Newton-Raphson equation

Neumann boundary conditions, electronic states, adiabatic-to-diabatic transformation, two-state system, 304-309 Newton-Raphson equation, conical intersection location locations, 565 orthogonal coordinates, 567 Non-Abelian theory, molecular systems, Yang-Mills fields nuclear Lagrangean, 250 pure vs. tensorial gauge fields, 250-253 Non-adiabatic coupling ... 

We now have the expressions for the gradient and the Hessian matrix. The corresponding Newton-Raphson equations can then be written down in matrix form as ... 

One method, which avoids the problem with undesired negative eigenvalues of the Hessian, and which introduces an automatic damping of the rotations, is the augmented Hessian method (AM). To describe the properties of this method, let us again consider the Newton-Raphson equation (4 4) ... 

The super-CI method can alternatively be given in a folded form, which includes the coupling between the Cl and orbital rotations. This is done by adding the complementary Cl space, IK>, to the super-CI secular problem. As in the Newton-Raphson approach, it is more efficient to transform the equations back to the original CSF space, and thus work with a super-CI consisting of the Cl basis states plus the SX states. It is left to the reader as an exercise to construct the corresponding secular equation and compare it with the folded one-step Newton-Raphson equations (4 22). 

The best-known presentations are by Tomich (32), Holland (8), and Orbach et al. (33). These vary in their choice of Newton-Raphson equations and independent variables and each may solve a different range of columns, These methods have been shown to work well for wide-boiling mixtures including refinery fractionators, absorber-stripper columns, and reboiled absorbers. 

Substitution of this value of Ax into either of the Newton-Raphson equations yields... 

The 2N Newton-Raphson method may be applied to any type of distillation column or to any system of interconnected columns. Absorbers, strippers, reboiled absorbers, and distillation columns are treated in Sec. 4-1. Selected numerical methods for solving the 2N Newton-Raphson equations are presented in Sec. 4-2. In Sec. 4-3, two methods for solving problems involving systems of columns interconnected by recycle streams are presented. 

In the following formulation of the Newton-Raphson equations, the independent variables are taken to be the N-stage temperatures 7 and the N-ratios of the total-flow rates Lj/Vj. When the gas and liquid phases form ideal solutions, the procedure described is an exact application of the Newton-Raphson method. 

The Newton-Raphson equations for solving the 2N functions F, G/ for the 2N independent variables 0jy 7 may be represented by the matrix equation... 

Three procedures are presented in this chapter for solving the Newton-Raphson equations. Procedure 1 is presented below and procedures 2 and 3 are presented in Sec. 4-3. 

Procedure 1. Solution of the 2N Newton-Raphson Equations by Use of the Calculus of Matrices and LU Factorization... 

Another formulation of the Newton-Raphson method was proposed by Newman20 in 1963 in which the total-flow rates of the liquid Lj were taken as the independent variables and the corresponding sets of temperatures needed to satisfy the component-material balances and equilibrium relationships was found by successive application of the Newton-Raphson equations. The compositions and temperatures so obtained were used to solve the enthalpy balances explicitly for a new set of liquid rates. The procedure was then repeated by commencing with this most recent set of liquid rates Lj. ... 

NUMERICAL METHODS FOR SOLVING THE IN NEWTON-RAPHSON EQUATIONS... 

For the general case of n independent equations in n unknowns, the Newton-Raphson method may be formulated in terms of n functions in n unknowns. For the kth trial, the resulting set of Newton-Raphson equations may be represented as follows... 

Broyden s algorithm consists of successively updating of the jacobian matrix of the Newton-Raphson equations by use of the correction matrix xCy7, that is,... 

For example, the Newton-Raphson equation for any one function, say fjk (the equilibrium function for plate j and component fc, where k denotes a particular one of the c components), is... 

The complete set of Newton-Raphson equations may be stated in the following matrix form... 

The complete set of Newton-Raphson equations may be solved by transforming the matrix shown in Fig. 5-1 into the upper triangular matrix shown in Fig. 5-2. The triangularization procedure, to be described next, is based on gaus-sian elimination. 

If the approximations presented in this section are made, the number of nonzero elements in the shaded submatrices of Fig. 5-1 is smaller, and the effort required to evaluate many of the partial derivatives is significantly reduced. However, the general algorithm just presented for solving the Newton-Raphson equations may be used regardless of whether or not any or all of the approximations presented in this section are used. 

Advantage of this relationship may be taken by consideration of the following terms of the Newton-Raphson equation for the function G ... 

The [N(2c + 1) + 2] Formulation of the Newton-Raphson Equations for a Conventional Distillation Column... 

Since all derivatives may be evaluated numerically in Broyden s method,5 the necessity for programming the expressions needed for the derivatives appearing in the Newton-Raphson equations is avoided by use of these methods. The wide variety of thermodynamic packages which are available make these approaches very attractive. 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

An algorithm is given elow for solving the Newton-Raphson equations by use of only the LU factorization of J0 and the Broyden update terms given by Eqs. (5-29), (5-30), and (5-31). As shown in App. 5-1, this algorithm is based on the successive application of Householder s formula to Eq. (5-29). 

Solution of this example as originally implemented by Broyden5 would have required an excessive amtfnnt of computer time. The example is described by 452 Newton-Raphson equations whose jacobian matrix contains only 3532 nonzero elements out of a possible 204,304. Of these 3532 nonzero elements, 798 are known to be constants, generally 1 or — 1 due to the linearity of the equations. 

State the Newton-Raphson equations for the functions in Prob. 6-9, and formulate a calcula-tional procedure for computing corrected values of xt and x2 after improved values of 0, and 02 have been found on the basis of assumed sets of values for x, and x2. 

For a given set of 7 s and Lj/Vfs, it is desired to find the set of rj/s which satisfy the material balances [Eqs. (8-8) and (8-11)] and which make each Rj = 0. The solution set of rjjs may be found by use of the Newton-Raphson method which consists of the repeated solution of the Newton-Raphson equations... 

For each choice of 7 s and Ofs, the component-material balances and total-material balances were solved for the set of rjf s that made each of the R/s equal to zero. The desired set of rj/s were found by use of the Newton-Raphson method. In this case, the Newton-Raphson equations are of the form... 

Another significantly faster procedure for solving the Newton-Raphson equations, called the Broyden-Bennett method, was proposed by Hess et al.10 The Broyden-Bennett algorithm may be used to solve the equations for an absorber-type column accompanied by a chemical reaction in the following manner. 

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