulting algorithm is readily applied to systems which are characterized by nonideal solution behavior, it is an exact application of the Newton-Raphson method for those systems in which ideal or near ideal solution behavior exists throughout the column. The algorithm presented is recommended for absorption-type columns which exhibit ideal or near ideal solution behavior. 

In Chaps. 2 through 5, the theta methods and variations of the Newton-Raphson method are applied to all types of single columns and systems of columns in the service of separating both ideal and nonideal solutions. Applications of the techniques presented in Chaps. 2 through 5 to systems of azeotropic and extractive distillation columns are presented in Chap. 6. An extension of these same techniques as required for the solution of problems involving energy exchange between recycle streams is presented in Chap. 7. Special types of separations wherein the distillation process is accompanied by chemical reactions are treated in Chap. 8. 

As applied to the solution of a new set of T) values from the energy equation (15-5), the recursion equation for the Newton-Raphson method is... 

The Crank-Nicolson method for the numerical solution of the boundary value problem defined by Eqs.(6)-(8) is valid for the case of discontinuous coefficients, k(T) and pCp(T), as obtains, or nearly obtains, at Tg. This method provides the temperature distribution at time t+ot, given the distribution at time t, as the solution of a certain non-linear system of algebraic equations, which are solved with the use of the Newton-Raphson method and the Thomas Algorithm. The Crank-Nicolson method is more easily (and more generally) applied to the heat equation with boundary conditions of the kind given by Eq.(8), rather than by Eq.(7). For the numerical solutions by the Crank-Nicolson method discussed below, then, the boundary condition. 

In this case, however, it is not possible to solve the nonlinear Equation A10.38 analytically, but a numerical algorithm, for example, the Newton-Raphson method for the solution of nonlinear equations, should be applied (Appendix 1). A general way to solve the nonlinear regression problem Equation A10.36 is to vary the value of a systematically by an optimum search method until the minimum is attained. This method is called nonlinear regression, and it is illustrated in Figure A 10.6. 

Applying the Newton-Raphson method in a nonlinear finite element system will yield results only in the pre-collapse range, but it will fail to give information about the post-collapse response. To circumvent this limitation, a constraint can be added into the finite element system, which relates the load increment and the incremental displacements within each iteration (Fig. 12). This technique allows the calculation of the whole equilibrium path, even beyond the critical limit points. A number of different solution algorithms have been proposed in the literature (Riks 1979 Crisfield 1981 Ramm 1981 Bathe and Dvorkin 1983). 

Method of Solution In the Newton-Raphson method with synthetic division, Eq. (1.42) is used for evaluation of each root. Eqs. (1.53)-(1.55) are then applied to perform synthetic division in order to extract each root from the polynomial and reduce the latter by one degree. When the nth-degree polynomial has been reduced to a quadratic... 

NLP methods provide first and second derivatives. The KKT conditions require first derivatives to define stationary points, so accurate first derivatives are essential to determine locally optimal solutions for differentiable NLPs. Moreover, Newton-Raphson methods that are applied to the KKT conditions, as well as the task of checking second-order KKT conditions, necessarily require second-... 

The steepest-descent method does converge towards the expected solution but convergence is slow in the vicinity of the minimum. In order to scale variations, we can use a second-order method. The most straightforward method consists in applying the Newton-Raphson scheme to the gradient vector of the function/to be minimized. Since the gradient is zero at the minimum we can use the updating scheme... 

Many separations which would be difficult to achieve by conventional distillation processes may be effected by a distillation process in which a solvent is introduced which reacts chemically with one or more of the components to be separated. Three methods are presented for solving problems of this type. In Sec. 8-1, the 0 method of convergence is applied to conventional and complex distillation columns. In Sec. 8-2, the 2N Newton-Raphson method is applied to absorbers and distillation columns in which one or more chemical reactions occur per stage. The first two methods are recommended for mixtures which do not deviate too widely from ideal solutions. For mixtures which form highly nonideal solutions and one or more chemical reactions occur per stage, a formulation of the Almost Band Algorithm such as the one presented in Sec. 8-3 is recommended. 

Application. Both the Tomich and the 2iV Newton-Raphson methods are proven methods and have been applied often. The Tomich method was part of the GMB system of The Badger Company, Cambric, Massachusetts, and is in many in-house simulators. Both methods are best for wide- or middle-boiling s iarations. Because one of the equations in the 2N Newton-R hson method is a dew- or bubble-point equation, it may work better for middle or more narrow-boiling mixtures than the Tomich method. Both methods have also been eq>plied to absorber-strippers, thou an SR method is still the best method for the most wide-boiling absorber-strippers. Because of the full Jacobian more numbers to manipulate), for columns over 50 stages these methods will use excessive computer time and memoiy. Also, the solution of the Jacobian is prone to failure when the number of stages is high, and so these methods are not recommended for tall columns. 

Linearity vs nonlinearity. The meaning of linearity and superposition can be demonstrated by writing Equation 6-42 first for pressure P] and then for P2. The sum Pi + P2, by direct substitution, also satisfies Equation 6-42. This is not so with Equation 6-49 because the presence of P in ( )pm/kP causes Pi + P2 to satisfy an equation other than Equation 6-49. Thus, superposition does not hold for nonlinear systems like Equation 6-49, classic superposition methods for liquids do not apply to gases. On the other hand. Equation 6-49 takes a form nearly identical to that of Equation 6-42. For the purposes of numerical simulation. Equation 6-49 can be treated identically as for linear flows, provided we regard the m/P as a fictitious compressibility that is updated using the latest available values at the previous time step. This allows us to use the linear solver TRIDI at each time step, and avoids time consuming Newton-Raphson methods. This solution is numerically stable. 

In this section we consider how Newton-Raphson iteration can be applied to solve the governing equations listed in Section 4.1. There are three steps to setting up the iteration (1) reducing the complexity of the problem by reserving the equations that can be solved linearly, (2) computing the residuals, and (3) calculating the Jacobian matrix. Because reserving the equations with linear solutions reduces the number of basis entries carried in the iteration, the solution technique described here is known as the reduced basis method. ... 

In 1690, Raphson turned Newton s method into an iterative one, applying it to the solution of polynomial equations of degree up to ten. His formulation still did not use calculus instead he derived explicit polynomial expressions for f(x) and f (x). 

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