Numerical methods Newton-Raphson method

The experimental results imply that the main reaction (eq. 1) is an equilibrium reaction and first order in nitrogen monoxide and iron chelate. The equilibrium constants at various temperatures were determined by modeling the experimental NO absorption profile using the penetration theory for mass transfer. Parameter estimation using well established numerical methods (Newton-Raphson) allowed detrxmination of the equilibrium constant (Fig. 1) as well as the ratio of the diffusion coefficients of Fe"(EDTA) andNO[3]. 

Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer, in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate/given x. One can estimate the value of the first derivative at Xq using... 

The computational procedure can now be explained with reference to Fig. 19. Starting from points Pt and P2, Eqs. (134) and (135) hold true along the c+ characteristic curve and Eqs. (136) and (137) hold true along the c characteristic curve. At the intersection P3 both sets of equations apply and hence they may be solved simultaneously to yield p and W for the new point. To determine the conditions at the boundary, Eq. (135) is applied with the downstream boundary condition, and Eq. (137) is applied with the upstream boundary condition. It goes without saying that in the numerical procedure Eqs. (135) and (137) will be replaced by finite difference equations. The Newton-Raphson method is recommended by Streeter and Wylie (S6) for solving the nonlinear simultaneous equations. In the specified-time-... 

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)]. 

Programs are written in QUICKBASIC for the Newton-Raphson method with analytical or numerical derivatives and the Wegstein method. The particular equation is entered following line 30. The one used here is from problem PI.02.01. 

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. 

According to the assumptions implied by the kinetic equation (9.3), the volume V and the concentrations cK,2 refer to organic phase. Moreover, Eq. (9.14) assumes that there is no change of volume due to mixing. This is a reasonable assumption in view of the data presented in Table 9.2. The system of Eqs. (9.11) to (9.14) is square and can be solved numerically, for example using the Newton-Raphson method. Table 9.4 presents typical results, for a reactor of 10 m3 operated at various temperatures. A large excess of i-butane (B) is necessary to achieve the required transformation. Butene is almost completely converted, while isobutane conversion is much lower. For this reason, the recycle contains mainly the excess isobutane. Moreover, the main reaction is favored by low temperatures. 

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. 

Find the value of 7 from Eq. (5) and compare with the assumed value. Apply the Newton-Raphson method with numerical derivatives to ultimately find the correct value of 7) and the corresponding value of ft. ... 

The derivatives in Eq. (22) are calculated numerically this requires K direct model evaluations. To solve the set of nonlinear equations (22), it is advantageous to use the Newton-Raphson method possessing quadratic rates of convergence and requiring no more than four to seven iterations [45]. 

One approach is to solve the equation x = P tanh x numerically, using the Newton-Raphson method or some other root-finding scheme. (See Press et al. (1986) for a friendly and informative discussion of numerical methods.)... 

The secant method is the Newton-Raphson method with a numerical version of the derivative based on the last... 

The Newton-Raphson method requires that you differentiate the function with respect to all the variables. The secant method avoids that mathematical step and uses a numerical difference to calculate the derivative ... 

The Newton-Raphson method requires differentiation of all data points with respect to the parameters. For flxed-geometry properties (like energy derivatives), the force field derivatives can be obtained analytically (32). For other types of properties, an approximate analytical solution can be obtained by assuming that the shift in geometry is small upon parameter change (45). However, the most general and safest method is to obtain the derivatives numerically (15). The drawback is that the method is substantially slower than calculating analytical derivatives. 

The Newton-Raphson method shows good convergence for parameters that display a strong penalty function curvature and are not too strongly interdependent. However, there are usually some parameters that will not be well converged by the method. This problem has been alleviated in some recent parameterization efforts (20,44) by alternating between optimization methods. In numerical schemes, the absolute second derivative of the penalty function with respect to each parameter is available from direct differentiation. It is assumed that a parameter will be badly determined by a Newton-Raphson step if this value is very low or negative. The 10-20 worst parameters are selected and subjected to a separate simplex optimization. 

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. 

Highly nonideal solutions are characterized by the fact that the activity coefficients and the partial molar enthalpies are strongly dependent upon composition. In order to compute the partial derivatives of these quantities which are needed in the application of the Newton-Raphson method, it is convenient to choose compositions or component-flow rates as members of the set of independent variables. Numerous choices of the independent variables have been made.6, lf 8 13,15 17 19-20 To demonstrate the formulation of the Newton-Raphson method, the choice of independent variables proposed by Naphtali and Sandholm17 is used. The Almost Band Algorithm may be formulated for other choices of independent variables as shown by Gallun and Holland.7,8 9... 

There follows the development of a class of methods proposed by Broyden2 for the purpose of overcoming some of the disadvantages of the Newton-Raphson method. One serious disadvantage of the Newton-Raphson method is the time required to evaluate all of the elements of the jacobian matrix Bk. Even if the functions f are sufficiently simple for their partial derivatives to be obtained analytically, the amount of labor required to evaluate all n2 of these may be excessive. If the partial derivatives of the yj s are approximated numerically, the amount of labor required constitutes a serious disadvantage. 

This method, which is also called the Newton-Raphson method, is an iterative procedure for obtaining a numerical solution to an algebraic equation. An iterative procedure is one that is repeated until the desired degree of accuracy is attained. The procedure is illustrated in Fig. 4.9. We assume that we have an equation written in the form... 

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