Newton-Raphson iterative technique

NEWTON-RAPHSON ITERATIVE TECHNIQUE, THE FINAL ITERATIONS ON EACH ROOT ARE PER,FORMED USING THE ORIGINAL FOLYNOMIAL RATHER THAN THE RF-DUCEU POLYNOMIAL TO AVOID ACCUMULATED ERRORS IN THE REDUCED POLYNOMIAL. 

From this equation, Mw is known implicitly and can be calculated using the Newton-Raphson iteration technique (35) ... 

A Newton-Raphson iteration technique is highly advantageous for treating the non-linear boundary conditions. 

Equation (49) contains two possible sources of nonlinearities material nonlinearity due to Schapery s law, and geometric nonlinearity arising from the large displacement (and small strain) formulation. In order to obtain a solution to this nonlinear equation at any time step, the Newton-Raphson iterative technique is used. The incremental displacement Aw obtained at the end of the iteration is used to update the total displacement for the time step,... 

For the explanation of symbols and details of the time approximation, see pp. 299-303 in Reddy. ( 7) xhe Newton-Raphson iterative technique is used to solve for the concentration c " at each time step. 

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. ... 

Fig. 4.4. Comparison of the computing effort, expressed in thousands of floating point operations (Aflop), required to factor the Jacobian matrix for a 20-component system (Nc = 20) during a Newton-Raphson iteration. For a technique that carries a nonlinear variable for each chemical component and each mineral in the system (top line), the computing effort increases as the number of minerals increases. For the reduced basis method (bottom line), however, less computing effort is required as the number of minerals increases.

The most commonly used numerical techniques are related to Newton-Raphson iteration. The guess for iteration k + 1 is determined from the value at iteration k, using ... 

As can be seen in Table I, algorithm MC-A does not converge in one iteration but almost quadratically in the same number of iterations as the full Newton-Raphson exponential technique. This is due, as it is shown in Table II, to the existence of a small coupling between the MO and Cl rotations after the first iteration which is significative at the degree of precission used (l.OxlO 8 a.u.). It must be noticed that for this... 

The one unknown in this equation is 0, which is calculated by an iterative technique such as Newton-Raphson s technique. The liquid and vapor mole fractions are then updated using the following equations ... 

This last equation can be solved by Newton-Raphson iteration or some other numerical technique. The first time through, it is simplest to set all activity coefficients equal to 1.0. Equation (19.11) is then solved for ttih then (19.9) is solved for mAc-, (19.7) for ttihac. and finally, (19.5) for mu. Notice that this takes us back through each of the equations outlined by a box these were the equations produced each time we first eliminated a variable. 

One simplifying approximation can be made for all non-zero concentrations of NaOH, the pH should be basic and we can omit from the charge balance (19.21). The above 7 equations can then be reduced to 1 non-linear equation in 1 unknown, which can be solved by a numerical technique such as Newton-Raphson iteration. Suitable numerical equation solvers are now available as software for personal computers. The range of solutions to these equations for different NaOH concentrations and temperatures is illustrated in Figure 19.1. At very high NaOH concentrations we would also have to consider the doubly deprotonated species H2Si04. ... 

Various procedures are available for accelerating the convergence of the modified Newton-Raphson iterations. Figure AIE.l shows the technique of computing individual acceleration factors, and <52 are known. Then, assuming a constant slope of the response curve, and from similar triangles, the value of <53 is computed ... 

If Equation 15-34 is to be written for each (i,j,k) node and solved at the new time step (n+1), we obtain a complicated system of algebraic equations that is costly to invert computationally. When it cannot be locally linearized, the full but sparse matrix is solved using even more expensive Newton-Raphson iterations. Thus, we employ approximate factorization techniques to resolve the system into three simpler, but sequential banded ones. In this approach. 

The kinetic equations such as Equation 7.37, Equation 7.38, Equation 7.40, and Equation 7.41 are known as transcendental equations, whose direct solution cannot be obtained. Such a kinetic equation is generally solved by the use of approximation techniques such as Newton-Raphson iterative method and nonlinear least-squares method. But, these methods have limitations of a different nature. For instance, the nonlinear least-squares method, which is most commonly used in such kinetic studies, tends to provide less reliable values of calculated kinetic parameters with increase in the number of such parameters. 

As stated, the most commonly used procedure for temperature and composition calculations is the versatile computer program of Gordon and McBride [4], who use the minimization of the Gibbs free energy technique and a descent Newton-Raphson method to solve the equations iteratively. A similar method for solving the equations when equilibrium constants are used is shown in Ref. [7],... 

The unknowns in Eq. (38) are p and 2n. These are found by demanding hm to obey the image constraint equations (18) and normalization (3). Because the unknowns enter in a nonlinear way, the resulting M + 1 equations were solved by an iterative technique—Newton-Raphson relaxation (see, e.g., Hildebrand, 1956). Empirical cases are studied in Section XI. 

Equation 2.14 may be solved by an iterative technique such as the Newton-Raphson method. The derivative of the function with respect to v at iteration A , / (V " ) equals the value of the function at iteration divided by the negative... 

The summations in Equations 2.20 and 2.21 are carried over all components in the system. If the temperature is fixed, the bubble point and dew point pressures may be calculated directly from Equations 2.20 and 2.21. If, however, the pressure is fixed and the bubble point or dew point temperature is required. Equation 2.20 or 2.21 must be solved by an iterative technique such as the Newton-Raphson method because the equations are implicit in the temperature. 

The material balance equation for component 3 can be solved separately once V is determined. The above three simultaneous equations are first solved for Yy Y2, and t /- They are nonlinear, involving products yy, and ti/Vj. Although they may be solved by elimination, a more general method is Newton-Raphson s multivariable iterative technique. First, rewrite the equations in the residual form ... 

The stage temperatures are determined from the calculated stage compositions by bubble point calculations. The calculations are carried out iteratively using Muller s algorithm (Wang et al., 1966). The authors have determined that convergence by this method is more reliable than the Newton-Raphson technique. 

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