By Newton-Raphson method

EX232 2.3.2 Reaction equilibrium by Newton-Raphson method M14,M15,M31... 

Example 2.3.2 Equilibrium of chemical reactions by Newton-Raphson method... 

Example 1.2 Finding a Root of an nth-Degree Polynomial by Newton-Raphson Method Applied to the Soave-Redlich-Kwong Equation of State. Develop a MATLAB function to calculate a root of a polynomial equation by Newton-Raphson method. Calculate the specific volume of a pure gas, at a given temperature and pressure, by using the Soave-Redlich-Kwong equation of state... 

Solving the polynomial by Newton-Raphson method while abs(x0 - x) > tol iter < maxiter iter = iter + 1 xO = x ... 

The polynomial may have no more than a pair of complex roots, A root of nth-degree polynomial is determined by Newton-Raphson method. This root is then extracted from the polynomial by synthetic division. This procedure continues until the polynomial reduces to a quadratic. 

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are ... 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

Equation 5-197 is a polynomial of the third degree, and by employing either a numerieal method or a spreadsheet paekage sueh as Mierosoft Exeel, the roots (C ) of the equation ean be determined. A developed eomputer program PROGS 1 using the Newton-Raphson method to determine was used. The Newton-Raphson method for the roots of Equation 5-197 is... 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

By iteration, the general expression for the Newton Raphson method may be written (if f can be evaluated and is continuous near the root) ... 

Finally, for formulation D the flows in the tree branches can be computed sequentially assuming zero chord flows. This initialization procedure was used by Epp and Fowler (E2) who claimed that it led to fast convergence using the Newton-Raphson Method. 

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-... 

Shown in Fig. 5.3d are free energies estimated from the same forward and backward simulation runs using Bennett s optimal estimator, obtained by solving (5.50) using a Newton-Raphson method. Unlike the direct exponential estimator (which... 

Geochemists, however, seem to have reached a consensus (e.g., Karpov and Kaz min, 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 1983) that Newton-Raphson iteration is the most powerful and reliable approach, especially in systems where mass is distributed over minerals as well as dissolved species. In this chapter, we consider the special difficulties posed by the nonlinear forms of the governing equations and discuss how the Newton-Raphson method can be used in geochemical modeling to solve the equations rapidly and reliably. 

To obtain the initial equilibrium concentrations of the various ions, the solution is taken to contain Fe2(S04)g, FeSO, H2SO4 and a small amount of CuSO. Leach liquor is recycled after the recovery step so traces of CuSO are always present. Analytical concentrations of these substances and the equilibrium constants for each equilibrium reaction must be known. Mass balances for Fe(III), Fe(II), Cu(II) and SO 2- and a charge balance supplement the mass action equations. This nonlinear set of equations can be solved by the well-known Newton-Raphson method (6). 

The dispersion coefiicients can now be foimd from the Ni/q by a nonlinear calculation procedure, such as the Newton-Raphson method, utilizing the expressions for Nt, Eqs. (56) or (57). In the general case, values of Pli, Psit Pl2, Pr2, and the physical dimensions of the apparatus are substituted into Eq. (56) or (57), and then Pl and Pr can be found from the simultaneous (nonlinear) solution of the expressions for Ni and N2. The variances of the dispersion coefficients could also be found from the variances of the Ki by standard statistical methods. 

These M + 1 equations in M + 1 unknowns p and Xm may be solved by the Newton-Raphson method, in which the unknowns are iteratively adjusted until the right and left sides of the equations agree. The object spectrum number-count set hm and noise em are then computed by substitution of p and the Xm into Eqs. (31) and (32). 

This is a system of M + 1 equations in the M + 1 unknowns fi and Xm. It may be solved by regarding the known left-hand sides as target values in a relaxation program that iteratively homes in on the solution. The Newton-Raphson method works fine for this job. With an initial trial solution of all Xm = 0 and fi = N, the final solution is attained in usually about 10 iterations. 

We need only a single user subroutine starting at line 900, which is completely analogous to the corresponding one required by the Newton-Raphson method. 

The four relations cited in Step 22 are solved simultaneously by trial to find the temperature of the gas. Usually it is in the range 1500-1800°F. The Newton-Raphson method is used in the program of Table 8.18. Alternately, the result can be obtained by interpolation of a series of hand calculations... 

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