Quasi-Newton equation

In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation and its analogue = Aq must hold (where - g " and... 

In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation = HAq and its analogue H Ag - = Aq - must hold (where Ag - = g - g and similarly for Aq - ). These equations, which have only n components, are obviously insufficient to determine the n(n + l)/2 independent components of the Hessian or its inverse. Therefore, the updating is arbitrary to a certain extent. It is desirable to have an updating scheme that converges to the exact Hessian for a quadratic function, preserves the quasi-Newton conditions obtained in previous steps, and—for minimization—keeps the Hessian positive definite. Updating can be performed on either F or its inverse, the approximate Hessian. In the former case repeated matrix inversion can be avoided. All updates use dyadic products, usually built... 

The development of an SC procedure involves a number of important decisions (1) What variables should be used (2) What equations should be used (3) How should variables be ordered (4) How should equations be ordered (5) How should flexibility in specifications be provided (6) Which derivatives of physical properties should be retained (7) How should equations be linearized (8) If Newton or quasi-Newton hnearization techniques are employed, how should the Jacobian be updated (9) Should corrections to unknowns that are computed at each iteration be modified to dampen or accelerate the solution or be kept within certain bounds (10) What convergence criterion should be applied ... 

In Chapter 4 the Gauss-Newton method for systems described by algebraic equations is developed. The method is illustrated by examples with actual data from the literature. Other methods (indirect, such as Newton, Quasi-Newton, etc., and direct, such as the Luus-Jaakola optimization procedure) are presented in Chapter 5. 

In the quasi-Newton method (secant method) the approximate model analogous to Equation (5.7) to be solved is... 

The quasi-Newton approximates fix) as a straight line (examine Figure 5.3) as xq —>xp, m approaches the second derivative of f(x). Thus Equation (5.9) imitates Newton s method... 

Quasi-Newton methods start out by using two points xP and jfl spanning the interval of jc, points at which the first derivatives of fix) are of opposite sign. The zero of Equation (5.9) is predicted by Equation (5.10), and the derivative of the function is then evaluated at the new point. The two points retained for the next step are jc and either xP or xP. This choice is made so that the pair of derivatives / ( ), and either/ (jc ) or/ ( ), have opposite signs to maintain the bracket on jc. This variation is called regula falsi or the method of false position. In Figure 5.3, for the (k + l)st search, x and xP would be selected as the end points of the secant line. 

Quasi-Newton. The application of Equation (5.10) yields the following results (examine Figure E5.2b). Note how the shape of / (x) implies that a large number of iterations are needed to reach jc. Some of the values of / (jc) and jc during the search are shown in the following table notice that jc remains unchanged in order to maintain the bracket with/ (jc) > 0. 

Procedures that compute a search direction using only first derivatives of/provide an attractive alternative to Newton s method. The most popular of these are the quasi-Newton methods that replace H(x ) in Equation (6.11) by a positive-definite approximation W ... 

We have referred to quasi-Newton methods rather than the quasi-Newton method because there are multiple definitions that can be used for the function F in this expression. The details of the function F are not central to our discussion, but you should note that this updating procedure now uses information from the current and the previous iterations of the method. This is different from all the methods we have introduced above, which only used information from the current iteration to generate a new iterate. If you think about this a little you will realize that the equations listed above only tell us how to proceed once several iterations of the method have already been made. Describing how to overcome this complication is beyond our scope here, but it does mean than when using a quasi-Newton method, the convergence of the method to a solution should really only be examined after performing a minimum of four or five iterations. 

Therefore, the shelf life is the root smaller than 28.90. A simple and practical tool to compute the roots of Equation (12) is perhaps solving the following equivalent problem. Find such that it minimizes the absolute value of /( ). This root is obtained by using the quasi-Newton line search (QNLS) algorithm [13]. The computer program requires an initial point and we recommend using the value... 

The quasi-Newton methods estimate the matrix = H-1 by updating a previous guess of C in each iteration using only the gradient vector. These methods are very close to the quasi-Newton methods of solving a system of nonlinear equations. The order of convergence is between 1 and 2, and the minimum of a positive definite quadratic function is found in a finite number of steps. 

The solutions of the above non-linear equations can be obtained using the Broyden quasi-Newton method [18], Based on the mass balance, the concentrations of other component, such as CO, H2, H2O and so on, can be calculated from the key components [9],... 

The concentration and temperature profiles are calculated from the above non-linear equations using the Broyden quasi-Newton method. The effectiveness factors for the catalyst pellet may be expressed as... 

The ij element of the Jacobian represents the partial derivative of equation i with respect to variable j. If analytical derivatives are not available, elements of the Jacobian are obtained by perturbation of the state variable, requiring n + 1 function evaluations for an -equation system of equations. Various quasi-Newton techniques provide approximations to the Jacobian and do not require as many function evaluations, thus reducing computational time. 

Schubert, L. K., "Modification of a Quasi-Newton Method for Nonlinear Equations with Sparse Jacobian", Math. Comp. (1970) 2 27-30. 

Get a new set of stripping factors by solving the energy balances and specification equations as the independent functions of the quasi-Newton technique of Broyden (Sec. 4.2.6). The derivatives of the Jacobian matrix are generated numerically and must include steps 4, 5, and 6 each time the independent variables are perturbed, The Jacobian is not recalculated after the first trial in the loop but is instead updated by Broyden s equation. 

Equation (26) is the "Quasi-Newton" condition it is fundamental to all the updating formula. There have been many updates proposed, and we briefly review some of the more important ones. The simplest are based on... 

Other than the problem of the selection of the value of h, the only additional disadvantage of a quasi-Newton method is that additional function evaluations are needed on each iteration k. Eq. (L, 12) can be applied to sets of equations if the partial derivatives are replaced by finite difference approximations. 

Equation 15 was used as a constraint with a value between 12 and 13 for Z (n-decane conversion), during optimization of the reaction variables, using a Non-linear Quasi-Newton search method with tangential extrapolation for estimates, forward differencing for estimation of partial derivatives, a tolerance of 0.05 and precision of 0.0005. The search was also constrained by boundary conditions 1 to -1 for the reaction variables x, and solved for maximization of Y . 

Step 6. Compute the SR s from the defining equation given beneath Eq. (5-39). Compare these values with the last assumed values. If they do not agree within an acceptable tolerance, assume a new set of values and return to step 2. A quasi-Newton method was used to find the new set of SR s. 

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