Modified Newton-Raphson Method

This procedure is essentially applying the single variable Newton-Raphson method n times, once for each variable. Each time the other variables are held constant. The approach is, given 

Qf-ilOy is evaluated atxi and y. With x, and j /i is reused to calculate x-. Then fi and the most recent values of x and y are used to calculate [3]. Notice that the 

The real disadvantage in the modified Newton-Raphson method is the need to know a priori which function has the steeper slope at the solution point. 

In practice, a graph of the functions can help to identify the curve with the steeper slope at the solution point. If a graph is not available, then one has to resort to an arbitrary choice of functions, and if divergence occurs, switch the roles of/i and/2. 

In summary, whether we are faced with a single nonlinear equation or a system, it is very likely that only one starting value will be available. A systematic selection of that value can improve the likelihood of a convergent solution. Equation 9.7 and Equation 9.9 provide such a systematic approach. 

Since there is a difficulty of having to solve a completely new sytem of equations at each iteration an approximation can be introduced [(Fig. 3.8 Plate 3.2)]. This can be done to 

This method is elaborately explained in the text [(Fig. 3.9)]. It is realised that the solution for u is known when the load term is zero. Once the starting point is known, it will be useful to study the ii as / is incremented. For small inerements of F, convergence is highly likely. For the loading process, the intermediate computed results would be useful information. Thus the method begins with 

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Usually, modified Newton-Raphson methods with relaxation are applied. Additional iteration loops are necessary for the determination of the dynamic pressure losses in ducts and duct fittings. 

How can we accelerate this process We can use Fox s method (which is a modified Newton-Raphson method for two-point boundary-value differential... 

The result (10.25) is a nonlinear equation. We can introduce a modified Newton-Raphson method (Owen and Hinton 1980) for solving (10.25). Then we rewrite (10.25), and (10.26)-( 10.29)by inttoducing variables with superscripts k and k — l, which implies the values of the variables at each iteration step, and we have... 

Equation 8 now can be solved using the modified Newton-Raphson method. Once the displacements are obtained, the member forces can be calculated accordingly. 

The Newton-Raphson method may be computationally expensive in a multi dof problem because a new global stiffness matrix is used in each iterative step. In the Modified Newton-Raphson method, the same global stiffness matrix is used in all the iterative steps within an increment. This method requires more iterations to achieve convergence but each iteration is computed far more quickly. 

When the function to be fitted to data does not depend linearly on the parameters, recursive methods must be used. A slightly modified version of the Newton-Raphson method (Chapter 3) will be used (Hamilton, 1964). Let jc be the vector of the n unknowns Xj and y — f(x) the m-vector of observable functions y, = /(jc). The analytical form of the functions f(x) may be the same or not. Let the vector / represent the m observations / of these functions. A vector jc is sought which minimizes the scalar c2 such that... 

The Newton-Raphson approach is another minimization method.f It is assumed that the energy surface near the minimum can be described by a quadratic function. In the Newton-Raphson procedure the second derivative or F matrix needs to be inverted and is then usedto determine the new atomic coordinates. F matrix inversion makes the Newton-Raphson method computationally demanding. Simplifying approximations for the F matrix inversion have been helpful. In the MM2 program, a modified block diagonal Newton-Raphson procedure is incorporated, whereas a full Newton-Raphson method is available in MM3 and MM4. The use of the full Newton-Raphson method is necessary for the calculation of vibrational spectra. Many commercially available packages offer a variety of methods for geometry optimization. 

A number of iterative methods exist, as described in Appendix L. TK Solver uses a modified Newton-Raphson iterative procedure (see Sec. L.2), which is satis-fectory for a wide variety of problems. 

Modifying the fiow equations to speed up the Newton-Raphson method... 

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. 

This conventional form of the Newton-Raphson method was very sensitive to the set of starting values selected for the rjf s, and it was difficult to find a set for which the calculational procedure would converge to the solution set. In order to eliminate the sensitivity of the Newton-Raphson method to the starting values selected for the rjfs, it was necessary to modify the procedure in the following manner. 

This formulation of the Newton-Raphson method for columns with infinitely many stages is analogous to the 2N Newton-Raphson method for a column with a finite number of stages. First the procedure is developed for a conventional distillation column with infinitely many stages for which the condenser duty Qc (or the reflux ratio Lx/D) and the reboiler duty QR (or the boilup ratio VN/B) are specified and it is required to find the product distribution. Then the procedure is modified as required to find the minimum reflux ratio required to effect the specified separation of two key components. 

The quantities x, and yj are the iterants, whereas gi and g2 are formed exactly the way Equation 9.5 was developed. Two common methods for finding roots to nonlinear systems are (1) Newton-Raphson and (2) the modified Newton-Raphson. Both approaches are briefly discussed in the subsections below. 

This last equation is a nonlinear algebraic equation in D, it can be solved using the Newton-Raphson method [if the differentiation is very lengthy and cumbersome, which is the case here, then in the Newton-Raphson method, you can use the modified Newton-Raphson by approximating BF/BDaf by - T")/ (Z)" — )") or, easier and sure to converge, use the bisectional... 

The simplest numerical method for a detailed geometrically and material nonlinear (GMN) analysis is the Newton-Raphson scheme (Crisfield 1979 Bathe 1995), which can be found in three forms (i) the full Newton-Raphson, which is the most accurate, but also the most time consuming, since the tangent stiffness of the structure has to be calculated and factorized within each iteration in the solution procedure (ii) the modified Newton-Raphson, which differs from the full Newton-Raphson in that the calculation and the factorization of the tangent stiffness matrix take place only in some iterations within each step, thus requiring in most cases a larger number of iterations per step but... 

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